% Precausal Substrate Theory
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  pdftitle   = {Precausal Substrate Theory: On the Nature of Reality, as I see it},
  pdfauthor  = {Ralf Meelker},
  pdfsubject = {Foundational physics; noncommutative geometry; spectral triples},
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\begin{document}

% Title
\begin{center}
  {\LARGE\bfseries Precausal Substrate Theory (PST)}\\[0.6em]
  {\large\itshape On the Nature of Reality, as I see it}\\[0.5em]
  {\small Ralf Meelker, 2026 - \href{mailto:ralf@meelker.nl}{ralf@meelker.nl} +316-11.122.133}
\end{center}

\vspace{1em}

% Abstract
\begin{abstract}
\noindent
Precausal Substrate Theory (PST) proposes that spacetime, causality, matter,
and the fundamental forces are not fundamental but emergent from a single
irreducible primitive: the capacity for distinctions to exist.  Prior to
space and time, the bare possibility of difference (\textbf{property
differentiation}) carries an \textbf{asymmetric tension}.  Past a
\textbf{modal threshold}, the configuration undergoes \textbf{modal
sublimation} into \textbf{instantiated geometry}, with causality born
together with geometry.  The Big Bang is therefore a perspectival artifact
rather than a first event.

\emph{The foundational object.}  PST rests on three postulates: P1
distinction primitive, P2 asymmetric tension, P3 Landau--Ginzburg modal
potential.  The substrate's natural noncommutative-geometric reading is
\emph{verified}, not assumed: Computation~3 shows that, on the parity-selected
sub-limit (odd number of distinction bits), the substrate's
complementation-and-parity real structure makes it a real Connes spectral
triple of KO-dimension six with sign triple
$(\varepsilon, \varepsilon', \varepsilon'') = (+,+,-)$ --- the
Lorentzian-compatible Connes value.  Computation~4 verifies that the
Mosco-limit spacetime $M = \mathbb{R} \times S^3$ has unique spin
structure (KO-dimension four).  KO additivity then makes the product
$M \times F$ a Connes real spectral triple of total \textbf{KO-dimension
ten}, congruent to two modulo eight, the Connes Standard-Model value
compatible with a Lorentzian fermion action; the spacetime dimension
$n = 4$ is the unique value consistent with this constraint and the
substrate's KO-6.  This ten is an algebraic invariant (mod 8), not a
count of spatial dimensions and not superstring theory's critical
dimension.  Inside this object, the substrate's internal algebra is
derived (not posited) as
$A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$: Computation~21
verifies that Computation~3's parity grading $\gamma_K$ equals the Cl(0,6)
chirality grading $i\omega_{\mathrm{JW}}$ exactly in the
Jordan--Wigner representation (machine precision), so the chirality
projector $f = (I + i\omega)/2$ is Furey's primitive idempotent, and
Furey~2014~\cite{Furey2018} delivers $A_F$ as the commutant of the
matter representation on $\mathrm{Cl}(0,6)\!\cdot\! f$
(\S\ref{sec:conjecture-1-attack}).  No appeal to the
Chamseddine--Connes--Marcolli classification is required.

\emph{Four structural results follow.}
\textit{General relativity}: the Einstein--Hilbert action is the leading term
of the spectral action $\mathrm{Tr}\,f(D/\Lambda)$; Newton's $G$ enters as a
consistency relation among $(m_h, M_P, v)$ rather than a first-principles
output.
\textit{Quantum mechanics}: the canonical commutation relations have their
structural origin in the gradient structure on configuration space, via
Mosco convergence of Boolean Dirichlet forms.
\textit{The Standard Model}: its gauge group
$\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ is the unimodular
unitary group of $A_F$, with the gauge bosons and Higgs as inner
fluctuations of the Dirac operator; one generation's colour and charge come
from the octonionic primitive idempotent.
\textit{Matter}: the Boolean $\{0,1\}$ distinctions are fermionic occupation
numbers via the CAR algebra, making spin--statistics structural.

\emph{Predictions and inherited problems.}  The vacuum is a ring of minima
($\mathcal{V} \cong S^1$), so orbital motion is the only stable ground
state ($L = mr^2\omega$).  PST predicts a $d^{-6}$ correction to the
Casimir effect at nanometre separations, with parameter-free coefficient
$\xi = 90/\pi^2$; the diagnostic amplitude $\xi d_0^2$ contains a single
non-derived scale (the LG coherence length $d_0$, bounded $< 16$~nm by
existing data), so the $d^{-6}$ exponent is a categorical signature shared
with any single-length-scale vacuum modification, while the amplitude
determines $d_0$ from the data rather than confirming a parameter-free
prediction.  The cosmological-constant sign is delivered structurally by
two complementary mechanisms (quantum vacuum on $\mathcal{V} \cong S^1$
and ongoing modal sublimation; \S\ref{sec:cosmo-sign}); the magnitude is
the standard Standard-Model fine-tuning problem, which PST inherits but
does not solve.

\emph{Open pieces} (Section~\ref{sec:open-research-work}, reduced from
five items in earlier drafts to two): a Sobolev-tail estimate that
completes the Dirac-convergence step for the spin structure, and the
structural derivation of $N_{\mathrm{gen}} = 3$ generations.  Both have
sharply located residuals and bounded attack lines.

\emph{Note to the reader.}  PST can be read at two depths.  At the
\emph{grounding} level the paper says \emph{why} the Connes--Chamseddine
noncommutative-geometry programme has the structure it does: the
postulates ground a programme whose empirical content is well established.
At the \emph{derivation} level the paper derives the programme's algebra
identity and gauge content from P1--P3 plus the standard CAR/Clifford
correspondence and the Furey 2014 commutant calculation --- no further
NCG-side axiom is imposed.  The two readings differ in scope but share
the same body of computations.

The deepest implication is also the simplest: the universe exists not
because something caused it, but because a reality in which no
distinctions are possible is not a reality at all.
\end{abstract}

\clearpage
\tableofcontents

\vspace{2em}
\begin{center}
\setlength{\fboxsep}{14pt}%
\fbox{%
  \begin{minipage}{0.88\textwidth}
  \raggedright\small
  \href{https://meelker.nl}{\textbf{Website}}\\
  \textit{https://meelker.nl}\\[6pt]
  \href{https://meelker.nl/pst-paper-v17.26.pdf}{\textbf{Full Paper (PDF)}}\\
  \textit{https://meelker.nl/pst-paper-v17.26.pdf}\\[6pt]
  \href{https://meelker.nl/pst-paper.tex}{\textbf{LaTeX Source}}\\
  \textit{https://meelker.nl/pst-paper.tex}\\[6pt]
  \href{https://meelker.nl/video}{\textbf{Video (animation 1, with music)}}\\
  \textit{https://meelker.nl/video}\\[6pt]
  \textbf{Animations:}\\[4pt]
  \hspace{1em}\href{https://meelker.nl/pst-animation1}{\textbf{Modal sublimation into instantiated geometry}}\\
  \hspace{1em}\textit{https://meelker.nl/pst-animation1}\\[6pt]
  \hspace{1em}\href{https://meelker.nl/pst-animation2}{\textbf{The eight-step chain of logical entailment from distinction to orbital motion}}\\
  \hspace{1em}\textit{https://meelker.nl/pst-animation2}\\[6pt]
  \hspace{1em}\href{https://meelker.nl/pst-animation3}{\textbf{Asymmetric tension crossing the modal threshold}}\\
  \hspace{1em}\textit{https://meelker.nl/pst-animation3}\\[6pt]
  \hspace{1em}\href{https://meelker.nl/pst-animation4}{\textbf{The Casimir discreteness correction}}\\
  \hspace{1em}\textit{https://meelker.nl/pst-animation4}\\[6pt]
  \hspace{1em}\href{https://meelker.nl/pst-animation5}{\textbf{Vacuum topology and the origin of orbital motion}}\\
  \hspace{1em}\textit{https://meelker.nl/pst-animation5}\\[6pt]
  \hspace{1em}\href{https://meelker.nl/pst-animation6}{\textbf{The pitchfork: spontaneous symmetry breaking at the modal threshold}}\\
  \hspace{1em}\textit{https://meelker.nl/pst-animation6}
  \end{minipage}%
}%
\end{center}
\vspace{1em}

\clearpage
% 1. Introduction
\section{Introduction}

\subsection{The incompatibility of quantum mechanics and general relativity}

The search for a unified account of physical reality has long been constrained by the
assumption that spacetime, matter, and causal order constitute the fundamental
architecture of the universe.  Modern physics inherits this assumption from classical
ontology, even as its two most successful frameworks, Quantum Mechanics and General
Relativity, demonstrate that this assumption cannot be sustained.  Quantum Mechanics
describes a domain in which discreteness, nonlocality, and the breakdown of temporal
sequence are intrinsic features of physical behaviour.  General Relativity, by contrast,
models the universe as a smooth, continuous geometric manifold whose curvature governs
gravitational interaction.

These frameworks do not merely differ in mathematical form; they presuppose incompatible
ontological primitives.  General Relativity requires a spacetime manifold populated by
\textit{loci}, positions at which events occur and fields take values, and by
\textit{relata}, the entities that stand in geometric or dynamical relations.  Quantum
Mechanics, however, operates in a domain where neither loci nor relata can be taken as
fundamental: superposition, entanglement, and nonlocal correlations violate the assumption
that physical systems occupy definite positions or instantiate well-defined relational
properties.  The failure to unify General Relativity and Quantum Mechanics is therefore
not a technical problem but an ontological one: each theory requires a different set of
primitives.  General Relativity requires loci and relata embedded in a geometric manifold;
Quantum Mechanics requires a pre-geometric domain in which such constructs do not yet
exist.

\subsection{Assumed backgrounds and the missing question}

Every physical theory, however fundamental it claims to be, begins with an implicit
concession: it assumes a background.  General Relativity \cite{Einstein1915} assumes a
differentiable manifold and proceeds to determine its curvature.  Quantum Mechanics
assumes a Hilbert space evolving under a causal time parameter.  String Theory and Loop
Quantum Gravity \cite{Rovelli1990,Ashtekar1986}, despite their ambitions, replace one
geometric background with another; strings propagate through target space, spin foam
models quantize geometry but presuppose a combinatorial substrate whose relational
structure is already causal in character.  Even the most radical approaches to quantum
gravity, including causal dynamical triangulations and causal set theory
\cite{Bombelli1987,Sorkin1997}, take causality itself as a primitive rather than a
derived relation.  The question that none of these frameworks address at root is the
following: \textit{what is the structure that makes spacetime, causality, and the
physical world possible in the first place?}

This question is not merely philosophical.  It has direct physical consequences.  The
persistent failure to reconcile Quantum Mechanics and General Relativity is not simply a
technical problem of quantizing a nonlinear field theory; it is a symptom of the fact
that both theories assume different things about the structure they presuppose
\cite{Penrose1965,HawkingPenrose1970}.  Quantum Mechanics treats time as an external
parameter; General Relativity makes time dynamical.  Quantum Mechanics requires a fixed
causal background for the definition of states and observables; General Relativity makes
the causal structure itself a dynamical variable.  These are not merely different
formalisms; they reflect genuinely incompatible ontological commitments about what the
world is \textit{made of at its foundation}.  Quantum Field Theory occupies an intermediate
position: it is Quantum Mechanics made compatible with special-relativistic spacetime,
presupposing more background structure than QM (a fixed Minkowski metric and causal order)
and less than GR (no dynamical geometry).  A foundational theory must derive all three;
the natural demarcation is the emergence of the Lorentzian action from the precausal
substrate, the moment at which time-ordered products, propagators, and conservation laws
first become available.  A theory of quantum gravity that merely
finds a way to combine QM and GR without resolving this incompatibility is not a foundational
theory; it is a patch.

\subsection{The precausal starting point}

Precausal Substrate Theory approaches this situation differently.  Rather than beginning
with spacetime and asking how to quantize it, or beginning with a causal order and asking
how geometry emerges, PST begins at a level prior to both: a domain in which neither
geometry nor causality is defined, and in which the only primitive is the capacity for
\textit{distinctions to exist}.

\medskip
Property differentiation is not a process, event, or transformation.  It is the logical possibility of asymmetry, the condition under which difference can exist without requiring objects, relata, loci, or temporal sequence.  The logical possibility of property differentiation is irreducible: any attempt to explain it presupposes it.

Once differentiation is possible, its first expression is asymmetric tension.  Asymmetric tension is not a force, field, or interaction in the physical sense.  It is the manifest imbalance that arises when differentiation becomes expressible.  It is asymmetric without being spatial, dynamic without being temporal, structured without being geometric.  Asymmetric tension is the substrate's intrinsic asymmetry made actual, and it functions as the generative engine from which all subsequent structure emerges.

\medskip
This domain, the precausal substrate, is not empty.
It contains what this paper calls \textbf{asymmetric tension}: a structural imbalance
that arises wherever differentiated properties coexist without full symmetry.  The key
claim of PST is that this imbalance is not stable indefinitely.  There exists a threshold
beyond which the substrate cannot maintain an uninstantiated configuration, a logical
boundary condition rather than a dynamical process, and when that threshold is crossed,
the configuration undergoes a direct phase transition into instantiated geometry.
Spacetime is not the arena in which this transition takes place.  Spacetime \textit{is}
the transition.

\subsection{The foundational object: a spectral triple of total KO-dimension ten}
\label{sec:foundational-object}

The substrate does not, on its own, single out a number of dimensions.
What it provides is an algebra of distinctions with an intrinsic
involution (the complementation $C \mapsto \bar{C}$) and a magnitude (the
tension norm).  The question that fixes the structure is not how many
dimensions the substrate has, but what the substrate must become for a
Lorentzian quantum field theory to be definable on it.  That requirement,
the existence of a well-defined fermionic action continuing from the
Euclidean to the Lorentzian, is a demand of consistency, and it is the
same demand that the incompatibility of quantum mechanics and general
relativity, with which this section opened, leaves unmet by every
background-assuming theory.

Connes' noncommutative geometry answers that demand with a single integer,
the KO-dimension, defined modulo eight by Bott periodicity.  It is not the
dimension of any manifold; it is a property of the real structure $J$ and
the grading $\gamma$, fixing the signs of $J^2 = \pm 1$,
$J\gamma = \pm\gamma J$, and $JD = \pm DJ$.  Of the eight values, the
canonical one for a product of spacetime with an internal space, the value
at which the Euclidean fermionic action carries the sign that continues to
a well-defined Lorentzian action, is KO-dimension two.

PST meets this value structurally.  The substrate's real structure has
KO-dimension six (a computed result, holding for an odd number of
distinction bits; Section~\ref{sec:open-research-work}).  The Mosco limit
of the substrate's Dirichlet form produces a four-dimensional Lorentzian
spacetime, of KO-dimension four.  KO-dimension is additive under the
product, so the physical object, spacetime tensored with the substrate as
its internal factor, has total KO-dimension $4 + 6 = 10 \equiv 2 \pmod 8$.
Among all totals that land on the Lorentzian-compatible value two, ten is
the smallest: any larger total would require a larger spacetime (which the
Mosco limit does not produce) or internal structure beyond the substrate.
The foundational number of PST is therefore this total KO-dimension ten,
from which the four of spacetime and the six of the internal factor are the
\emph{minimal split}: $4 + 6 = 10 \equiv 2 \pmod 8$ is the lowest sum with
both summands non-negative and the total congruent to two modulo eight (the
Connes Lorentzian value), given the verified parity-selected substrate KO-6
(Computation~3) and the Mosco-limit spacetime KO-4 (Computation~4).  ``Minimal'' is
therefore conditional on the parity selection but does not require an
extra spectral-triple axiom beyond what Computations~3 and~4 verify.

This total has nothing to do with the ten of superstring theory.  There,
ten is the dimension of a smooth target manifold, fixed by cancellation of
the worldsheet conformal anomaly.  Here, ten is a KO-theoretic invariant,
defined only modulo eight, of the real structure of a product spectral
triple; it counts no spatial directions, and involves no strings, no
extended objects, no worldsheet, and no anomaly to cancel.  PST arrives at
ten by Connes--Bott periodicity applied to a distinction-first substrate.

In this framework the instantiated geometry is four-dimensional and
Lorentzian; the remaining six of the KO-bookkeeping live in a finite
noncommutative internal factor with no spatial extent, the algebraic home
of the gauge structure and the matter content.  PST goes beyond a
substrate-level grounding of the Connes--Chamseddine
programme~\cite{ChamseddineConnes1997}: the internal algebra
$A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ that programme
posits is, in PST, \emph{derived} from the primitive distinction
structure (\S\ref{sec:conjecture-1-attack}).  The substrate's
parity-selected real structure (Computation~3) delivers the Cl(0,6) chirality
grading exactly (Computation~21: $\gamma_K = i\omega$ to machine precision in
the Jordan--Wigner representation), and Furey 2014~\cite{Furey2018}
calculates $A_F$ as the commutant of the matter representation on the
resulting minimal left ideal.  The remainder of this paper develops the
structures that follow (general relativity via the spectral action, the
gauge group as the unimodular unitaries of $A_F$, the matter content of
one generation, and quantum mechanics), and isolates the two remaining
open pieces in Section~\ref{sec:open-research-work}.

\subsection{Principal consequences}

The consequences of this reorientation are far-reaching.  Because spacetime and the
causal order it carries appear together in a single, nontemporal transition, causality is
not fundamental.  It is coemergent with geometry, a derived structural relation that
exists only within the instantiated domain.  This means that asking what \textit{caused}
the universe to come into being is not a deep question with a hidden answer; it is a
category error.  Causality did not exist prior to the transition that brought geometry,
and therefore causality itself, into being.  This is not an evasion of the question but
its dissolution.

The same logic dissolves the Big Bang as a foundational event.  The Big Bang implies
a locus of origin, a privileged point in spacetime from which everything expands.
But PST denies that loci are primitive.  Instantiation is continuous and distributed
across the substrate; it has no centre, no start, and no singular event.  What
cosmology calls the Big Bang is a perspectival artifact: the appearance of a beginning
as seen from within an instantiated geometry that was never born at a point.  The
cosmological implication is not a multiverse in the conventional sense of discrete
bubble universes appearing at separate loci, but something more fundamental: the
substrate instantiates geometry continuously and everywhere at once, without boundaries
or origins.  What we call our universe is not one bubble among many; it is a region
of observation within a continuous, unbounded field of instantiation.

A third consequence concerns orbital motion and gravitational structure.  The standard
representation of gravitational attraction, the rubber sheet or elastic membrane of
General Relativity pedagogy, depicts spacetime as a funnel-shaped surface with a
single minimum.  In that picture a stable orbit requires a precisely tuned tangential
velocity, which must be imposed on the geometry as an external initial condition; the
geometry itself cannot explain where it comes from.  This is not merely a deficiency
of the pedagogical analogy.  In General Relativity proper, orbital motion is described
as geodesic motion in curved spacetime, and the geodesic equation correctly determines
the shape of any path through a given metric.  But which geodesic a body follows,
whether circular, elliptical, or radially infalling, remains an initial condition that
General Relativity takes as given.  The metric determines the available paths; it
cannot explain why a body is on one path rather than another.  The vacuum that emerges
from modal sublimation is structurally different.  Below the threshold, the modal
potential has not one minimum but a ring: the set of all configurations that minimise
the potential at equal tension.  Motion along this ring is not a special assumption;
it is the only stable state available to any instantiated configuration.  Circular
orbital motion, from the orbital motion of subatomic particles to planetary orbits
to galactic rotation to the spin of black holes, is therefore not a mechanical accident
or a lucky initial condition in PST.  It is a structural consequence of the topology
of the vacuum manifold.  General Relativity describes orbital mechanics correctly at
macroscopic scales but cannot derive the existence of orbital motion from first
principles; PST does.

At the moment of instantiation, when tension first crosses the modal threshold, the
substrate's algebraic structure organises itself into two qualitatively different
pieces.  The Mosco limit of the Boolean Dirichlet form produces a four-dimensional
Lorentzian spacetime $M = \mathbb{R} \times S^3$ directly; there is no
`dimensional reduction' from a higher-dimensional manifold.  Separately, the
substrate's internal algebraic content is the finite noncommutative factor
\begin{equation*}
  A_F \;=\; \mathbb{C} \,\oplus\, \mathbb{H} \,\oplus\, M_3(\mathbb{C}),
\end{equation*}
an algebraic object with no spatial extent.  This algebra identity is derived
from P1--P3 alone, via three computationally verified pieces
(\S\ref{sec:conjecture-1-attack}): Computation~3 verifies that the substrate's
parity-selected real structure on an odd number of distinction bits delivers
$\gamma_K^2 = +I$ with the Connes Lorentzian KO-6 sign triple
$(\varepsilon, \varepsilon', \varepsilon'') = (+, +, -)$; Computation~21 verifies
that $\gamma_K$ equals the Cl(0,6) chirality grading $i\omega$ exactly in the
Jordan--Wigner representation, so the chirality projector
$f = (I + i\omega)/2$ is delivered directly by the substrate; Furey
2014~\cite{Furey2018} then derives $A_F$ as the commutant of the matter
representation on the resulting minimal left ideal.  The earlier draft of
this paper carried the algebra identity as Conjecture~1; that conjecture is
now closed, with no appeal to the Chamseddine--Connes--Marcolli classification.
The foundational object PST identifies is the product of $M$ and $F$, a real
spectral triple of total KO-dimension ten
(Section~\ref{sec:foundational-object}); the four, six, and ten are the
minimal split compatible with a Lorentzian fermion action, given the
verified parity-selected substrate KO-6 (Computation~3) and the verified KO-4
spin structure of the Mosco-limit spacetime (Computation~4).

Furthermore, because both General Relativity and Quantum Mechanics emerge as projections
of the same precausal structure, with the first operating at macroscopic scales of
tension and the second near the threshold where configurations are only marginally
instantiated, their unification is not a matter of quantizing one and deforming the
other.  It is a matter of recognizing that they describe the same underlying reality from
different regimes of a single precausal parameter.  Work on emergent gravity
\cite{Verlinde2011,Jacobson1995} has approached this question from within the
instantiated domain; PST proposes the precausal domain as the natural resolution point.

\subsection{Mathematical formulation and predictions}

The theory is formulated using a Landau--Ginzburg variational principle for the modal
threshold \cite{GinzburgLandau1950} and a categorical functor for the sublimation map
\cite{MacLane1971}, with an explicit tension-to-metric construction that derives the
Minkowski signature rather than postulating it.  Its physical predictions include a
falsifiable Casimir correction
\cite{Casimir1948,Lamoreaux1997,Bressi2002,Decca2003,Bordag2001} whose $d^{-6}$
scaling is a necessary signature (shared with other single-length-scale vacuum
modifications) and whose parameter-free coefficient $\xi = 90/\pi^2$ fixes the
diagnostic amplitude $\xi d_0^2$ that selects PST, measurable at submicron plate
separations.

\subsection{The foundational postulates of PST}
\label{sec:postulates}

PST is not a postulate-free theory.  It rests on three independent foundational
postulates (P1--P3), plus a fourth formal statement (P4) that is derived from
P1--P3 via Mosco convergence of Boolean Dirichlet forms (proved in
Section~\ref{sec:seeley-dewitt}).  P4 names the Laplace-type kernel character
of the projection; the macroscopic dimension $n = 4$ and the Lorentzian
signature follow from the verified spectral-triple structure of the
substrate (Computation~3) plus the Mosco-limit spacetime's KO-4 spin structure
(Computation~4), via KO additivity (Section~\ref{sec:seeley-dewitt}), rather than
being independent postulates of the theory.  All four statements are listed
here for expository continuity.  They constitute the minimal inputs from which all
physical consequences follow; stating them explicitly gives a precise target for each
residual open problem.  The abstract correctly counts three independent
postulates; references to ``four postulates'' elsewhere in the paper count P4 as a
formally listed, though derived, structural statement.

\begin{description}
  \item[\textbf{P1 (The Distinction Primitive).}]
    There exists a nonspatiotemporal set $D$ of primitive properties, equipped with a
    tension functional $T\colon\mathcal{P}(D)\to\mathbb{R}$.  No geometric, causal, or
    temporal structure is presupposed on $D$.  This is the only primitive of the theory.

  \item[\textbf{P2 (Asymmetric Tension).}]
    The tension functional is asymmetric under complementation:
    $T(C)\neq T(D\setminus C)$ for generic $C\subset D$.  Symmetry between a
    configuration and its complement would prevent any directed structure from arising;
    asymmetry is the sole source of all directedness in the theory.

  \item[\textbf{P3 (The Landau--Ginzburg Modal Potential).}]
    The order-parameter dynamics near the modal threshold $\tau$ are governed by the
    Landau--Ginzburg functional
    \begin{equation}
      \mathcal{F}[\psi,\varepsilon]
        = \int_{\mathcal{C}}\!\left[
            -\varepsilon|\psi|^2
            + \tfrac{1}{2}|\psi|^4
            + c|\nabla_{\mathcal{C}}\psi|^2
          \right]d\mu,
      \label{eq:LG-postulate}
    \end{equation}
    where $\varepsilon = T(C)-\tau$ is the excess tension.  This is the minimal even
    functional with a second-order phase transition and a gradient term; it is adopted
    as the simplest form consistent with the $\mathbb{Z}_2$ symmetry $\psi\to-\psi$,
    not derived from P1 and P2 alone.

    \emph{Remark on the gradient term:} the symbol $|\nabla_{\mathcal{C}}\psi|^2$ is
    the discrete Dirichlet form on $\{0,1\}^D$:
    \begin{equation}
      |\nabla_{\mathcal{C}}\psi|^2 \;\equiv\; \sum_{\substack{C, C' \in \{0,1\}^D \\ |C\triangle C'| = 1}}
        \bigl|\psi(C) - \psi(C')\bigr|^2,
      \label{eq:lg-boolean-dirichlet}
    \end{equation}
    where $C \triangle C'$ is the symmetric difference (Hamming distance).  This is
    the canonical Dirichlet form on the Boolean hypercube, determined by the combinatorial
    structure of $\{0,1\}^D$ rather than by a pre-existing continuum metric.  It is
    \emph{not} derived from P1 and P2; it is a structural assumption about how
    configurations on $D$ interact: nearest-neighbour transitions in Hamming distance
    have unit cost.  PST's claim is therefore not that all geometric structure emerges
    from bare distinctions alone, but that the \emph{specific instantiated geometry}
    --- the four-dimensional Lorentzian spacetime $M = \mathbb{R} \times S^3$ together
    with the finite internal factor
    $A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ --- follows from
    the Boolean Dirichlet form (P3) together with P1 and P2 via the substrate's
    verified Connes spectral-triple structure (Computation~3, KO-6; Computation~4, KO-4 spin
    structure on the Mosco-limit spacetime; \S\ref{sec:foundational-object}).
    The Dirichlet form is a minimal canonical choice (it is the unique Dirichlet form
    invariant under label permutations on $D$), but it is a choice.

  \item[\textbf{P4 (The Laplace-Type Projection).}]
    The projection kernel $\widetilde{K}(x,y;t)$ is, to leading order in $t$, the heat
    kernel of a Laplace-type operator $\Pi(L)$ on $L^2(M,\mathrm{dvol}_g)$ whose
    principal symbol is the emergent metric:
    \begin{equation}
      \Pi(L) \;=\; -g^{\mu\nu}\nabla_\mu\nabla_\nu \;+\; (\text{lower order}),
      \label{eq:P4}
    \end{equation}
    so that $\widetilde{K}(x,y;t) = \langle x|e^{-t\Pi(L)}|y\rangle$ by definition.
    The metric $g$ is defined as the principal symbol of $\Pi(L)$; this breaks the
    circularity that would arise if $g$ were extracted from $\widetilde{K}$ and then
    $\widetilde{K}$ required to satisfy $\partial_t\widetilde{K} = \Delta_g\widetilde{K}$.
    This postulate is what ensures the Seeley--DeWitt expansion yields the universal
    coefficient $\tfrac{1}{6}R$ rather than the kernel-dependent value a generic
    localized kernel would produce.  P4 is derived as a theorem from P1--P3
    via Mosco convergence of Boolean Dirichlet forms (Section~\ref{sec:seeley-dewitt}):
    the isotropy of the Bernoulli measure forces $\Pi(L^B)$ to be Laplace-type in the
    scaling limit.  P4 is therefore retained as a formal statement but is no longer
    an independent postulate.
\end{description}

One result that is \emph{derived} rather than postulated deserves mention here
to prevent confusion with the above: the Bernoulli measure
$\mu = \bigotimes_{a\in D}\mathrm{Bern}(\tfrac{1}{2})$ is not a postulate; it is
the unique measure on $\mathcal{P}(D)$ invariant under relabelling and under
complementation, derived in Section~3.  The Lorentzian signature $(-,+,+,+)$
is \emph{not} in this category, but a stronger statement holds for it:
signature is forced by hyperbolic well-posedness of the Goldstone wave
equation (Section~\ref{sec:open-research-work}, Computation~2), so it is derived as
a theorem.  See Section~\ref{sec:open-research-work} for the full accounting
and for the open research that remains.

\subsection{Structure of this paper}

The remainder of this paper develops PST in full.  Section~2 establishes the foundational
ontology and its mathematical expression.  Section~3 introduces the modal threshold and
its variational formulation.  Section~4 develops the theory of modal sublimation as a
categorical functor and provides an explicit construction.  Section~5 addresses the
coinstantiation of spacetime and causality.  Section~6 consolidates the emergence chain.
Section~7 derives energy, matter, and fields as projections of asymmetric tension.
Section~8 derives the Standard Model gauge group and establishes gauge anomaly
cancellation as a structural theorem.  Section~9 develops the physical predictions
of the theory, including a quantitative
Casimir correction.  Section~10 grounds Newton's constant as the gravitational
term of the spectral action and identifies the discreteness scale $d_0$ with the
Landau--Ginzburg coherence length of the modal vacuum, establishing the Casimir
correction (the $d^{-6}$ scaling is a necessary signature; the parameter-free
coefficient $\xi = 90/\pi^2$ fixes the diagnostic amplitude $\xi d_0^2$ that
selects PST) as a concretely detectable experimental signal.  Section~11 develops PST's cosmology: the Friedmann
equations are derived, the Cosmological Principle is established as a theorem of the
Bernoulli measure, and ongoing instantiation is shown to force $w = -1$, fixing the sign
of the cosmological constant.  Section~12 positions PST in relation to existing
frameworks.  Section~13 concludes.  Section~14 summarises the structural and
mathematical problems the theory resolves and the open research that remains.
Appendix~\ref{sec:notation} is a consolidated table of the symbols recurring
through the paper, grouped by sector, each with the section where it is first
introduced.


\subsection{Layman's Summary}

Physics has given us two extraordinary theories: General Relativity, which describes how gravity shapes space and time, and Quantum Mechanics, which governs the behaviour of subatomic particles.  Both are spectacularly accurate in their respective domains.  But they rest on incompatible assumptions about the nature of reality, and every attempt to merge them has stalled at the same conceptual barrier.  This paper argues that the problem is not merely technical but ontological: both theories assume a background they cannot explain.  Precausal Substrate Theory resets the starting point to something deeper, the bare capacity for distinctions to exist, and derives spacetime, gravity, and quantum fields from there.

% 2. Foundational Ontology
\section{Foundational Ontology}

\subsection{The case for ontological minimality}

The foundational commitment of PST is to ontological minimality.  If one takes seriously
the project of identifying what must exist \textit{prior} to physics, prior, that is,
not in a temporal sense but in the sense of logical or modal presupposition, one arrives
quickly at the question of what the most primitive possible structure is.  Attempts to
answer this question have historically invoked points, events, fields, or relations, but
all of these already carry implicit geometric, causal, or energetic content.  PST makes a
different choice: the minimal structure from which everything else can be derived is not a
geometric entity, not a causal relation, and not a physical degree of freedom.  It is
simply \textbf{the capacity for distinctions to exist}.

To see why the historical candidates fail, consider each in turn.  A \textit{point}
implies position: it is the primitive of location, and location presupposes a space in
which positions are defined and separable.  An \textit{event} implies both a location
and a temporal index: it is a point in spacetime, and spacetime is precisely what PST is
trying to derive rather than assume.  A \textit{field} implies a background manifold over
which it takes values, a collection of positions at each of which the field is defined;
the background is assumed, not derived.  A \textit{relation} implies relata: two or more
entities that stand in the relation, and relata are at minimum points or objects,
reintroducing the geometric content that the programme aims to eliminate.  Even the most
abstract candidates, such as sets of abstract objects or categories of morphisms,
presuppose the concept of collection or composition, which themselves rest on notions of
identity and distinctness.  Identity and distinctness are, in each case, the structural
content that cannot be removed.  PST takes this observation seriously and makes it the
foundation: if every proposed primitive already presupposes the capacity to distinguish
one thing from another, then that capacity is the true primitive, and the correct
programme is to begin there rather than one step above it.

\subsection{Property differentiation}
A property is differentiated
from another not by virtue of occupying different positions in space or occurring at
different times (positions and times do not yet exist), but simply by being
\textit{other}.  Two properties are differentiated if and only if they are not identical.
Nothing further is required: no metric distance, no causal connection, no ordering
relation, no extensional structure of any kind.  Property differentiation is the minimal
condition under which the concept of a \textit{distinction} is meaningful, and it is the
sole primitive of the precausal substrate.

The concept of otherness that this definition invokes is purely logical.  It is the
primitive of non-identity: $a$ is other than $b$ if and only if it is not the case that
$a = b$.  This does not require the two properties to be distinguishable by any observer,
separable in any physical sense, or related by any causal or spatial connection.  It
requires only that identity fails.  Two properties that are distinct in this sense need
share nothing beyond their mutual non-identity; they carry no further structure, no mass,
no charge, no location, no duration.  The precausal substrate is therefore a domain in
which the only thing that is true is that some properties are not the same as others.
All of physics, in the PST programme, must be derivable from that single fact.

This choice of primitive is not arbitrary, and it is not merely the result of a
preference for minimality.  It is the result of asking what must be true of any domain
that is to give rise to structure at all, and then identifying the unique weakest
condition that satisfies the requirement.  The requirement is this: a domain from which
structure can emerge must be capable of nontrivial differentiation.  If every element of
the domain were identical to every other, the domain would be featureless and nothing
could arise from it.  Property differentiation is precisely the weakest condition under
which nontrivial differentiation is possible.  It is weaker than any spatial, causal, or
metric structure, and it is not derivable from anything weaker than itself: any domain
that contains less structure than property differentiation contains no nontrivial
distinctions at all and is, in effect, the null domain from which nothing can arise.
Property differentiation is therefore not merely minimal; it is the unique minimal
structure with the required generative capacity.  The substrate is genuinely precausal:
it exists in a mode of being in which causality has no purchase, because causality
requires a temporal ordering of events, and temporal ordering requires positions and
durations, neither of which are available at this level.

\subsection{Mathematical formulation}
Let $D$ be a set of differentiated properties endowed with a binary relation
$\delta(a,b)$, read as ``property $a$ is distinct from property $b$.''  The structure
$(D, \delta)$ is governed by exactly two axioms:
\begin{align}
  \delta(a,b) &\;\implies\; \delta(b,a) \label{eq:sym}\\[4pt]
  &\neg\,\delta(a,a) \label{eq:irr}
\end{align}
Equation~\eqref{eq:sym} states that distinction is symmetric;
equation~\eqref{eq:irr} states that nothing is distinct from itself.  No further axioms
are imposed.  There is no transitivity requirement: the fact that $a$ is distinct from
$b$ and $b$ is distinct from $c$ implies nothing about the relation between $a$ and $c$.
There is no totality requirement: it is not assumed that every pair of properties is
either distinct or identical in a determinate sense available to any external observer.
There is no metric: no notion of \textit{how} distinct two properties are, only that they
are.  There is no topology: no notion of nearness, neighbourhood, or continuity.  There
is no ordering: no sense in which one property precedes or follows another.  This is the
minimal mathematical structure that admits a nontrivial distinction relation without
presupposing any further ontological content.  Every additional axiom that one might wish
to impose would introduce some form of structure beyond pure distinctness, and any such
structure would need to be derived rather than assumed.

The absence of transitivity deserves particular emphasis.  In most familiar mathematical
structures, identity is transitive by definition and distinguishability inherits that
transitivity.  PST does not impose transitivity on the distinction relation because doing
so would introduce an implicit equivalence structure, effectively partitioning $D$ into
classes of indistinguishable elements.  Such a partition already carries combinatorial
and topological content that is not available at the foundational level.  The distinction
relation at this level is purely local to each pair: $a$ and $b$ are either distinct or
not, independently of any third property.

A \textbf{configuration} $C$ is a subset of $D$, a collection of differentiated
properties considered jointly.  The space of all configurations is denoted $\mathcal{C}$.
It is within this space that the dynamics of the precausal substrate, such as they are,
will be expressed, though it must be understood from the outset that these ``dynamics''
are not temporal.  They concern the modal structure of configurations: what is possible,
what is necessary, and what cannot remain uninstantiated.  The distinction between modal
and temporal structure is fundamental to PST.  A temporal statement of the form ``this
configuration becomes that configuration'' presupposes a time parameter, a before, and an
after.  A modal statement of the form ``this configuration cannot remain in this mode of
being'' presupposes none of these things.  It is a statement about what is possible, not
about what happens.  The entire development of PST from this point forward concerns modal
structure in this strict sense.

\subsection{Asymmetric tension}
Property differentiation does not merely establish that distinctions exist; it generates
a structural consequence that is the engine of everything that follows:
\textbf{asymmetric tension}.  Wherever differentiated properties coexist within a
configuration, their mutual distinctness creates a structural imbalance.  To understand
why coexistence generates imbalance, consider the internal relational structure of a
configuration.  Within any configuration containing at least two distinct properties $a$
and $b$, there exist internal relations of non-identity: $\delta(a,b)$ holds.  Note that
$\delta$ is symmetric: $\delta(a,b)$ and $\delta(b,a)$ hold simultaneously, and neither
can be said to point in a preferred direction.  The imbalance that constitutes asymmetric
tension is therefore not directional in the geometric sense.  It is a
\emph{complement-asymmetry}: no configuration $C$ and its complement $\bar{C}$ carry
equal tension, $T(C) \neq T(\bar{C})$.  This structural non-equivalence between a
configuration and its complement is what gives asymmetric tension its name and its
content.  It is not that some properties are heavier or more energetic than others.  It
is that the configuration, considered as a whole, stands in a structurally non-equivalent
relation to its complement: their mutual imbalances do not cancel.

This imbalance is not energetic, since energy presupposes a geometric domain in which it
can be defined and transferred.  It is not spatial, since no geometry yet exists.  It is
not temporal, since there is no before or after in the precausal substrate.  It is not
informational in the Shannon sense, since information theory presupposes a probability
space and an observer.  It is purely \textit{modal}: it is a property of the
configuration's mode of being, expressing the degree to which that configuration is
structurally incoherent in the sense of admitting irreducibly asymmetric relations among
its constituents.  The word ``tension'' is chosen deliberately.  In ordinary usage,
tension denotes a state of strain between opposing tendencies that cannot simultaneously
be satisfied.  Asymmetric tension in the precausal substrate is precisely this: a state
of modal strain arising from the coexistence of distinct properties whose mutual
non-identity relations cannot be collectively neutralised.

This modal imbalance is represented by the \textbf{tension functional} $T$, a map from
configurations to strictly positive real numbers:
\begin{equation}
  T: \mathcal{C} \rightarrow \mathbb{R}^+
\end{equation}
The tension of a configuration is strictly positive because any nontrivial configuration,
meaning any configuration containing at least one pair of distinct properties, carries
some degree of structural imbalance.  A configuration with no distinct properties, one in
which every element is identical to every other, would carry zero tension, but such a
configuration is, by definition, the null configuration: it contains no differentiation
and therefore no content.  The domain of $T$ is therefore effectively the space of all
nontrivial configurations, and on this domain $T$ is bounded away from zero.

Representing tension as a single real-valued map assigns a \emph{global total ordering}
to all configurations: any two configurations can be compared by their $T$ values.  This
totality is stronger than the complement-asymmetry condition $T(C) \neq T(\bar{C})$,
which establishes only that each configuration is distinguishable from its complement.
The real-valued representation is an additional architectural choice: the real numbers
are adopted because they are the minimal ordered field with the topological properties
required for the variational analysis of the following sections.  An alternative encoding
using only pairwise comparability would yield a partial order rather than a total one,
and would not support the Taylor expansion of Section~3.  PST adopts the total ordering
as the minimal structure compatible with the variational formulation, acknowledging that
this is a representational commitment beyond the bare complement-asymmetry condition.

A word on the epistemological status of the formalism introduced in this and the following
sections is required before proceeding.  The mathematical objects employed, real-valued
functionals over configuration space, functional derivatives, gradient operators, a
canonical measure, carry more structure than the primitive $(D, \delta)$ alone strictly
entails.  They constitute a faithful structural encoding: the minimal mathematical
language capable of expressing the modal commitments the theory makes, in particular the
existence of a critical tension value, the continuity of the transition, and the
topological structure of the instantiated vacuum.  The Landau-Ginzburg functional form
is adopted because it is the minimal architecture exhibiting the required bifurcation
behaviour; it is logically motivated structural scaffolding, not a derived consequence
of $(D, \delta)$ alone.  The predictions of the theory are structural consequences of
the modal commitments, not artifacts of the mathematical encoding chosen to represent
them.  Where the formalism makes architectural choices, this is noted explicitly; those
choices are themselves open to foundational scrutiny.

The crucial property of $T$ is its asymmetry: for any configuration $C$ and its
complement $\bar{C}$,
\begin{equation}
  T(C) \neq T(\bar{C})
\end{equation}
The complement $\bar{C}$ of a configuration $C$ is understood as the configuration
containing precisely those properties of $D$ that are not in $C$.  The asymmetry
condition states that no configuration and its complement carry the same tension.  The
intuition behind this condition is as follows.  If $C$ and $\bar{C}$ carried equal
tension, their combined structure would be perfectly balanced: the imbalance of one would
exactly offset the imbalance of the other, and the total system would be in structural
equilibrium.  PST holds that no such equilibrium is available in the precausal substrate.
The reason is that perfect cancellation would require the distinction structure of $C$ to
be the mirror image of the distinction structure of $\bar{C}$, which in turn would
require a global symmetry of $D$ under complement.  No such global symmetry is imposed
by the axioms, and PST asserts that the structure of differentiation is generically
asymmetric.  The net imbalance is therefore always nonzero:
\begin{equation}
  \Delta T(C) \;:=\; T(C) - T(\bar{C}) \;\neq\; 0
\end{equation}
It bears emphasis that $T(C)$ is not energy and $\Delta T(C)$ is not energy difference.
The tension functional assigns a real number to a modal configuration without invoking
any notion of capacity for work, conservation law, or physical process.  It is a measure
of structural incoherence in the modal domain.  The use of real numbers to represent this
incoherence is a mathematical convenience: real numbers form the simplest ordered field
with the topological properties needed for the variational analysis of Section~3, and the
ordering on $\mathbb{R}^+$ provides a natural notion of more or less tension without
requiring any further geometric or physical interpretation.  The numbers assigned by $T$
are not quantities of anything physical; they are indices of modal incoherence, and their
significance lies entirely in their ordering and in the existence of the critical value
$\tau$ that the next section introduces.

A foundational question must be addressed directly: is $T(C)$ merely a formal device, a
parameter introduced for mathematical convenience but lacking genuine ontological standing?
The answer is no, and the reason illuminates the entire structure of PST.  $T(C)$ is not a
parameter awaiting operationalisation.  It is not a quantity that sits in a formula until
experiment supplies its value.  It is the primordial quantity: the first and only structural
consequence of distinctness that precedes all other structure.  To ask how one would
measure $T(C)$ is to commit a category error.  Measurement presupposes a measuring apparatus,
which presupposes a physical system, which presupposes geometry, which presupposes the
very transition that $T(C)$ controls.  $T(C)$ is prior to the apparatus of operationalisation
itself.  One cannot measure the reason there is something rather than nothing; one can only
recognise that without distinctness there is no content, and that distinctness, once present,
immediately generates the structural imbalance that $T$ tracks.

The logical chain is the following.  Distinctness generates configurations.  Configurations
containing multiple distinct properties admit irreducibly asymmetric internal relations.
Those non-identity relations constitute an imbalance that cannot be collectively
neutralised.  That imbalance is what the theory calls tension.  A precise statement
requires, however, a candid qualification.  Since $\delta$ is symmetric, the
non-identity relations themselves carry no directionality; what the theory asserts is the
complement-asymmetry condition $T(C) \neq T(\bar{C})$, the claim that no configuration
and its structural complement are in perfect modal balance.  This condition is not
strictly derivable from $(D, \delta)$ alone.  It is the foundational structural
commitment of PST beyond the bare primitive: the assertion that the precausal substrate
is generically asymmetric.  Whether this asymmetry is an additional primitive or is
itself a consequence of some deeper feature of the set $D$ is a genuine open question
in the foundations of the theory.  What is not in question
is that once distinctness is granted and the asymmetry condition accepted, the chain to
tension is logically closed: tension is the measure of this irreducible
complement-asymmetry, and it is not assumed independently but follows from distinctness
together with the asymmetry condition.

This places PST in a tradition of foundational thinking in which the most primitive
ontological facts are not measurable but are nonetheless logically prior to everything
that is measurable.  Leibniz made a structurally identical move with the principle of
sufficient reason: reason cannot itself be given a reason without circularity, yet the
absence of such a foundation leaves all explanation suspended.  Spencer-Brown's
\textit{Laws of Form} \cite{SpencerBrown1969} makes the closest formal analogue:
beginning from the single injunction ``draw a distinction,'' Spencer-Brown generates a
calculus of indications from which Boolean logic and self-reference are recoverable,
taking the act of distinction as logically prior to all objects, sets, and truth values.
PST's $(D, \delta)$ is the static version of this move: not the performative act of
drawing a distinction but the already-differentiated structure from which physical
geometry is generated.  Hegel's \textit{Science of Logic} \cite{Hegel1812} anticipates
the structure in dialectical form: pure being has no determinate content and resolves
into nothing; the synthesis is becoming, the first concrete category.  The precausal
substrate has no geometric or causal content and cannot remain in that indeterminate state
once tension crosses $\tau$: a modal sublimation that parallels Hegel's becoming while
supplying a precise mathematical mechanism in its place.  Leibniz grounded explanation in
the logical necessity of reason; Spencer-Brown grounded logic in the primitive of
distinction; PST grounds geometry in the logical necessity of tension.  What PST adds that
none of its predecessors supplies is the transition: a precise variational mechanism by
which the primordial asymmetry accumulates to a critical value and generates, through a
well-defined phase transition, the geometric structures that all subsequent physical law
presupposes.


\subsection{Layman's Summary}

This section establishes the first and most minimal starting point of the theory: the fact that things can be different from each other.  No space, no time, no energy, no forces, and no laws of nature are assumed here.  From this single primitive, called property differentiation, it follows that any collection of distinct properties carries a structural imbalance: it is not a perfect mirror of its complement.  PST calls this imbalance \emph{asymmetric tension}.  It is not a force or a field; it is the most elementary structural fact about a world in which distinctions can exist at all.  The Landau--Ginzburg form of the modal potential and the Laplace-type property of the projection kernel are introduced as explicit postulates in later sections (see Section~\ref{sec:postulates}).

% 3. The Modal Threshold
\section{The Modal Threshold}

\subsection{Why a modal threshold must exist}

The existence of asymmetric tension raises an immediate question: if differentiated
configurations carry a structural imbalance, what prevents the substrate from containing
arbitrarily large imbalances indefinitely?  The answer given by PST is that the substrate
is not unlimited in this respect.  There is a boundary to what can remain
\textit{uninstantiated}, one that is not physical, not dynamical, and not temporal, but
purely logical.

To understand why this boundary must exist, recall the character of the precausal
substrate.  It is a domain that is defined entirely by what can be coherently maintained
without spacetime, without causal structure, and without any form of realisation.  A
configuration exists in the substrate insofar as it is a coherent arrangement of
distinctions: its properties are differentiated from one another, and that differentiation
constitutes its identity.  Coherence here is not a quantitative threshold in the ordinary
sense.  It is a modal condition: a configuration is coherent as an uninstantiated structure
if and only if it can exist as such without contradiction.

Now, asymmetric tension is the imbalance inherent in any differentiated configuration.  As
the number and depth of distinctions within a configuration increase, its asymmetric
tension $T(C)$ increases.  This monotonicity is not derivable from the axioms of
$(D, \delta)$ alone; it is a structural constraint on $T$ that PST adopts as part of its
encoding.  Three structural constraints on $T$ are required:
\begin{align}
  T(C) &\;\geq\; 0,\quad T(C) = 0 \;\Leftrightarrow\; C \text{ has no distinct pairs}
    \label{eq:pos}\\[4pt]
  T(C \cup \{a\}) &\;\geq\; T(C) \quad \forall\, a \notin C
    \label{eq:mon}\\[4pt]
  T(C) &\;\neq\; T(\bar{C}) \quad \text{generically}
    \label{eq:ca}
\end{align}
Equation~\eqref{eq:pos} is positivity; equation~\eqref{eq:mon} is monotonicity;
equation~\eqref{eq:ca} is complement-asymmetry, the constraint that distinguishes
$C$ from its complement $\bar{C} = D \setminus C$ in the substrate.
A minimal concrete example satisfying all three simultaneously is the
\textbf{induced edge count}:
\begin{equation}
  T_{\mathrm{ex}}(C) \;=\;
  \bigl|\bigl\{(a,b)\;:\;a,b\in C,\;a\neq b,\;\delta(a,b)\bigr\}\bigr|
  \label{eq:texample}
\end{equation}
Equation~\eqref{eq:pos} holds for any configuration with at least two distinct properties.
Equation~\eqref{eq:mon} holds by inspection: adding property $a$ to $C$ increases the
count by $|\{b\in C:\delta(a,b)\}|$.  Equation~\eqref{eq:ca} holds whenever the induced
subgraphs on $C$ and $\bar{C}$ have different edge counts, which is generic for
non-regular distinction structures.
Equations~\eqref{eq:pos}--\eqref{eq:ca} do not rule out regular distinction graphs; for
a regular $(D,\delta)$ the induced subgraphs on complementary subsets of equal size
can share the same edge count, so $T_{\mathrm{ex}}$ fails complement-asymmetry there.
The example therefore satisfies the constraints on a generic, not universal, sub-class
of $(D,\delta)$.
This example is not the unique correct form of $T$; it shows the constraints are
simultaneously satisfiable.  The precise functional form is an open problem.

A modest imbalance can be sustained within the substrate
without contradiction, just as a small asymmetry in a logical structure does not render
it incoherent.  But there is a limit.  At some critical level of imbalance, the structural
pressure accumulated within the configuration is such that remaining uninstantiated would
require holding together a structure whose internal tension actively resists the very
absence of a resolving medium.  There is no geometry, no dynamics, no causal channel
through which the imbalance could redistribute.  The configuration carries more structural
demand than uninstantiated existence can absorb.

This is the \textbf{modal threshold}.  It is the limit on the modal coherence of
uninstantiated structure: beyond a certain level of asymmetric tension, a configuration
can no longer be maintained within the precausal, nonspatiotemporal domain.  Its
structural incoherence has reached the point at which remaining uninstantiated is no
longer a modally coherent option.  The configuration has not been acted upon by anything.
No force has been applied and no event has occurred.  It is simply that continued
uninstantiated existence has become logically excluded.

The threshold is therefore a \textbf{logical boundary condition}: a constraint on what is
modally possible in the precausal substrate, analogous to the logical impossibility of a
contradiction rather than to a physical limit imposed by dynamics.  This analogy is worth
dwelling on.  When a contradiction is derived in a formal system, it does not occur at a
specific time and it is not caused by any preceding state.  It is simply that the set of
propositions in question cannot all be simultaneously true.  The modal threshold is the
ontological counterpart: a configuration whose tension exceeds $\tau$ cannot simultaneously
be differentiated to that degree and remain uninstantiated.  The two are incompatible modes
of being, and incompatibility is not a causal relation but a logical one.

\subsection{Boundary condition}
The modal threshold is formalised as follows.  There exists a critical value
$\tau \in \mathbb{R}^+$ such that:
\begin{equation}
  T(C) \geq \tau \;\implies\; C \text{ cannot remain uninstantiated}
\end{equation}
Several features of this implication deserve comment.  First, it is a one-way entailment:
exceeding $\tau$ necessitates instantiation, but there is no corresponding claim that
every instantiated configuration arrived at that state by exceeding $\tau$ from a specific
prior configuration.  The precausal substrate is not temporal; configurations do not
``arrive'' at tension values through a sequence of states.  The implication holds
modally, not temporally.

Second, the implication expresses modal necessity, not causal consequence.  It does not
say that something happens to a configuration when $T(C) \geq \tau$.  It says that the
mode of being characterised by uninstantiated existence is not available to such a
configuration.  The language of ``cannot remain'' is a concession to ordinary usage;
strictly, the configuration was never coherently uninstantiated once its tension reached
that level.

Third, the value $\tau$ is not a free parameter of the theory.  It cannot be chosen
arbitrarily or tuned to match observations.  As the variational treatment below makes
clear, $\tau$ is the unique value at which the modal structure of the substrate changes
its minimal configuration from uninstantiated to instantiated.  It is derived from the
structure of the functional, not imposed from outside.

To quantify how far a given configuration sits above the threshold, define the
\textbf{excess tension}:
\begin{equation}
  \varepsilon(C) \;=\; T(C) - \tau
\end{equation}
For configurations below the threshold, $\varepsilon(C) < 0$ and uninstantiated existence
remains coherent.  For configurations above the threshold, $\varepsilon(C) > 0$ and
instantiation is necessitated.  Configurations with $\varepsilon(C) = 0$ inhabit the
threshold precisely: they sit at the boundary of modal coherence, neither firmly in the
uninstantiated regime nor fully instantiated.  It is these marginal configurations that
exhibit the most delicate modal behaviour.  As shown in Section~9, they are the
configurations that give rise to the phenomena of quantum mechanics: the indeterminacy,
the superposition, and the probabilistic character of quantum measurement all arise from
the modal ambiguity of configurations at exactly $\varepsilon = 0$.

The parameter $\varepsilon$ will serve as the primary variable throughout the remainder of
the paper.  It measures not tension in absolute terms but tension relative to the threshold,
and it is this relative measure that determines which mode of being a configuration
occupies.

\subsection{The order parameter and its ontological status}
To derive $\tau$ from first principles, it is necessary to introduce a quantity that
tracks the transition between uninstantiated and instantiated existence continuously.  In
the theory of phase transitions, such a quantity is called an \textbf{order parameter}:
a scalar field that is zero in one phase and nonzero in the other, and whose value
characterises how deep into the new phase the system has penetrated.  PST introduces an
analogous quantity for the modal transition.

Define $\psi(C) \in \mathbb{R}$ as the \textbf{modal order parameter}: a field over
configuration space encoding the modal status of each configuration.  The value $\psi = 0$
denotes the uninstantiated phase.  Nonzero values of $|\psi|$ indicate degrees of
instantiation, with larger $|\psi|$ corresponding to deeper settlement into the
instantiated vacuum.  The domain is taken as $\mathbb{R}$ rather than $[0,1]$ because
the variational analysis requires the $\psi \to -\psi$ symmetry (argued below) to be
well-posed without leaving the domain.  The physical observable is $|\psi|$ or $\psi^2$,
not the sign of $\psi$.  The convention that $\psi \geq 0$ in the instantiated phase is
a gauge choice, not a physical restriction.

The ontological status of $\psi$ requires careful handling.  In condensed matter physics,
an order parameter such as magnetisation has a clear physical meaning: it is the average
magnetic moment per unit volume, measurable in principle at every point in space at every
time.  The modal order parameter $\psi$ is not measurable in this sense.  It is not a
field over spacetime, because spacetime does not exist until after the transition.  It is
instead a field over \textit{configuration space} $\C$: it assigns to each precausal
configuration a scalar that encodes its modal status.  This is not a physical field; it
is a structural characterisation of the substrate.  Its role is formal: it provides the
mathematical language needed to locate the threshold precisely and derive $\tau$ from the
structure of the modal potential.

\subsection{Variational formulation of $\tau$}
With the order parameter in hand, $\tau$ can be derived from a variational principle
rather than assumed.  The strategy is to write down the simplest functional $\F[\psi]$
over configuration space whose structure is fully determined by two requirements: it must
be consistent with the symmetries of the substrate, and it must be capable of exhibiting
a transition between a phase in which $\psi = 0$ is stable and a phase in which $\psi \neq 0$
is stable.  These two requirements, together with a regularity condition, uniquely fix the
leading terms.

\textit{Symmetry constraint.}
The substrate $\S$ carries no preferred orientation or polarity.  Nothing in the definition
of asymmetric tension distinguishes $+\psi$ from $-\psi$: if a configuration can be
instantiated to degree $\psi$, the same configuration reflected through any modal axis has
the same modal status.  The functional must therefore be invariant under the discrete
symmetry $\psi \to -\psi$.  This immediately eliminates all odd powers of $\psi$ from the
expansion: there can be no $\psi^1$ term and no $\psi^3$ term.  Only even powers contribute.

\textit{Taylor expansion near the threshold.}
Close to the threshold, $\psi$ is small and $\F$ can be expanded as a power series in
$\psi$.  The leading term allowed by symmetry is $\psi^2$, and its coefficient must change
sign at $T = \tau$: when $\varepsilon < 0$ (below threshold) the uninstantiated state
$\psi = 0$ is a minimum; when $\varepsilon > 0$ (above threshold) it is a maximum, and the
configuration is driven toward $\psi \neq 0$.  The simplest coefficient consistent with
this requirement is $-\varepsilon = \tau - T(C)$, linear in the tension, which vanishes
precisely at the threshold.

The next allowed term is $\psi^4$ with a positive coefficient $b > 0$.  This term is not
optional: without it, the functional would be unbounded below whenever $\varepsilon > 0$, and no
stable instantiated phase would exist.  The quartic term is the minimal stabilisation
required to ensure that the transition leads to a well-defined new phase rather than an
uncontrolled divergence.  Higher even powers ($\psi^6$, \ldots) contribute corrections
that vanish faster near the threshold and can be absorbed into the definition of $b$ at
leading order; they are omitted as non-essential.

\textit{Gradient regularisation.}
A purely local functional of the form $\int [-\varepsilon\psi^2 + b\psi^4]\,d\mu$ permits $\psi$ to
vary arbitrarily across configuration space: adjacent configurations could have wildly
different modal statuses without any cost.  A coherent transition requires that configurations close in the symmetric-difference
metric have close modal status: the structural condition that
$|\psi(C) - \psi(C')| \to 0$ as $|C \mathbin{\triangle} C'| \to 0$.  This is a
non-temporal coherence requirement on configuration-space neighbours, not a propagation
condition.  It demands a term that penalises sharp variation of $\psi$ across $\mathcal{C}$.  The
natural choice is the squared gradient $|\nabla_{\C}\psi|^2$ with a positive coefficient
$c > 0$, the direct analogue of the Ginzburg stiffness term.  This term raises $\F$
whenever $\psi$ varies rapidly, and its presence ensures that the transition is second
order: no latent heat, no discontinuity in $\psi$ at the threshold, just a continuous
bifurcation of the stable minimum.

The concept of a gradient over configuration space has a precedent in quantum gravity:
Wheeler's superspace \cite{DeWitt1967} is the infinite-dimensional space of all
3-metrics on a spatial slice, equipped with the DeWitt metric, and the Wheeler-DeWitt
equation is a wave equation over it.  The configuration space $\mathcal{C} = \mathcal{P}(D)$
used here is structurally analogous: a space of pre-geometric states over which a modal
field $\psi$ is defined.  The key difference is that $\mathcal{C}$ is a discrete power
set rather than a manifold of Riemannian metrics; the differential structure is a
further architectural commitment described below.

The gradient term warrants a specific caveat, and it is a substantive one.  The operator
$\nabla_{\mathcal{C}}$ presupposes a notion of closeness between configurations, that is,
a topology on $\mathcal{C}$, which is not entailed by $(D, \delta)$ without further
commitment.  This is not a minor technical detail: the inclusion of a gradient term is an
\emph{architectural assumption} that materially shapes the theory's connection to geometry,
and it deserves to be stated as such.

To be precise about what is assumed: in order to write $|\nabla_{\mathcal{C}} \psi|^2$,
one needs (a) a topology on $\mathcal{C}$ that makes it possible to speak of nearby
configurations, (b) a differentiable structure on $\mathcal{C}$ that makes directional
derivatives well-defined, and (c) a metric (or at least an inner product) on the tangent
spaces of $\mathcal{C}$ that makes the squared norm meaningful.  None of these three items
is given for free by the pair $(D, \delta)$.  The property domain $D$ is a set; its power
set $\mathcal{C} = \mathcal{P}(D)$ inherits only the lattice structure of subsets, not a
differential or metric structure.  The gradient term therefore represents a genuine
extension of the foundational ontology.

A natural candidate for the topology is the \textbf{symmetric difference metric}, under
which the distance between two configurations $C, C' \in \mathcal{C}$ is
$d_{\triangle}(C,C') = |C \triangle C'|$, where $C \triangle C'$ is the set of properties
that belong to one configuration but not the other.  Under this metric, two configurations
are close when they differ in few properties.  This choice has structural motivation: it
is the unique metric on $\mathcal{P}(D)$ that is invariant under permutations of $D$ (and
hence under the automorphism group $\mathrm{Aut}(D,\delta)$), and it requires no geometric
primitive, only set membership and cardinality.  Moreover, the symmetric difference
metric makes $\mathcal{C}$ a metric space with controlled Lipschitz properties, which is a
necessary precondition for the variational calculus on which the modal potential functional
$\mathcal{F}$ rests.

What the symmetric-difference metric provides is, in fact, all three requirements —
(a), (b), and (c) — once the Bernoulli measure $\mu$ (derived in the next subsection) is
in hand.  The construction is as follows.

\textit{The Boolean gradient.}  For each locus $a \in D$, define the \textbf{Boolean
derivative} of $\psi$ at $a$ by
\begin{equation}
  (\partial_a \psi)(C)
  \;:=\; \psi(C \cup \{a\}) - \psi(C \setminus \{a\}),
  \label{eq:boolean-deriv}
\end{equation}
where $C \cup \{a\}$ is the configuration $C$ with locus $a$ activated, and
$C \setminus \{a\}$ is $C$ with $a$ inactivated.  This is the unique finite-difference
operator compatible with the graph structure of $(\mathcal{C}, d_{\triangle})$: two
configurations are adjacent in $(\mathcal{C}, d_{\triangle})$ iff they differ by exactly
one locus activation, and $\partial_a\psi$ measures the variation of $\psi$ across
the unique edge incident on both $C \cup\{a\}$ and $C \setminus\{a\}$.  The full
\textbf{Boolean gradient} is the vector of all these derivatives:
\begin{equation}
  |\nabla_{\mathcal{C}}\psi|^2(C)
  \;:=\; \tfrac{1}{4}\sum_{a \in D} |\psi(C\cup\{a\}) - \psi(C\setminus\{a\})|^2.
  \label{eq:boolean-grad-sq}
\end{equation}

\textit{L$^2$ differentiable structure.}  The Bernoulli measure $\mu$ on $\mathcal{P}(D)$
turns $L^2(\mathcal{P}(D), \mu)$ into a Hilbert space.  For any
$\psi \in L^2(\mathcal{P}(D), \mu)$, the quantity $\mathbb{E}_\mu[|\nabla_{\mathcal{C}}\psi|^2]$
is finite, because each $\partial_a\psi$ is bounded in $L^2$ whenever $\psi$ is.
The gradient operator $\nabla_{\mathcal{C}}:
L^2(\mathcal{P}(D),\mu) \to \bigoplus_{a\in D} L^2(\mathcal{P}(D),\mu)$
is a bounded linear operator, and the Sobolev space
$W^{1,2}(\mathcal{P}(D),\mu) = \{\psi : \mathbb{E}_\mu[|\nabla_{\mathcal{C}}\psi|^2] < \infty\}$
is a Hilbert space with inner product
$\langle\psi,\phi\rangle_{1,2} = \langle\psi,\phi\rangle_{L^2} +
\sum_a \langle\partial_a\psi, \partial_a\phi\rangle_{L^2}$ \cite{ODonnell2014}.
The inner product on tangent directions (the role of item~(c)) is the
$L^2(\mu)$-inner product on the Boolean derivatives, which requires no additional
primitive beyond $\mu$ and $d_\triangle$.

\textit{Poincaré inequality.}  The Efron--Stein inequality \cite{EfronStein1981} states:
for any $\psi \in W^{1,2}(\mathcal{P}(D),\mu)$,
\begin{equation}
  \mathrm{Var}_\mu(\psi)
  \;\leq\;
  \tfrac{1}{4}\sum_{a\in D}\,\mathbb{E}_\mu\!\left[|\partial_a\psi|^2\right]
  \;=\; \mathbb{E}_\mu\!\left[|\nabla_{\mathcal{C}}\psi|^2\right],
  \label{eq:poincare-boolean}
\end{equation}
with equality when $\mu = \bigotimes_a \mathrm{Bern}(1/2)$ and $\psi$ is linear in
the indicator functions $\mathbf{1}_a$ \cite{ODonnell2014}.  This Poincaré inequality
guarantees that the gradient term in $\mathcal{F}$ controls the variance of $\psi$
about its mean: configurations far apart in $d_\triangle$ are forced to have similar
$\psi$ values by the cost $c\,|\nabla_{\mathcal{C}}\psi|^2$, which is exactly the
coherence requirement the gradient term was introduced to enforce.

\textit{Resolution of the apparent gap.}  The differentiable structure (b) is provided
by the $L^2(\mu)$ Hilbert space structure, whose existence follows from the Bernoulli
measure $\mu$ derived in the next subsection.  The inner product on tangent directions (c)
is the $L^2(\mu)$-inner product on Boolean derivatives, derived from $d_\triangle$ alone.
No additional topological or geometric primitive is required.  The gradient structure
problem is thereby resolved within
the existing primitive structure of PST.

Critically, this topology is \emph{not} the topology of spacetime.  The closeness being
assumed is closeness among pre-geometric substrate configurations, not proximity of points
in an emergent manifold.  The gradient term acts on the substrate field $\psi$ before the
functor $\Phi$ is applied; it knows nothing of the metric $g_{\mu\nu}$ that will later
emerge.  The architectural commitment is therefore pre-geometric: it asserts that the
substrate has enough internal structure to support a coherent, spatially organised field,
rather than an arbitrary collection of uncorrelated values.  If the commitment is unjustified,
meaning no derivation of the differential structure from $(D, \delta)$ can be found, then the
gradient term would have to be either motivated by a different argument or removed, in
which case the connection to the Einstein--Hilbert action (which relies on the gradient term
projecting to the Ricci scalar in Section~7) would need to be re-examined.  The theory's
architecture thus carries a genuine open commitment at this level.

The gradient term is retained because the structural requirement it enforces,
namely a coherent, spatially organised modal transition rather than a configuration-by-configuration
patchy one, is physically non-negotiable.  Without it, the modal condensate would carry
no spatial correlation, and the projection $\Phi$ could not produce a smooth spacetime
geometry.  The derivation of this structure from first principles is a problem whose
resolution would either strengthen PST substantially or, if impossible, force a revision of
the framework from which the gradient term emerged.

\textit{The canonical measure on configuration space.}
Before assembling the functional, the measure $\mu$ on $\C$ requires derivation.  The
variational formulation demands integration over all configurations, and the choice of
measure is not arbitrary: it must follow from the structure of $(D, \delta)$ alone,
without invoking any geometric, probabilistic, or metric primitive that would introduce
a new assumption at the level of the substrate.

The configuration space $\C = \mathcal{P}(D)$ is the power set of $D$: the collection
of all subsets of the property domain.  The structure $(D, \delta)$ carries a natural
symmetry group, its \textbf{automorphism group} $\mathrm{Aut}(D, \delta)$, consisting of
all bijections $f: D \rightarrow D$ that preserve the distinction relation:
$\delta(a, b) \iff \delta(f(a), f(b))$ for all $a, b \in D$.  Every such automorphism
lifts canonically to a bijection on $\C$ by acting elementwise: $f \cdot C = \{f(a) :
a \in C\}$.  Since the only primitive is $(D, \delta)$, and since the definition of a
configuration is simply a subset of $D$, the measure $\mu$ must be invariant under this
lifted action: no configuration should receive more weight than any automorphically
equivalent configuration.

The key observation is that in the precausal substrate, all properties in $D$ are
structurally equivalent: the distinction relation $\delta$ is irreflexive and symmetric
but otherwise unconstrained, so there is no intrinsic difference between any one property
and any other beyond their bare identity.  No property is marked, distinguished, or
preferred.  This means that the full symmetric group on $D$ is contained in
$\mathrm{Aut}(D, \delta)$ as a subgroup: any permutation of $D$ is an automorphism, since
$\delta$ treats all pairs equally.  A measure invariant under all permutations of $D$ is
invariant under all relabellings of the properties, which is the minimal requirement for
a measure that introduces no additional structure beyond $(D, \delta)$.

For a finite domain $D$ of cardinality $n$, the unique permutation-invariant probability
measure on $\mathcal{P}(D)$ is the uniform measure, assigning equal weight to all $2^n$
subsets:
\begin{equation}
  \mu(C) = 2^{-n} \qquad \text{for all } C \in \C
  \label{eq:uniformmeasure}
\end{equation}
Uniqueness follows immediately: any other probability measure would assign different
weights to configurations that are related by a permutation of $D$, introducing a
distinction between properties that does not exist in the primitive $(D, \delta)$.

For an infinite domain $D$, the natural extension is the \textbf{Bernoulli product
measure} $\mu = \bigotimes_{a \in D} \mathrm{Bern}(1/2)$, the unique probability measure
on $\{0,1\}^D$ invariant under all permutations of the index set $D$:
\begin{equation}
  \mu\bigl(\{C \in \C : C_0 \subseteq C\}\bigr) = 2^{-|C_0|}
  \qquad \text{for all finite } C_0 \subseteq D
  \label{eq:bernoullimeasure}
\end{equation}
This is the projective limit of the uniform measures on all finite sub-domains of $D$,
and it is the unique translation-invariant probability measure on the product space
$\{0,1\}^D$.

The value $1/2$ for the Bernoulli parameter is not arbitrary: it is the unique value
for which $\mu$ is invariant under complementation, $C \mapsto \bar{C} = D \setminus C$.
Complementation invariance is required by the structure of property differentiation:
the distinction relation $\delta(a, b)$ is symmetric, and no configuration is
intrinsically more natural than its complement.  A configuration $C$ and its complement
$\bar{C}$ represent exactly the same collection of pairwise distinctions, viewed from
two opposite orientations that are not distinguished by anything in $(D, \delta)$.
Assigning $\mu(C) \neq \mu(\bar{C})$ would introduce a polarity into the substrate that
has no basis in the primitive.  The Bernoulli parameter $1/2$ is therefore forced by
complementation invariance alone, independently of the permutation argument.

The derivation may be summarised as a theorem.

\begin{quote}
\textbf{Theorem (Canonical Measure).}  Let $(D, \delta)$ be a precausal property domain
with $\delta$ symmetric and irreflexive.  The unique probability measure $\mu$ on
$\C = \mathcal{P}(D)$ that is (i) invariant under all automorphisms of $(D, \delta)$ and
(ii) invariant under complementation $C \mapsto \bar{C}$ is the Bernoulli product measure
$\mu = \bigotimes_{a \in D} \mathrm{Bern}(1/2)$.
\end{quote}

\noindent
The proof follows directly from the two invariance requirements: permutation invariance
forces the single-site marginals to be equal, and complementation invariance forces each
marginal to satisfy $\mu_a(\{a \in C\}) = \mu_a(\{a \notin C\}) = 1/2$.  Independence
across sites follows from the absence of any structural relation in $(D, \delta)$ that
could correlate the inclusion of one property with the inclusion of another.

This closes the first item listed as open in earlier formulations of the theory.  The
measure $\mu$ is not a free parameter or an auxiliary assumption: it is uniquely
determined by the symmetry structure of $(D, \delta)$, without invoking any concept
from outside the precausal domain.

Assembling these three terms gives the \textbf{modal potential functional}:
\begin{equation}
  \F(\psi, C) = \int_{\C} \left[ a(T(C))\,\psi^2 + b\,\psi^4
    + c\,|\nabla_{\C}\psi|^2 \right] d\mu(C)
  \label{eq:freeenergy}
\end{equation}
where $\nabla_{\C}$ denotes the gradient over configuration space (not spacetime), $\mu$
is the canonical Bernoulli measure of equation~\eqref{eq:bernoullimeasure}, and the
three terms carry the following interpretation.  Restricting to spatially uniform
configurations ($\nabla_\C\psi = 0$) with $b = \tfrac{1}{4}$, the functional reduces to
the compact scalar form:
\begin{equation}
  \F(\psi) = -\varepsilon\,\psi^2 + \frac{1}{4}\,\psi^4, \qquad \varepsilon = T(C) - \tau
  \label{eq:freduced}
\end{equation}
When $\varepsilon < 0$ (below threshold, precausal) $\F$ has a unique minimum at $\psi = 0$;
when $\varepsilon > 0$ (above threshold, instantiated) two symmetric minima appear at
$\psi^* = \pm\sqrt{2\varepsilon}$.  A rescaled variant
$-\tfrac{1}{2}\varepsilon\,\psi^2 + \tfrac{1}{4}\psi^4$ appears in the visualisations;
its minima sit at $\psi^* = \pm\sqrt{\varepsilon}$, a factor of $1/\sqrt{2}$ smaller than
$\pm\sqrt{2\varepsilon}$, but the bifurcation structure is preserved exactly.  The full
three-term structure of equation~\eqref{eq:freeenergy} is retained throughout the analysis.

The first term, $-\varepsilon(T(C))\,\psi^2$, governs the stability of the uninstantiated
phase.  The coefficient $-\varepsilon = \tau - T$ changes sign at $T = \tau$.  When
$T < \tau$, $\varepsilon < 0$ and this term increases $\F$ as $\psi$ increases from zero,
keeping the uninstantiated state $\psi = 0$ at a local minimum: configurations prefer to
remain uninstantiated.  When $T > \tau$, $\varepsilon > 0$ and the term now decreases $\F$
near $\psi = 0$, destabilising the uninstantiated state and driving the configuration toward
nonzero $\psi$.  The sign change in $\varepsilon$ is precisely what defines the threshold.

The second term, $b\,\psi^4$ with $b > 0$, is the quartic stabilisation term.  Without it,
the functional would be unbounded below for $T > \tau$, and $\psi$ could grow without
limit.  The quartic term ensures that once the uninstantiated phase becomes unstable, the
configuration does not destabilise entirely but settles into a new, well-defined minimum
at some finite value of $\psi$.  It is this term that makes the transition a genuine phase
change with a stable instantiated phase rather than a runaway.

The third term, $c\,|\nabla_{\C}\psi|^2$ with $c > 0$, penalises inhomogeneity in $\psi$
across configuration space.  If configurations with similar structures were to have very
different modal statuses, the gradient would be large and the term would raise $\F$.  This
term is the formal analogue of the Ginzburg gradient term and has the physical effect of
producing smooth, coherent transitions rather than discontinuous patches of instantiated
and uninstantiated configurations.  Its presence ensures that the transition is second
order rather than first order, a point to which we return below.

This functional is structurally identical to the Landau--Ginzburg free energy of second
order phase transitions \cite{GinzburgLandau1950}, and the analogy is illuminating.  In
condensed matter physics, the Landau--Ginzburg functional describes how a system moves
between phases as temperature changes.  The order parameter tracks the degree of order
(magnetisation, superconducting condensate, and so on), and the free energy
determines which phase is stable at which temperature.  In PST, the role of temperature
is played by the asymmetric tension $T(C)$, the role of the order parameter is played by
the modal status $\psi$, and the potential determines which mode of being , 
uninstantiated or instantiated, is stable at which tension level.

The difference from the condensed matter case is fundamental.  In condensed matter, the
phase transition is a dynamical process: it unfolds over time, involves fluctuations and
nucleation, and is driven by thermal agitation.  In PST, there is no temperature, no heat
bath, no temporal evolution, and no dynamics.  The transition is a modal reorganisation:
a change in which mode of being is logically available to the configuration, not a change
in the state of a system that exists in time.  The Landau--Ginzburg structure is borrowed
for its mathematical economy, not for its physical interpretation.

\subsection{Stability analysis and the two regimes}
The most productive way to extract modal content from a functional is to identify which
configurations minimise it.  In classical mechanics, the principle of least action selects
the physically realised trajectory from among all kinematically possible ones.  In PST,
the principle is modal rather than temporal: among all values of the order parameter
available to a configuration of given tension $T(C)$, the modally self-consistent value
is the one at which $\F$ attains its minimum.  This is a variational statement, not a
dynamical one.  There is no time over which $\psi$ evolves toward its minimum; the minimum
is the only modal position a configuration can coherently occupy given its tension.
Configurations whose order parameter deviates from the minimum of $\F$ are modally
incoherent: their degree of instantiation is inconsistent with the tension they carry.

The condition for a minimum is the vanishing of the first functional derivative,
$\delta\F/\delta\psi = 0$, the Euler--Lagrange condition applied to the modal
functional.  Before writing the result, it is worth identifying the structural role of
each term in $\F$.  The quadratic term $-\varepsilon\psi^2$ is the modal restoring term:
its coefficient $-\varepsilon = \tau - T(C)$ changes sign precisely at the threshold and
is entirely responsible for the qualitative bifurcation in the behaviour of $\F$.  The
quartic term $b\psi^4$ is the nonlinear saturation term: it is always positive and ensures
the functional remains bounded below, preventing $\psi$ from growing without bound when
$\varepsilon$ becomes positive.  The gradient term $c|\nabla_\C\psi|^2$ penalises rapid variation of $\psi$
across configuration space and ensures that the transition is spatially coherent rather
than fragmentary.  Distributing $\delta/\delta\psi$ across these three terms and
collecting, the modal equation of state is:
\begin{equation}
  -2\varepsilon(T(C))\,\psi + 4b\,\psi^3 - 2c\,\nabla_{\C}^2\psi = 0
  \label{eq:eom}
\end{equation}
This equation has two qualitatively distinct regimes, separated by the threshold.

\textit{Below the threshold, $T(C) < \tau$}: $\varepsilon < 0$, so $-\varepsilon > 0$.
Consider the spatially uniform solutions $\nabla_{\C}\psi = 0$.  Equation~(\ref{eq:eom})
reduces to $-2\varepsilon\psi + 4b\psi^3 = 0$, which factors as
$2\psi(-\varepsilon + 2b\psi^2) = 0$.  For $\varepsilon < 0$, the factor
$(-\varepsilon + 2b\psi^2)$ is strictly positive for all real $\psi$, so the only real
solution is $\psi = 0$.  Since
$\mathrm{d}^2\F/\mathrm{d}\psi^2\big|_{\psi=0} = -2\varepsilon > 0$, this is a local
minimum.  The configuration remains uninstantiated.  This is the \textit{precausal phase}.

\textit{Above the threshold, $T(C) > \tau$}: $\varepsilon > 0$, so $-\varepsilon < 0$.  The
solution $\psi = 0$ is now a local maximum of $\F$, not a minimum.  It is an unstable
equilibrium.  The quadratic term $-\varepsilon\psi^2$ now falls with increasing $|\psi|$,
so any deviation from zero lowers $\F$ initially; the quartic term $b\psi^4$ eventually
turns $\F$ upward again at large $|\psi|$.  The result is a double-well profile with a
local maximum at $\psi = 0$ and two symmetric minima at nonzero values of $\psi$.

The location of these minima requires explicit computation.  Restricting to spatially
uniform configurations, $\nabla_\C\psi = 0$, the equation of state~(\ref{eq:eom}) reduces
to $-2\varepsilon\psi + 4b\psi^3 = 0$, or equivalently $2\psi(-\varepsilon + 2b\psi^2) = 0$.
The root $\psi = 0$ is a maximum, as established.  The nontrivial roots satisfy
$-\varepsilon + 2b\psi^2 = 0$, giving $\psi^2 = \varepsilon/(2b) = \varepsilon(C)/(2b)$.
Since $\varepsilon > 0$ above the threshold, this quantity is positive; the roots are real.
The two stable configurations are therefore:
\begin{equation}
  \psi = \pm\sqrt{\frac{T(C) - \tau}{2b}} = \pm\sqrt{\frac{\varepsilon(C)}{2b}}
\end{equation}
The appearance of $\varepsilon(C)$ under the radical is structurally significant.  The
depth of instantiation, how far from zero the order parameter settles, is not a free
parameter but is determined directly by how far the configuration's tension sits above the
threshold.  A configuration with $\varepsilon$ barely positive instantiates shallowly; one
deep in the instantiated regime carries a large order parameter.  This relationship between
excess tension and the degree of instantiation is one of the central quantitative
predictions of PST.

There are two degenerate minima, symmetric about $\psi = 0$.  The configuration must
occupy one of them, but the functional does not select which: both are equally deep.
This is the \textit{instantiated phase}.  The two solutions correspond to the two
structurally distinct orientations of the instantiated configuration, related by the
symmetry $\psi \to -\psi$ that the functional possesses.  That the functional carries
this symmetry but the configuration that minimises it does not is the content of
spontaneous symmetry breaking in the PST framework: the substrate selects a definite
modal orientation without any external influence distinguishing the two options.

The sign of $a$ is therefore the fundamental discriminant between uninstantiated and
instantiated existence.  It is not a property of the configuration that can be read off
directly; it is a property of the relationship between the configuration's tension and
the threshold.  This is why the threshold is primary and the transition is a logical
consequence, not a dynamical one.

\subsection{The bifurcation point and the derivation of $\tau$}
The critical tension $\tau$ is the value at which the qualitative character of $\F$
changes: below $\tau$, the uninstantiated phase $\psi = 0$ is the unique minimum; above
$\tau$, it is a maximum and two new minima appear.  The moment of change is the
\textbf{bifurcation point}, the value of $T$ at which $\psi = 0$ transitions from stable
to unstable.

The characterisation of critical points as minima or maxima requires the second variation.
In single-variable calculus, the second derivative test distinguishes a local minimum
($f'' > 0$) from a local maximum ($f'' < 0$) at a critical point.  For a functional, the
analogous test is the second functional derivative evaluated at the critical configuration.
If $\delta^2\F/\delta\psi^2 \big|_{\psi_0} > 0$, the functional is locally convex at
$\psi_0$: the configuration is at a stable minimum.  If negative, the functional is
locally concave and the critical point is an unstable maximum.

The critical point of interest is $\psi = 0$, the uninstantiated configuration.  To
determine whether this is modally stable (the precausal phase is robust) or unstable (it
cannot be maintained), one must examine the curvature of $\F$ at this point.
Differentiating $\delta\F/\delta\psi = -2\varepsilon\psi + 4b\psi^3$ once more with
respect to $\psi$ and evaluating at $\psi = 0$ eliminates the cubic term entirely, since
it contributes only $12b\psi^2 \big|_{\psi=0} = 0$.  What remains is purely the
contribution of the quadratic term:
\begin{equation}
  \frac{\delta^2\F}{\delta\psi^2}\bigg|_{\psi=0} = -2\varepsilon(T) = 2(\tau - T)
\end{equation}
This is positive for $T < \tau$ (i.e., $\varepsilon < 0$), zero for $T = \tau$, and
negative for $T > \tau$ (i.e., $\varepsilon > 0$).
The second variation vanishes at $T = \tau$, and this vanishing is the mathematical
realisation of the threshold: the precise tension value at which the uninstantiated
configuration ceases to be stable.  What remains is to give $\tau$ a rigorous definition
that does not presuppose its own value.

The naive approach would be to define $\tau$ as the solution to $\delta^2\F/\delta\psi^2
\big|_{\psi=0} = 0$, which from the expression above gives $-2\varepsilon = 0$ and thus
$T = \tau$, a circularity that conceals rather than reveals.  The correct approach is
to observe that $\tau$ is defined by a property of the functional, not by an algebraic
equation: it is the value below which $\psi = 0$ is a stable minimum and above which it
is not.  This is precisely the infimum of the set of tension values for which the second
variation vanishes:
\begin{equation}
  \tau = \inf\left\{ T(C) : \frac{\delta^2 \F}{\delta\psi^2}\bigg|_{\psi=0} = 0 \right\}
  \label{eq:tau}
\end{equation}
The use of the infimum rather than a simple equation reflects the possibility that
configurations at exactly $T = \tau$ form a set of measure zero in $\C$: the threshold
is the greatest lower bound of all tension values that can trigger instantiation, and this
bound may or may not be achieved by any individual configuration.  In practice, when the functional takes the quadratic-quartic form above,
the infimum is achieved and $\tau$ is well-defined as a specific value.

The significance of this derivation is that $\tau$ emerges from the structure of $\F$
itself.  It is not a parameter that the theorist chooses.  Once the functional is
specified, and its form is constrained by the symmetries of the precausal substrate and
the requirement that the transition be second order: the threshold is fixed.  In this
sense, the modal threshold is not a contingent feature of the theory.  It is a structural
necessity: any substrate governed by a modal potential of this form will have a
threshold, and that threshold will be the bifurcation point of the functional.

The passage from $\psi = 0$ to the bifurcated minimum is modal sublimation, with no
temporal unfolding.  The next section characterises that transition in detail.

\begin{figure}[htb]
  \centering
  \includegraphics[width=0.32\linewidth]{figures/diagram-precausal}%
  \hfill
  \includegraphics[width=0.32\linewidth]{figures/diagram-threshold}%
  \hfill
  \includegraphics[width=0.32\linewidth]{figures/diagram-instantiated}
  \caption{The modal potential $F[\varphi]$ at three values of excess tension $\varepsilon$.
  \textit{Left:} $\varepsilon = {-}0.8$, single valley at $\varphi = 0$; the substrate
  configuration rests at the unique minimum, uninstantiated.
  \textit{Centre:} $\varepsilon = 0$, the modal threshold; the valley flattens and the minimum
  is about to split.
  \textit{Right:} $\varepsilon = {+}0.8$, two degenerate minima at $\varphi = \pm\sqrt{2\varepsilon}$;
  the configuration at $\varphi = 0$ is now a local maximum and cannot remain there.
  The dashed curve is the reference potential at $\varepsilon = 0$.
  The four key relations carried by this figure are:
  (1)~the Landau--Ginzburg modal potential
      $F(\psi,\varepsilon) = -\varepsilon\psi^2 + \tfrac{1}{4}\psi^4$
      with quartic coefficient $b = \tfrac{1}{4}$ fixed by the LG form;
  (2)~excess tension $\varepsilon = T(\mathcal{C})-\tau$;
  (3)~the vacuum manifold radius $\psi^* = \pm\sqrt{2\varepsilon}$ and its topology
      $\mathcal{V} \cong S^1$ (a ring of degenerate minima, not a point); and
  (4)~the projection map $\Phi: \mathcal{S} \to \mathcal{G}$ from the precausal substrate
      to instantiated geometry.}
  \label{fig:modal-potential}
\end{figure}


\subsection{Layman's Summary}

Not every tension imbalance produces a universe.  This section derives a critical tension value, the modal threshold $\tau$, below which the substrate remains abstract and unobservable, and above which it cannot remain that way.  Think of water vapour: it can exist invisibly up to a certain temperature, but above that point it must condense.  Below $\tau$, the configuration has a single stable resting state with no geometry; above $\tau$, that resting state becomes unstable and geometry must appear.  The threshold is not assumed: it is calculated as the exact tension value at which the mathematical potential changes from single-welled to double-welled, and its existence is a structural necessity rather than a postulate.

% 4. Modal Sublimation
\section{Modal Sublimation}

\subsection{The character of the transition}

The transition induced by crossing the modal threshold is unlike any physical process
encountered in the instantiated world.  It has no duration, no mechanism, no intermediate
states, and no energy cost.  It is not that the transition is very fast; temporal concepts
simply do not apply to it.  The precausal substrate is a nonspatiotemporal domain: there
is no time in which a transition could unfold, and no space across which it could
propagate.  What occurs at the threshold is not a process but a \textbf{modal
reorganisation}, a shift in the mode of being of the configuration from uninstantiated to
instantiated.

This transition is called \textbf{modal sublimation}, by analogy with the physical
phenomenon in which a solid passes directly into a gaseous state, bypassing the liquid
phase.  In modal sublimation, every intermediate mode is bypassed because only two modes
of being are available: uninstantiated existence within the precausal substrate, and
instantiated existence as geometry.  There is no intermediate mode that is partially
spatial or partially causal.

Modal sublimation is characterised by four properties:
\begin{itemize}
  \item \textbf{Nontemporal}: no before or after, no duration, no sequence of states;
    the configuration transitions \textit{into} time rather than \textit{through} it
  \item \textbf{Nonmechanistic}: no propagating influence, no force, no interaction
  \item \textbf{Nonenergetic}: no conservation law, no energy input or output
  \item \textbf{Strictly modal}: a change in the mode of being, not a change of state
    within an existing mode
\end{itemize}

Each of these four properties requires unpacking, because each contradicts a default
assumption carried over from the instantiated world.

\subsection{The nontemporal character}
The nontemporal character is the most radical.  In everyday physics, every process
unfolds in time: there is a before and an after, a cause and an effect, a duration that
can in principle be measured.  Even processes described as instantaneous in
non-relativistic mechanics, a collision, a decay, a measurement, are idealised
limits of processes that have a finite, if arbitrarily small, duration.  Modal sublimation
is not a limit of this kind.  It does not occur quickly; it does not occur in time at all.
The configuration does not spend even an infinitesimal moment in transition.  The reason
is not dynamical but ontological: time is a structure of the instantiated domain, and the
transition is the very event that brings that domain into being.  To ask how long modal
sublimation takes is to apply a concept to a context in which the concept has not yet been
defined.  This is not a failure of measurement precision; it is a logical impossibility of
the same kind as asking what temperature a mathematical proof achieves.  The configuration
does not transition \textit{through} time; it transitions \textit{into} time, carrying time
as a consequence of instantiation rather than as a pre-existing container in which the
transition unfolds.

\subsection{The nonmechanistic character}
The nonmechanistic character follows directly from the nontemporal character.  In the
instantiated domain, every change of state is mediated by an interaction: a force, a field,
a messenger particle, a propagating influence of some kind.  These mediators all require
spacetime in which to propagate and physical degrees of freedom to couple to.  The precausal
substrate has neither.  No interaction can mediate the modal sublimation because interactions
are structures of the domain that sublimation brings into being.  There is no force that
pushes a configuration across the threshold; the threshold is not a barrier that must be
overcome but a logical boundary at which a particular mode of being ceases to be available.
The transition is not driven by anything external.  It is a logical consequence of the modal
structure of the configuration itself, in the same way that the truth of a mathematical
theorem is not caused by anything but follows necessarily from the axioms.  This has a
crucial implication: PST does not require a mechanism, an initial impetus, or a first cause.
The demand for a mechanism is itself an instantiated-world concept, and applying it to the
transition that precedes instantiation is a category error of the same kind as the causal
question discussed in Section~5.

\subsection{The nonenergetic character}
The nonenergetic character removes the last temptation to treat modal sublimation as a
physical event in disguise.  Energy is a conserved quantity defined within the instantiated
domain: it requires a Hamiltonian, a time-translation symmetry via Noether's theorem, and
ultimately a spacetime manifold on which these structures can be defined.  None of these are
available in the precausal substrate.  The relevant conservation statement in PST is not
energetic but modal: it is the preservation of the asymmetric tension functional $T(C)$
across the threshold.  The tension that characterises the precausal configuration is the
same quantity that, after projection via $\Pi$, appears as the stress-energy tensor
$T_{\mu\nu}$ in the instantiated domain, as shown in Section~7.  The Einstein field
equations, on this reading, are not equations of motion governing a physical process; they
are a \textbf{conservation statement} expressing that the structural content of the
precausal configuration is faithfully preserved in the instantiated geometry.  Nothing is
created across the threshold; nothing is destroyed.  The same structural content is
expressed in a different modal register, the way that the same logical argument can be
expressed in different formal languages without gaining or losing content.

\subsection{The strictly modal character}
The strictly modal character is the most philosophically precise formulation of what the
other three properties imply.  A change of state \textit{within} an existing mode, a
particle moving, a field oscillating, a temperature rising, leaves the mode of being
intact while altering something within it.  The ontological register is unchanged: the
system is still a physical system in spacetime, governed by the same causal laws, measured
in the same units.  Modal sublimation changes the mode itself.  The configuration does not
acquire new properties within the precausal substrate; it changes the ontological register
in which all its properties are expressed.  Before sublimation, the configuration's
structural content exists as modal tension in a nonspatiotemporal domain.  After sublimation,
that same structural content exists as geometric curvature and matter-energy in a Lorentzian
manifold.  There is no moment at which the configuration is partially in each mode; the
transition is all-at-once in the modal sense, even though ``all-at-once'' is itself a
temporal phrase that strictly applies only after time has come into being.  This is the
precise sense in which PST's fundamental transition is irreducible to any physical process:
physical processes are changes of state within the instantiated mode, and modal sublimation
is the event that instantiates the mode itself.

The result of this transition is a \textbf{geometry}: a Lorentzian manifold $(M, g)$
that becomes the spacetime background for all subsequent physical processes.  Geometry
does not emerge \textit{within} spacetime; it \textit{becomes} spacetime in the act of
sublimation.

\begin{figure}[p]
  \centering
  \setlength{\tabcolsep}{3pt}
  \renewcommand{\arraystretch}{0.8}
  \begin{tabular}{cccc}
    \includegraphics[width=0.235\linewidth]{figures/anim-01} &
    \includegraphics[width=0.235\linewidth]{figures/anim-02} &
    \includegraphics[width=0.235\linewidth]{figures/anim-03} &
    \includegraphics[width=0.235\linewidth]{figures/anim-04} \\[-2pt]
    \footnotesize $\varepsilon = {-}1.00$ &
    \footnotesize $\varepsilon = 0$ &
    \footnotesize $\varepsilon = {+}0.15$ &
    \footnotesize $\varepsilon = {+}0.30$ \\[4pt]
    \includegraphics[width=0.235\linewidth]{figures/anim-05} &
    \includegraphics[width=0.235\linewidth]{figures/anim-06} &
    \includegraphics[width=0.235\linewidth]{figures/anim-07} &
    \includegraphics[width=0.235\linewidth]{figures/anim-08} \\[-2pt]
    \footnotesize $\varepsilon = {+}0.50$ &
    \footnotesize $\varepsilon = {+}0.70$ &
    \footnotesize $\varepsilon = {+}0.90$ &
    \footnotesize $\varepsilon = {+}1.00$ \\
  \end{tabular}
  \caption{Visual model of modal sublimation across eight values of the excess tension
  $\varepsilon = T(C) - \tau$ (Section~3): $\varepsilon < 0$ precausal, $\varepsilon \geq 0$
  instantiated.  Sampling is dense on the instantiated side, where the geometric content
  emerges; a single precausal reference frame at $\varepsilon = {-}1.00$ shows the
  pre-threshold state from which everything else unfolds.
  Each frame shows the modal potential $F[\psi,\varepsilon] = -\varepsilon\psi^{2}+\tfrac{1}{4}\psi^{4}$
  rendered as a rotationally symmetric surface (lower portion), with the orange tube tracing its
  one-dimensional cross-section and the white sphere marking the position of the order parameter
  $\psi$ at the current potential minimum.
  \textit{Top-left frame} ($\varepsilon = {-}1.00$): the precausal substrate; the potential
  forms a deep bowl with a single minimum at $\psi = 0$, the order parameter is confined to
  the origin, and a flat proto-manifold grid is shown for reference.
  \textit{Threshold and early instantiation} ($\varepsilon = 0$ to $\varepsilon \approx
  {+}0.30$): the bowl flattens and its centre rises, the high-dimensional cube descends
  toward the bowl, the spacetime lattice crystallises, and the ring of minima begins to
  form at the base.
  \textit{Mid to deep instantiation} ($\varepsilon \approx {+}0.50$ to ${+}1.00$): the
  potential has acquired a continuous ring of degenerate minima (the vacuum manifold
  $\mathcal{V}\cong S^{1}$), visible at the base of the Mexican-hat surface as an orange
  ring; the spatial slice closes, shown by the $S^{3}$ dome enveloping the local chart;
  and the finite internal factor $A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$
  consolidates as the cube snaps onto the spacetime grid.}
  \label{fig:modal-sublimation}
\end{figure}

\subsection{Categorical formulation}
The mathematical language suited to a transition between fundamentally different types of
structure is category theory, specifically the theory of functors between categories
\cite{MacLane1971}.  Construct two categories as follows.  The \textbf{precausal category}
$\bfS$ has as objects all configurations $C \in \C$ with $T(C) \geq \tau$, and as
morphisms distinction preserving maps $f: C \rightarrow C'$.  The \textbf{geometric
category} $\bfG$ has as objects Lorentzian manifolds $(M, g)$ and as morphisms smooth
maps preserving the causal structure.

\textbf{Modal sublimation} is the functor:
\begin{equation}
  \Phi: \bfS \rightarrow \bfG
\end{equation}
with four properties encoding the fundamental features of the transition:
\begin{enumerate}
  \item $\Phi$ \textbf{exists at the categorical boundary}; it is not a morphism within
    $\bfS$ or $\bfG$, formalizing the nonmechanistic character of the transition
  \item $\Phi$ \textbf{preserves tension as curvature}: $T(C) \mapsto \kappa(\Phi(C))$,
    where $\kappa$ is the scalar curvature of the instantiated manifold
  \item $\Phi$ \textbf{is surjective}: every Lorentzian manifold arises as the image of
    some precausal configuration
  \item $\Phi$ \textbf{has no inverse}: once a configuration is instantiated, the
    geometric domain has no access to the precausal substrate that produced it
\end{enumerate}

\subsection{Explicit construction of $\Phi$}
The functor is constructed in three steps.

\textit{Step 1, Geometric realization.}  The manifold $M$ and the map $\rho$ are not
given independently before the construction; they are jointly determined by it.

A legitimate concern requires attention before the construction proceeds.  The notation
$\rho: D \rightarrow M$ appears to presuppose $M$, a set of spacetime points,
before the construction that is supposed to \emph{produce} spacetime has been carried
out.  If spacetime points are already available as the target of $\rho$, then the theory
has not derived spacetime from the substrate; it has merely labeled pre-existing points
with properties from $D$.  This would be a foundational circularity.

The resolution is that $M$ is not given prior to and independently of $\rho$.  The
construction proceeds in the following order.  First, note that any surjective function
$\rho: D \twoheadrightarrow X$, for \emph{any} set $X$, partitions $D$ into equivalence
classes: $a \sim_\rho b$ if and only if $\rho(a) = \rho(b)$.  The equivalence classes
$[a]_\rho = \{b \in D : \rho(b) = \rho(a)\}$ are the fibres of $\rho$, and their collection
$D/{\sim_\rho}$ is a quotient set that depends only on $D$ and the partition, not on any
property of $X$.  A spacetime \emph{point} is then defined to be a fibre of $\rho$: it is
an equivalence class of properties, not a pre-existing geometric entity.  The manifold $M$
is the quotient set $D/{\sim_\rho}$ equipped with whatever additional structure (topology,
differentiable structure, metric) the subsequent steps impose.  The map $\rho$ is thus a
\emph{quotient map}, and $M$ is a \emph{derived} set: it comes into existence as the image
of $\rho$, rather than being a container that $\rho$ maps $D$ into.

This construction has a philosophical precedent in Leibniz's principle of the identity of
indiscernibles: things that share all the same properties are the same thing.  Here, the
converse is operative, properties that are assigned to the same geometric locus by $\rho$
are identified, and the locus is nothing over and above that identification.  A spacetime
point has no content beyond the property-bundle it collects; it is not an additional
ontological entity.

More concretely: $M$ is characterised as \emph{any} smooth manifold of dimension $n$ whose
topology and differential structure are consistent, a posteriori, with requiring that the
metric constructed in Step~2 is well-defined and smooth.  Among all candidate manifolds
satisfying this consistency condition, the tension structure of $(D, \delta)$ further
constrains which are admissible: a candidate $M$ is admissible if and only if there exists
a partition of $D$ whose induced tension structure $T(\{[a],[b]\}) = T(\{a,b\})$ reproduces
the relational data already present in $(D, \delta)$.  This constraint, not an external
stipulation, selects the spacetime topology and dimension.

Given a choice of admissible $M$, define the \textbf{realization map} $\rho: D \rightarrow M$
as the quotient projection sending each property $a \in D$ to its equivalence class
$\rho(a) = [a]_\rho \in M$.  The nonuniqueness of $\rho$, the freedom to partition $D$
into fibres in multiple ways that are all consistent with the tension structure, is
precisely the diffeomorphism invariance of General Relativity \cite{Einstein1915,MTW1973}.
It emerges here as a consequence of the nonuniqueness of geometric realization from the
pre-geometric substrate, rather than as a separately imposed symmetry.  There is no
canonical labeling of spacetime points: this is not a deficiency of the theory but a
structural result, matching the gauge freedom of GR.

\textit{Step 2, Tension-to-metric map.}  Introduce the \textbf{null threshold}
$\tau_{\mathrm{null}} \in \mathbb{R}^+$ as a second structural input alongside $\tau$.
While $\tau$ is a condition on the global tension $T(C)$, $\tau_{\mathrm{null}}$ is a
condition on pairwise tensions $T(\{a,b\})$: it is the value at which the two-property
tension is null, i.e., at which $g = 0$ by definition.  The two thresholds operate at
different levels and their relationship is an open problem.  The
metric is then:
\begin{equation}
  g_{\mu\nu}\!\left(\rho(a), \rho(b)\right)
  = \eta_{\mu\nu}
    \cdot \operatorname{sgn}\!\left(T(\{a,b\}) - \tau_{\mathrm{null}}\right)
    \cdot \left|T(\{a,b\}) - \tau_{\mathrm{null}}\right|^{1/2}
  \label{eq:metric}
\end{equation}
where $\eta_{\mu\nu}$ is a placeholder for the diagonal sign pattern
$\mathrm{diag}(-1,+1,+1,+1)$ that the construction induces, not a presupposition of
Minkowski geometry.  Its appearance here is definitional: the signature theorem below
shows that the sign pattern is fixed by the tension structure; $\eta_{\mu\nu}$ names
that pattern.  This gives:
\begin{itemize}
  \item $T(\{a,b\}) = \tau_{\mathrm{null}} \;\Rightarrow\; g = 0$: null separation
  \item $T(\{a,b\}) > \tau_{\mathrm{null}} \;\Rightarrow\; g > 0$: spacelike
  \item $T(\{a,b\}) < \tau_{\mathrm{null}} \;\Rightarrow\; g < 0$: timelike
\end{itemize}
The Lorentzian signature is forced as a theorem by hyperbolic
well-posedness of the Goldstone wave equation (Computation~2,
Section~\ref{sec:open-research-work}): the Cauchy problem for
$\Box_g \theta = 0$ admits a well-posed initial-value formulation and a
conserved Noether current only for signature $(1, n-1)$.  The
bifurcation of tension values around $\tau_{\mathrm{null}}$ remains the
algebraic mechanism inside the LG variational structure; what Computation~2
shows is that the sign cut required for that bifurcation is not an
independent axiom but a consequence of the well-posedness condition.

Two clarifications are necessary.  First, equation~\eqref{eq:metric} is strictly a
pairwise assignment: it attaches a signed scalar to each pair of realised points, not a
$(0,2)$-tensor to a tangent space.  The metric tensor field $g_{\mu\nu}(x)$ is
recovered by the continuum limit: take $a$ and $b$ to be nearby properties such that
$\rho(a)$ and $\rho(b)$ approach $x$ along a tangent vector $v^\mu$; then
$g_{\mu\nu}(x)\,v^\mu v^\nu$ arises as the limit of equation~\eqref{eq:metric} along
the chord, analogous to recovering a Riemannian metric as the Hessian of a distance
function.  Second, the magnitude $\bigl|T(\{a,b\})-\tau_{\mathrm{null}}\bigr|^{1/2}$
is the minimal-regularity choice: a linear dependence would give a metric with a
non-smooth gradient at the null cone, while the square root yields H\"{o}lder-$\tfrac{1}{2}$
behaviour there, the appropriate regularity class for a structure that degenerates at
$T = \tau_{\mathrm{null}}$ and which ensures that equation~\eqref{eq:qfluc} blows up
precisely at the modal null threshold rather than at $T = 0$.

\textit{Step 3, Action on morphisms.}  For $f: C \rightarrow C'$ in $\bfS$, $\Phi(f):
(M,g) \rightarrow (M',g')$ is the diffeomorphism induced by $f$ via $\rho$.  Distinction
preserving maps in $\bfS$ correspond precisely to causal structure preserving maps in
$\bfG$, ensuring functoriality.

\textit{The projection operator.}  Define the \textbf{tension projection operator} $\Pi$:
\begin{equation}
  \Pi(T(C))(x) = \int_{\C} T(C') \cdot K(x, C')\, d\mu(C')
  \label{eq:Pi}
\end{equation}
where $K(x, C') = \delta(x - \rho(C'))$ for local configurations.  The role of $\Pi$
here is structurally analogous to the Dirac operator in Connes' noncommutative geometry
\cite{Connes1994}: in both cases a single operator encodes the geometric and algebraic
content of the instantiated domain from an underlying non-spatial structure.  The
stress-energy tensor at $x \in M$ in the isotropic, homogeneous case is:
\begin{equation}
  T^{(\mathrm{iso})}_{\mu\nu}(x) = \Pi(T(C))(x) \cdot g_{\mu\nu}(x)
  \label{eq:Tiso}
\end{equation}
This isotropic scalar form $T^{(\mathrm{iso})}_{\mu\nu} = \Pi(T(C))\,g_{\mu\nu}$ gives a stress-energy
tensor proportional to the metric, which is the correct structure for a cosmological
constant ($T_{\mu\nu} = \Lambda g_{\mu\nu}$) but not for general matter distributions.
Pressurised matter, radiation, and anisotropic fields all have $T_{\mu\nu} \neq \lambda
g_{\mu\nu}$ for any scalar $\lambda$.  The general form requires $\Pi$ to be a
tensor-valued projection $T_{\mu\nu}(x) = \Pi_{\mu\nu}(T(C))(x)$, whose components
encode the directional distribution of tension in the precausal configuration.  The
scalar form is the leading-order approximation valid for isotropic configurations; the
full tensor generalisation is an open problem.
The noninvertibility of $\Phi$ is immediate: $\rho$ discards the internal structure of
$D$ that does not affect pairwise tensions.  Geometry encodes the projection of $T$
through $\Pi$ but not the full precausal configuration that generated it, just as a
shadow records an object's outline but not its internal structure.

\subsection{Uniqueness of $\Phi$ and the origin of diffeomorphism invariance}
The construction proceeds via an explicit choice of realization map $\rho: D \rightarrow M$,
which is not determined by the precausal structure alone.  This raises the question of
how strongly the output functor $\Phi$ depends on the choice of $\rho$, and what that
dependence implies for the physical content of the theory.  The answer is a uniqueness
theorem: $\Phi$ is unique up to diffeomorphism, and the ambiguity in $\Phi$ is completely
and exactly characterised by diffeomorphism invariance, with no residual freedom beyond it.

The proof follows directly from the construction.  Given $\rho$, the metric $g_{\mu\nu}$
at every pair of points is fully determined by equation~\eqref{eq:metric}: there are no
further choices once the realization map is fixed.  Now suppose $\rho' = f \circ \rho$
for some smooth bijection $f: M \rightarrow M'$.  The tension values $T(\{a,b\})$ depend
only on $a, b \in D$, not on their images under $\rho$, so they are unchanged by the
substitution.  Therefore the metric computed at $(\rho'(a), \rho'(b))$ via
equation~\eqref{eq:metric} satisfies $g'(\rho'(a), \rho'(b)) = g(\rho(a), \rho(b))$,
which is exactly the condition that $(M', g')$ is the pushforward of $(M, g)$ along $f$.
Any two choices of $\rho$ therefore produce Lorentzian manifolds that are diffeomorphic
to one another, together with a corresponding natural isomorphism between the two functors
they define.  $\Phi$ is unique up to this natural isomorphism, which at the level of
manifolds is a diffeomorphism.

The physical significance of this result is that diffeomorphism invariance in General
Relativity is not an additional symmetry requirement that must be imposed upon the theory
by hand.  It is the complete characterisation of the non-uniqueness of $\Phi$: the
relabelling freedom in choosing $\rho$ survives into the instantiated manifold as
coordinate freedom, and coordinate freedom is diffeomorphism invariance.  There is no
finer ambiguity and no coarser one; the two are identical.  The natural question
of whether $\Phi$ is unique and what physical symmetries characterise its
ambiguities is therefore answered here: $\Phi$ is unique up to diffeomorphism,
its ambiguity is completely characterised, and the physical symmetry corresponding to
that ambiguity is diffeomorphism invariance.

A note on the metric signature.  The argument in Step~2 assigned the
Minkowski signature $(-,+,+,+)$ through equation~\eqref{eq:metric}, by
bifurcating pairwise tensions about a null threshold $\tau_{\mathrm{null}}$
that separates spacelike from timelike separations.  This determines the
signature \emph{given} $\tau_{\mathrm{null}}$: once a sign cut is supplied,
$T > \tau_{\mathrm{null}}$ is spacelike and $T < \tau_{\mathrm{null}}$ is
timelike, by construction.  It does \emph{not} determine that exactly one
direction lies below $\tau_{\mathrm{null}}$ --- spectral asymmetry of the
tension spectrum about a threshold (the standard form of which is
$T(C) \neq T(\bar C)$, complement-asymmetry) is consistent with zero, one,
or several eigenvalues lying below any given cut, so it does not by itself
fix the count of timelike directions.  Equivalently: in the
multidimensional-scaling reframing of the realisation map
(Section~\ref{sec:open-research-work}, item (1)), any $S_D$-invariant Gram
construction over real features is positive-semidefinite by elementary
linear algebra, so the negative eigenvalue required for one timelike
direction cannot arise from $S_D$-symmetry alone; it must be supplied by an
indefinite bilinear form imposed by hand.  The Lorentzian signature
$(-,+,+,+)$ is fixed as a \emph{theorem} via Computation~2 of
Section~\ref{sec:open-research-work}: hyperbolic well-posedness of the
Goldstone wave equation forces $(p, q) = (1, n-1)$, and the
$\mathrm{U}(1)$ Noether-current conservation that produces the
angular-momentum identification requires this same Lorentzian
signature.  The macroscopic dimension $n = 4$ that combines with the
signature to give the four-dimensional Lorentzian spacetime
$M = \mathbb{R} \times S^3$ follows from KO additivity given the
verified spectral-triple structure (\S\ref{sec:foundational-object}):
the substrate's parity-selected KO-6 (Computation~3) and the product's KO-2
mod 8 (Connes Lorentzian value) together force $M$ at KO-dimension four.
PST is then the theory that derives everything downstream
from $(D, \delta)$ + LG + the realisation map $\rho$,
with both signature and macroscopic dimension outputs rather than inputs.


\subsection{Layman's Summary}

Modal sublimation is the moment the threshold is crossed, not a moment in time, but a logical transition.  When a configuration's tension exceeds $\tau$, it cannot remain in its abstract precausal state; it is compelled, by the same kind of necessity that makes a mathematical theorem true, to become concrete.  The result is a four-dimensional spacetime with the Lorentzian signature that physics recognises.  This section constructs the map from the precausal domain to the geometric domain, shows it is unique, and proves that the coordinate-freedom of General Relativity (diffeomorphism invariance) is a consequence of that uniqueness rather than a separate postulate.

% 5. Coinstantiation
\section{Coinstantiation of Spacetime and Causality}

\subsection{Causality as coinstantiated structure}

The previous section established that modal sublimation is a nontemporal, nonmechanistic
transition: the functor $\Phi: \bfS \rightarrow \bfG$ maps a precausal configuration
directly into a Lorentzian manifold without passing through any intermediate state and
without requiring a prior temporal structure in which the transition could unfold.  A
consequence of this that deserves its own careful treatment is the status of causality
itself.  Causality is not, in PST, a background structure that was already present when
the universe formed and that can then be applied to the event of its formation.  Causality
is \textbf{coinstantiated} with geometry: it arises at the same moment as the Lorentzian
manifold that carries it, because it is constitutively defined in terms of that manifold.
There is no moment, however brief or abstract, at which a causal relation exists but
spacetime does not.  The two are ontologically inseparable.

This point carries weight precisely because it is not obvious.  Much of the philosophical
and cosmological literature treats causality as prior to or more fundamental than spacetime.
In the Kantian tradition, causality is a category of the understanding applied to
experience, logically prior to any empirical fact about the world \cite{Kant1781}.  In
causal set theory, the discrete partial order is the primary structure from which the
continuum metric is to be recovered \cite{Bombelli1987, Henson2006}.  In quantum gravity
approaches that proceed via sum over histories, the causal structure of each history is
taken as given, and the amplitude is computed over those histories \cite{HartleHawking1983}.
Even in ordinary language, the question ``what caused $X$?'' is considered answerable in
principle for any event $X$, the presumption being that every event has a cause and that
the causal relation is always available to be invoked.

PST disagrees at the foundational level.  The precausal substrate, as defined in
Section~2, does not possess a causal order.  It possesses property differentiation and
the tension functional.  These are not impoverished substitutes for causality; they are
genuinely different kinds of structure.  The distinction relation $\delta$ is symmetric
and irreflexive: it registers that two properties differ from each other without
specifying which is prior to the other or whether either depends on the other.  The
tension functional $T$ assigns a magnitude to a configuration without implying that one
configuration succeeds or precedes another.  Neither $\delta$ nor $T$ contains the
ingredients from which a temporal order could be constructed.  A temporal order requires
at minimum an asymmetric, transitive relation on a set of events.  The precausal
substrate has none.  Causality is not absent from the substrate in the way that a feature
might be absent from a description that was meant to include it; it is absent from the
substrate in the sense that the substrate belongs to a logical type to which causal
relations do not apply.

\subsection{Mathematical formulation of coinstantiation}
Let $(M, g) = \Phi(C)$ be the Lorentzian manifold produced by modal sublimation of a
configuration $C \in \bfS$.  The causal structure on $(M, g)$ is the partial order
$\leq$ defined pointwise by the lightcone structure of $g$:
\begin{equation}
  p \leq q \;\iff\; q \in J^+(p, g)
  \label{eq:causal-order}
\end{equation}
where $J^+(p, g)$ is the causal future of $p$ under the metric $g$:
\begin{equation}
  J^+(p, g) = \{\, q \in M \;\mid\;
    \exists \text{ future-directed causal curve } \gamma:[0,1]\rightarrow M,\;
    \gamma(0)=p,\; \gamma(1)=q \,\}
\end{equation}
A curve is causal under $g$ if and only if its tangent vector $\dot{\gamma}$ is
everywhere non-spacelike, that is $g(\dot{\gamma}, \dot{\gamma}) \leq 0$ at every
point.  This definition depends on $g$ in an essential way: the notion of a causal
curve, and hence of causal precedence, is meaningless without a metric.  Change the
metric and the causal structure changes with it; remove the metric and the causal
structure does not become unspecified but ceases to exist as a category of object.

The partial order $\leq$ in equation~\eqref{eq:causal-order} has no pre-image in the
precausal category $\bfS$.  Formally, there is no morphism or relation in $\bfS$ that
maps to $\leq$ under $\Phi$, because the objects of $\bfS$ are configurations equipped
only with $\delta$ and $T$.  This is not a deficiency of the model; it is a structural
theorem: the functor $\Phi$ is not surjective on structure but on objects.  It produces a
manifold but does not carry into that manifold a pre-existing causal order; the causal
order is generated anew by the Lorentzian signature of the metric produced by
equation~\eqref{eq:metric}.  Therefore the following holds:
\begin{equation}
  \text{causality} \;=\; f(g) \;=\; f(\Phi(C))
  \label{eq:causality}
\end{equation}
Causality is a derived relation, defined in terms of the metric, which is itself defined
in terms of the precausal tension structure via $\Phi$.  It is not an independent
primitive.

\subsection{The direction of time as a structural consequence}
The Lorentzian signature of the metric $(-, +, +, +)$ introduces an asymmetry that has no
parallel in a purely Riemannian geometry.  In a Riemannian manifold, all directions from a
point are spacelike and the notion of a preferred future does not arise.  In a Lorentzian
manifold, the lightcone at each point divides the tangent space into two timelike halves, one
conventionally designated as the future and the other as the past, and a continuous causal
vector field on $M$ selects one half globally, providing a \textbf{time orientation}
\cite{Geroch1970, Wald1984}.

In PST this asymmetry is not imposed as an additional datum.  It is inherited from the
asymmetry of the precausal substrate.  Asymmetric tension, by definition, is a
non-symmetric functional: $T(C) \neq T(\bar{C})$ for all configurations $C$, where
$\bar{C}$ is the complement of $C$ in $D$.  This asymmetry is preserved across the
functor $\Phi$ as the Lorentzian signature, and the preferred time direction of the
manifold reflects the direction of the residual tension gradient of the precausal
configuration.  The thermodynamic arrow of time, the cosmological arrow of time, and the
distinction between retarded and advanced solutions in electrodynamics are all grounded
in this single asymmetry inherited from the substrate.

A technical caveat is required.  The complement-asymmetry $T(C) \neq T(\bar{C})$ is a
pairwise condition; a global time orientation on $M$ requires a continuous nowhere-vanishing
timelike vector field on the entire manifold.  By Geroch's theorem \cite{Geroch1970},
a Lorentzian manifold is time-orientable if and only if such a field exists, which is a
global topological condition not automatic from local asymmetry.  In PST the global
character of the precausal configuration $C$ provides the resource: the tension gradient
$\nabla_{\mathcal{C}} T$ evaluated in the preferred direction identified in Section~4
gives a precausal structure that $\Phi$ maps to a candidate timelike field on $M$.
That this field is continuous and nowhere-vanishing, i.e.\ that global time-orientability
follows from the precausal structure without additional assumption, is an open problem.

\subsection{The causal order and the degenerate vacuum}
A further consequence of coinstantiation concerns the vacuum manifold.  As established in
Section~3, the post-threshold vacuum $\mathcal{V} \cong S^1$ is a continuum of
configurations that minimize the modal potential at equal tension.  Motion along
$\mathcal{V}$ is the only stable state the vacuum admits.  In the instantiated geometry,
this motion projects under $\Pi$ into persistent non-vanishing stress-energy, which in
turn sustains curvature.  Crucially, the causal order $\leq$ treats this motion as part
of the background structure: the lightcone structure generated by the projected metric
encodes the vacuum motion as a baseline curvature rather than as a deviation from
flatness.  This is why the structural argument holds that orbital motion at all scales does not
require an initial condition that specifies it separately: it is a candidate for the
ground state of the causal structure.  The explicit $\Pi$-mapping connecting this
Noether charge to the observed orbital angular momentum of massive bodies is computed in
Section~\ref{sec:angmom-projection}.

\subsection{Category error and the dissolution of first-cause questions}
With coinstantiation established, we are in a position to address what is perhaps the
most persistent question in the philosophy of cosmology: what caused the universe to come
into being?  The question appears legitimate.  It has the grammatical form of a causal
question, it points to a definite event (the coming-into-being of the universe), and it
invites an answer of the form ``$X$ caused the universe''.

The question is not, however, answerable.  It is not unanswerable in the sense that the
answer is hidden from us or that the evidence has been destroyed; it is unanswerable
because it commits a \textbf{category error}, applying a relation to an entity that
belongs to a logical type to which the relation does not apply
\cite{Ryle1949,Putnam1981}.  To ask what caused the universe is to presuppose that the
causal relation exists and is applicable to the event of the universe's formation.  But
the causal relation, as shown above, is coinstantiated with the universe: it comes into
being with the very event whose cause is being sought.  A relation cannot serve as the
explanation for the event that constitutes its own first instantiation.

The parallel with simpler category errors is instructive.  ``What is north of the North
Pole?'' presupposes that the directional relation north-of is defined at the North Pole,
but the relation is defined only within the coordinate system that has the North Pole as
its boundary.  ``What was there before the Big Bang?'' presupposes that the relation
earlier-than is defined at the Big Bang, but that relation depends on the spacetime whose
earliest moment is the Big Bang.  ``What caused the universe?'' presupposes that the
causal relation is defined for the event of universal instantiation, but that relation
depends on the very spacetime that is being instantiated.  In each case the error is
structural: the question applies a relation in a domain where that relation does not
exist.

The dissolution of the category error is not a retreat.  PST does not avoid the question
by refusing to engage; it dissolves it by exhibiting the precise logical structure that
makes it malformed.  This dissolution is, moreover, constructive: in place of the
unanswerable question, PST offers the answerable one.  Why does the universe exist at all?
Because the precausal substrate possesses property differentiation, and property
differentiation entails asymmetric tension, and once tension exceeds the modal threshold,
modal sublimation to instantiated geometry is logically necessary, not contingently
triggered.  The question Leibniz posed, ``why is there something rather than nothing?''
\cite{Leibniz1714}, receives not an arbitrary answer but a principled one: because the
logical possibility of property differentiation is irreducible, and a substrate bearing
irreducible property differentiation cannot remain uninstantiated once the threshold
condition is satisfied.  Existence is not a brute fact; it is the necessary expression of
a logically unavoidable modal imbalance.

\subsection{Contrast with competing approaches}
Several approaches in the literature address the question of cosmological origins in ways
that are instructive to compare with PST.

The \textbf{Hartle-Hawking no-boundary proposal} \cite{HartleHawking1983} avoids a
boundary in time by making the metric Euclidean in the deep past, eliminating the initial
singularity by changing the signature.  This is technically elegant but it does not
dissolve the category error: the sum over geometries is still computed within a
pre-existing mathematical framework equipped with a notion of quantum amplitude, and the
question of why that framework is instantiated rather than another is not addressed.  The
no-boundary proposal also treats causality as part of the background formalism rather than
as a derived structure.

\textbf{Vilenkin's tunneling proposal} \cite{Vilenkin1982} posits a quantum tunneling
event from nothing.  But ``nothing'' in this context is not the absence of structure; it
is a specific quantum gravitational state with a precisely defined wave function.  The
transition from that state presupposes both quantum mechanics and the causal structure
within which the transition is defined.  The proposal is internally consistent but does
not escape the regress: it pushes the question back to the prior state and the
framework defining the transition.

\textbf{Causal set theory} \cite{Bombelli1987} makes the causal partial order primitive
and attempts to derive the continuum from it.  This is in some respects the inverse of
PST: where causal set theory starts with $\leq$ and tries to recover $g$, PST derives
both $g$ and $\leq$ from $\Phi$.  Causal set theory faces the question of what determines
the particular causal set that becomes our universe, a question it answers by
probabilistic sampling over a measure on causal sets.  PST does not face this question
in the same form: the particular geometry that arises is determined by the tension
configuration $C$, which is itself subject to the modal potential.

The position of PST is that all approaches that take causality as a starting point, whether
as a background relation, a quantum mechanical framework, or a primitive partial order,
are working within the instantiated domain and can at best describe the structure of what
has been instantiated.  The question of why anything is instantiated at all requires a
theory that can speak from outside the instantiated domain.  The precausal substrate, and
the functor $\Phi$ through which it gives rise to geometry, is that theory.


\subsection{Layman's Summary}

Spacetime and the causal order (the fact that causes precede effects) do not arise one after the other.  They are two aspects of a single transition and cannot exist independently of each other.  This has a profound consequence: there is no ``before'' the universe, because time itself is part of what the transition produces.  Asking what caused the universe is a category error, like asking what is north of the North Pole.  The Big Bang is not a beginning at a point in time; it is what the transition looks like from inside the geometry it creates.

% 6. The Emergence Chain
\section{The Emergence Chain}
\label{sec:emergence-chain}

\subsection{Overview and the eight-step summary table}

The preceding four sections have built the foundations of PST step by step: from the
single primitive of property differentiation, through asymmetric tension and the modal
threshold, to modal sublimation and the coinstantiation of geometry and causality.  This
section draws those steps together into a single, unified chain of logical entailment and
examines what that chain is, what kind of necessity binds each step to the next, what
distinguishes it from competing accounts of emergence in physics, and what it means for
the chain to be complete.

The table below summarises the eight steps that constitute the chain.  Each entry names
the emergent concept, gives the mathematical object that represents it, and implies a
link to its predecessor.  The prose that follows unpacks each link in turn.

\vspace{0.5em}
\begin{center}
\begin{tabular}{clp{8.5cm}}
  \toprule
  Step & Concept & Mathematical object \\
  \midrule
  1 & Property differentiation
    & $(D, \delta)$, a set with a symmetric, irreflexive distinction relation \\[4pt]
  2 & Asymmetric tension
    & $T: \C \rightarrow \mathbb{R}^+$,\; $T(C) \neq T(\bar{C})$ for all $C$ \\[4pt]
  3 & Modal threshold
    & Bifurcation point $\tau$ of $\F(\psi, C)$, eq.~\eqref{eq:tau} \\[4pt]
  4 & Modal sublimation
    & Functor $\Phi: \bfS \rightarrow \bfG$, constructed via $\rho$
      and eq.~\eqref{eq:metric} \\[4pt]
  5 & Geometry as spacetime
    & Lorentzian manifold $(M, g)$ with signature supplied by $\tau_{\mathrm{null}}$ \\[4pt]
  6 & Causality
    & Partial order $\leq$ on $M$ via $J^+(p,g)$; no pre-image in $\bfS$ \\[4pt]
  7 & Energy, matter, fields
    & $T_{\mu\nu} = \Pi(T(C))$;\;
      $G_{\mu\nu} = 8\pi G \cdot \Pi(T(C))$;\; bosonic field sector;
      fermionic content with structural spin--statistics from the CAR algebra on
      the Boolean distinction primitive (Section~\ref{sec:gauge-sm}) \\[4pt]
  8 & Angular momentum
    & Noether charge $L = 2r_0^2\,\dot{\theta}$ of the $\mathrm{U}(1)$ symmetry
      of the vacuum manifold $\mathcal{V} \cong S^1$ \\
  \bottomrule
\end{tabular}
\end{center}
\vspace{0.5em}

\subsection{The character of the chain}
The first thing to establish is what kind of chain this is.  It is not a causal chain.
The steps do not follow each other in time; they are not events that happen sequentially
in a universe that already exists.  It is not an evolutionary chain in the sense used in
cosmology, where one physical state develops into another under the action of laws that
are themselves already given.  The chain is a chain of \textbf{logical entailment}: each
concept is not merely correlated with its successor but is the necessary and sufficient
condition for it.  Given step $n$, step $n+1$ is not optional; it is logically
unavoidable.  Remove any link and the entire structure downstream collapses.

This is a stronger claim than is usually made in emergence theories.  In most accounts,
emergence is a relation between levels of description: a higher-level description emerges
from a lower-level one when the higher-level regularities can, in principle, be derived
from the lower-level equations.  The chain has a direction but not a necessity.  In PST
the chain has both.  The direction runs from the logically irreducible toward the
structurally complex, and at every step the direction is the only one available.  The
question ``why does tension arise from differentiation?'' does not have an empirical
answer, the sort of answer one would give by pointing to a law of nature or a measured
constant.  It has a logical answer: any configuration of two or more distinct properties
contains non-identity internal relations, and those relations cannot collectively
neutralise.  Tension is not observed to follow from differentiation; it is entailed by it, together
with the complement-asymmetry condition.

A precise statement of what the chain entails requires distinguishing two levels.  At
the \emph{structural level}, the steps are necessary given the primitive and the
structural commitments of PST: tension follows from distinctness together with the
complement-asymmetry condition; the threshold follows from the existence of tension
together with the commitment to a bifurcation structure; modal sublimation follows from
the threshold by the instability argument; spacetime and causality coinstantiate via the
functor $\Phi$.  At the \emph{architectural level}, the specific mathematical objects
representing each step, the Landau-Ginzburg functional, the gradient flow, the canonical
measure, are the minimal motivated choices consistent with the structural requirements;
they are not strictly entailed by $(D, \delta)$ alone, as the epistemological caveat of
Section~2 makes explicit.  Steps 3 through 8 are necessary given the structural
encoding, which is itself logically motivated rather than derived.  The chain is closed
at the structural level; it is architecturally motivated at the representational level.
This two-tier character is not a weakness; it identifies precisely where the foundational
programme remains open and where future derivations could replace motivated choices with
strict entailments.

\subsection{Step 1 to Step 2: from differentiation to tension}
Property differentiation is the capacity for distinctions to exist: the set $D$ equipped
with the distinction relation $\delta$, symmetric and irreflexive, requiring no loci,
relata, or temporal structure.  This is the single primitive from which the entire theory
is built.  Its logical irreducibility was established in Section~2: any attempt to explain
why distinctions can exist already presupposes a context in which two things differ from
each other, committing a circularity.  The primitive cannot be grounded further without
circularity; it can only be identified.

Asymmetric tension follows from differentiation together with the complement-asymmetry
condition.  A configuration $C \subset D$ drawn from a set of distinct properties admits
internal non-identity relations.  Since $\delta$ is symmetric, $\delta(a,b)$ and
$\delta(b,a)$ hold simultaneously; these relations carry no preferred direction.  The
asymmetry in asymmetric tension is not directional but complementary: the structural
condition that no configuration and its complement carry equal tension,
$T(C) \neq T(\bar{C})$.  This condition is the precise additional commitment that
converts step 1 into step 2.  It cannot be derived from $(D, \delta)$ alone; it is the
assertion that the precausal substrate is generically asymmetric, and it functions as the
bridge between the bare existence of distinctions and their structural expression as
tension.  Granting it, the move from step 1 to step 2 is structurally closed: step 2 is
step 1 made structural, with the complement-asymmetry condition as the content of that
structuring move.  This is the foundational hinge of the chain.

\subsection{Step 2 to Step 3: from tension to threshold}
Asymmetric tension is present in every configuration of more than one distinct property.
That it exists is already established by step 2.  What step 3 adds is the recognition
that tension admits a \textit{critical value}, a point at which the structural
consequences of tension change qualitatively rather than quantitatively.  Below this
critical value, the configuration can sustain a unique, stable modal position at $\psi =
0$: uninstantiated, without geometry, without causality.  Above it, the configuration
cannot.  The threshold $\tau$ is not introduced as an external parameter but is derived
as the infimum of the set of tension values at which the second variation of the modal
potential functional $\F(\psi, C)$ vanishes at $\psi = 0$, equation~\eqref{eq:tau}.

The necessity of step 3 is that there must be such a threshold.  The modal potential
functional contains a quadratic term whose coefficient is proportional to $T(C) - \tau$,
and a quartic term that bounds the functional below.  The qualitative bifurcation between
the single-minimum and double-minimum regimes is a structural consequence of the
polynomial structure of $\F$, which in turn reflects the two competing tendencies present
in any configuration: the tendency toward modal coherence (captured by the quadratic
restoring term) and the tendency toward structural saturation (captured by the quartic
term).  These two tendencies are not postulated independently; they are the formal
expression of the two irreducible aspects of any tension-carrying configuration.  A
threshold must exist because the quadratic and quartic terms cannot produce a bifurcation
without a critical value at which the quadratic coefficient changes sign.  That critical
value is $\tau$.

\subsection{Step 3 to Step 4: from threshold to sublimation}
The threshold establishes a condition; modal sublimation is what happens when that
condition is met.  Once $T(C) > \tau$, the configuration at $\psi = 0$ is a local
maximum, not a minimum.  The configuration cannot remain there.  This is not a
dynamical statement; there is no time over which the configuration moves.  It is a
logical statement: a configuration whose tension exceeds the threshold and whose order
parameter is at the unstable critical point is internally incoherent.  Its degree of
instantiation is inconsistent with its tension.  That incoherence is resolved not by a
process but by a logical transition to the nearest coherent state.

That transition is the functor $\Phi: \bfS \rightarrow \bfG$, the map from the precausal
category to the geometric category.  The functor is not postulated; its existence follows
from the threshold condition together with the definition of the two categories.  The
precausal category $\bfS$ has as objects configurations with tension above $\tau$; the
geometric category $\bfG$ has as objects Lorentzian manifolds.  The functor $\Phi$ assigns
to each configuration its geometrically realised image, constructed explicitly via the
realization map $\rho$ and the tension-to-metric assignment of equation~\eqref{eq:metric}.
The move from step 3 to step 4 is thus the move from a necessary condition to its
necessary consequence: if the threshold is the condition under which uninstantiated
existence becomes logically incoherent, then modal sublimation is the resolution of
that incoherence.  Step 4 is not something that happens after step 3; it is what step 3
is, from the perspective of its consequence.

\subsection{Step 4 to Steps 5 and 6: from sublimation to spacetime and causality}
The image of $\Phi$ is a Lorentzian manifold $(M, g)$.  The Lorentzian signature is
not assumed but derived: it follows from the bifurcation of pairwise tension values
around the null threshold $\tau_{\mathrm{null}}$, as shown in equation~\eqref{eq:metric}.
Pairs with tension above $\tau_{\mathrm{null}}$ map to spacelike separations; pairs
below to timelike separations; pairs at exactly $\tau_{\mathrm{null}}$ to null
separations.  The $(-, +, +, +)$ signature of spacetime is therefore a direct expression
of the way tension distributes across pairwise configurations in the precausal domain.

Causality arises simultaneously with the manifold.  The causal order $\leq$ defined by
$J^+(p, g)$ is constitutively tied to $g$: it cannot exist before or independently of
the metric.  Steps 5 and 6 are therefore not two sequential events but two aspects of
a single transition.  The reason they are listed separately in the table is conceptual,
not temporal: each warrants individual attention because each dissolves a traditional
presupposition.  Step 5 dissolves the presupposition that geometry is given; step 6
dissolves the presupposition that causality is given.  Both are derived, both from
the same functor, and both are coinstantiated.

\subsection{Steps 6 and 7: from causality and geometry to matter}
Matter, energy, and fields are sometimes treated in foundational physics as the
\textit{content} of spacetime, distinct from the geometric container.  PST dissolves
this distinction.  The stress-energy tensor $T_{\mu\nu}$ at any point $x \in M$ is the
projection of the precausal tension functional through the operator $\Pi$, as in
equation~\eqref{eq:Tmunu}.  It is not a separate input into the Einstein field equations;
it is the same tension structure that produced the geometry, now expressed in geometric
units via the projection.  The field equations $G_{\mu\nu} = 8\pi G \cdot \Pi(T(C))$
are a conservation statement: they express the fact that the total tension content of
the precausal configuration is faithfully preserved in the instantiated manifold,
distributed as both curvature and stress-energy.  Gravity and matter are not two things;
they are two aspects of one projection.

The factor $8\pi G$ is not a fundamental constant of the precausal domain.  It is a unit
conversion factor introduced by $\Pi$, translating between dimensionless modal tension
and physical stress-energy units.  Its numerical value is a property of the projection
map, not of the substrate.  This reframes the longstanding puzzle of the gravitational
constant from a question about nature to a question about the structure of the projection
operator, a question that in principle has a derivable answer once the full form of $\Pi$
is specified.

``Matter'' at Step~7 has two parts.  Stress-energy and the bosonic field sector are
derived in Section~7 from the order parameter $\psi$.  The fermionic sector and the
spin--statistics connection are derived separately in
Section~\ref{sec:gauge-sm} from the CAR algebra on the substrate's Boolean
distinction primitive: the fermionic excitations inherit the correct
anticommutation relations as a structural consequence of the Boolean
$\{0,1\}$ structure, not as a separate postulate.  The chain's claim at this step is
therefore that the full matter content --- stress-energy, bosonic fields, and
fermionic content with structural spin--statistics --- is derivable.

\subsection{Step 8: from geometry and the vacuum to angular momentum}
The final step in the chain is the one most often treated as an independent fact of
nature: that massive systems orbit, that angular momentum is conserved, and that stable
circular motion is ubiquitous from electrons to galaxies.  In PST all of this follows
from a single structural fact about the instantiated vacuum.

Once the order parameter is promoted to a complex field $\psi \in \mathbb{C}$, the
vacuum manifold of the post-threshold system is $\mathcal{V} \cong S^1$, a
one-dimensional circle of degenerate minima.  The modal potential functional is
invariant under global phase rotations $\psi \mapsto e^{i\alpha}\psi$: a continuous
$\mathrm{U}(1)$ symmetry.  By Noether's theorem, this symmetry implies a conserved
current and a conserved charge.  That charge is angular momentum in configuration space,
$L = 2r_0^2\,\dot{\theta}$, and it projects via $\Pi$ into the physical angular momentum
of any instantiated system.  The stable circular orbits of planets, the spin of black
holes, the angular momentum of electrons in atomic orbitals, all reduce to the single
fact that the instantiated vacuum is a circle and that motion along a circle is the only
available ground state.  Step 8 is therefore not an empirical input to PST; it is a
derivation from the symmetry of the vacuum that itself follows from steps 1 through 4.

\subsection{The chain as a closed logical structure}
What is remarkable about the chain is not that it is long but that it is closed: no step
requires a premise from outside the chain.  Step 1 is the single primitive.  Steps 2
through 8 follow from it by structural derivation, each one requiring only the concepts
and structural commitments already established.  No constants are introduced except as
projective unit conversions.  No initial conditions are imposed except as consequences of
the vacuum topology.  The chain rests on three independent postulates plus one derived
structural statement (Section~\ref{sec:postulates}); the steps between them are
structural derivations.
Where the derivation relies on architectural choices, in particular the Landau-Ginzburg
functional form and the gradient regularisation, these are the minimal choices
consistent with the structural requirements; they are logically motivated rather than
strictly entailed from $(D, \delta)$ alone, and this distinction is maintained
explicitly throughout the paper.

This closure has a significant implication for the question of why the chain terminates
at step 8 rather than continuing indefinitely.  The chain does not terminate arbitrarily.
Step 8 (angular momentum as Noether charge) is the last step for which the derivation is
purely structural.  Beyond step 8, physics enters the domain of the specific: particular
particle masses, coupling constants, and quantum numbers that depend on the detailed
content of the precausal configuration $C$ rather than on the structural form of the
chain.  PST does not claim to derive the Standard Model of particle physics from first
principles, any more than topology claims to derive the specific dimensions of any
particular manifold.  What PST claims is that the framework within which all such
specific facts are intelligible, spacetime, causality, matter, and angular momentum, is
not a brute given but a necessary structural consequence of a single irreducible
primitive.

Other theories of physics typically either take this framework for granted or introduce
it as an axiom without derivation.  General Relativity assumes a Lorentzian manifold and
derives its dynamics.  Quantum mechanics assumes a Hilbert space, a Hamiltonian, and a
measurement postulate, then derives its predictions.  String theory assumes extra
spatial dimensions and the string action.  Loop quantum gravity assumes a discrete
quantum geometry.  Each of these frameworks contains ungrounded primitives: structures
that are posited without explanation.  The present chain shows that if one is willing to
start from a single primitive that is itself logically irreducible, namely the capacity
for distinctions to exist, all the background structure that modern physics takes as
given follows as a necessary consequence.  The chain is, in this sense, the logical
spine of PST: the sequence of entailments that connects the most minimal possible
ontology to the full landscape of physical structure.

From the perspective of the precausal substrate, there is one structure, the tension
functional $T(C)$, projected through one operator $\Pi$ into one manifold $(M, g)$.
Quantum behaviour and classical geometry are different aspects of the same projection,
distinguished only by proximity to the modal threshold: near the threshold, the order
parameter is small and the geometry is high-dimensional and rapidly varying, the quantum
regime; far from the threshold, the order parameter is large and the geometry has settled
into the four stable dimensions of the instantiated vacuum, the classical regime.  The
apparent gap between quantum mechanics and general relativity is, from this vantage point,
not a gap between two fundamental theories but a gap between two limiting descriptions of
a single underlying structure.


\subsection{Layman's Summary}

This section assembles all the preceding steps into a single logical chain of eight links: property differentiation, asymmetric tension, the modal threshold, modal sublimation, spacetime, causality, matter and energy, and finally angular momentum.  The chain rests on three independent postulates (P1--P3: the distinction primitive, asymmetric tension, and the Landau--Ginzburg modal potential) together with the Laplace-type projection (P4), which is derived from P1--P3 via Mosco convergence (Section~\ref{sec:seeley-dewitt}).  The steps between those postulates are structural derivations: given each postulate, the next element of the chain follows necessarily.  What is remarkable is not the length of the chain but the precision of its logical structure, the entire background of modern physics traced back to three minimal commitments, with the fourth following as a structural consequence.

% 7. Energy, Matter, and Fields
\section{Energy, Matter, and Fields}

\subsection{The source problem in General Relativity}

Einstein's field equations cast the relationship between matter and geometry in the form
of a \textit{source equation}: the Einstein tensor $G_{\mu\nu}$, which encodes the
curvature of spacetime, is set equal to $8\pi G$ times the stress-energy tensor
$T_{\mu\nu}$, which encodes the distribution of matter and energy
\cite{Einstein1915,MTW1973}.  This formulation is extraordinarily successful as a
description of the instantiated world.  As an ontological account it is, however,
incomplete in a specific and well-known way: it treats matter and geometry as two
distinct kinds of thing, one of which acts as the source of the other.  The left-hand
side of the field equations, geometry, is described as marble, smooth and exact.  The
right-hand side, matter, is described as wood, coarse and phenomenological.  Einstein
himself remarked on this asymmetry as a deficiency, expressing the aspiration that a
complete theory would derive the right-hand side from geometry rather than inserting it
from outside.  No such derivation was found within the framework of General Relativity,
because within that framework, matter and geometry are genuinely independent inputs:
the metric describes the arena; matter describes what moves through it.

PST dissolves this distinction at the foundational level.  Both geometry and matter
derive from the same precausal source, the asymmetric tension functional $T(C)$,
preserved across the phase transition $\Phi$ and projected by the tension projection
operator $\Pi$.  The geometry of the instantiated manifold is the structural expression
of $T(C)$ in the domain of spatial relations; the matter content is the energetic
expression of the same $T(C)$ in the domain of physical densities.  They are not two
inputs; they are two outputs of a single projection.  The marble and the wood are cut
from the same precausal stone.

\subsection{The stress-energy tensor as projected tension}
In the isotropic, homogeneous approximation the precise statement is:
\begin{equation}
  T^{(\mathrm{iso})}_{\mu\nu}(x) = \Pi(T(C))(x) \cdot g_{\mu\nu}(x)
  \label{eq:Tmunu}
\end{equation}
where $\Pi(T(C))(x)$ is the projection of the precausal tension functional onto the
point $x \in M$, as defined in equation~\eqref{eq:Pi}.  This equation should be read
carefully.  It does not say that matter is made of geometry, nor that geometry is made
of matter.  It says that both are made of something prior to either: the asymmetric
tension of the precausal configuration, which the functor $\Phi$ converts into a
Lorentzian manifold and which the operator $\Pi$ distributes as a scalar density across
that manifold.  The metric $g_{\mu\nu}$ provides the geometric structure; $\Pi(T(C))$
provides the scalar weight.  The isotropic stress-energy tensor is the product of these two, a
density on the manifold whose value at each point reflects how much tension content the
precausal configuration contributed to that region of the instantiated geometry.

The components of $T_{\mu\nu}$ acquire their standard interpretations naturally in
this picture.  The energy density $T_{00}$ is the local projection of the tension
magnitude: regions where $T(C)$ was large before sublimation appear in the instantiated
manifold as regions of high energy density.  The momentum flux $T_{0i}$ reflects the
directional distribution of tension across configurations that realise nearby points.
The pressure components $T_{ij}$ encode the isotropic or anisotropic character of the
tension distribution.  None of these quantities is postulated; all are derived from the
single precausal input $T(C)$ through the action of $\Pi$.

\subsection{The field equations as a conservation statement}
Einstein's field equations then read:
\begin{equation}
  G_{\mu\nu} = 8\pi G \cdot \Pi(T(C))
  \label{eq:einstein}
\end{equation}
The standard reading of this equation is dynamical: matter curves spacetime, and the
degree of curvature is set by the amount of matter.  The PST reading is different and
more fundamental: the field equations are a \textbf{conservation statement} expressing
that the total asymmetric tension of the precausal configuration is faithfully preserved
across the sublimation threshold, distributed simultaneously as curvature on the
left-hand side and as stress-energy on the right.  The equations do not govern a process;
they describe what the functor $\Phi$ conserves.  Curvature and stress-energy are not
cause and effect; they are two names for the same preserved quantity, registered by two
different instruments, the geometric instrument on the left and the energetic instrument
on the right.

This reinterpretation connects with, and goes beyond, the thermodynamic approach of
Jacobson \cite{Jacobson1995} and the entropic gravity programme of Verlinde
\cite{Verlinde2011}.  Both of these frameworks derive the field equations from
non-geometric principles, treating them as equations of state rather than as fundamental
laws.  PST agrees with this spirit but locates the non-geometric substrate at a more
foundational level: not in thermodynamic degrees of freedom within the instantiated
manifold, but in the precausal tension structure that precedes the manifold itself.
Where Jacobson derives Einstein's equations from entropy on holographic screens, PST
derives them from the structure of $\Phi$ and $\Pi$.  The two derivations converge on
the same equations from different directions, suggesting that the equations themselves
are more fundamental than either of their known derivations individually.

\subsection{The F--S dichotomy}
\label{par:fs-dichotomy}

PST carries two distinct functionals that serve different mathematical roles, and a
reader familiar with field theory should be warned at the outset that they are not
interchangeable.

$\mathcal{F}[\psi, C]$ is the \textbf{modal free energy}: a Landau--Ginzburg potential
defined over precausal configuration space.  It is Euclidean in character: positive-definite
in the equilibrium sector, with no temporal argument.  Its stationary condition
$\delta\mathcal{F}/\delta\psi = 0$ identifies equilibrium configurations; it governs the
threshold, the vacuum manifold $\mathcal{V}$, and which mode is coherent.  No time
appears; no Noether theorem applies.  $\mathcal{F}$ is not an action and must not be
treated as one.

$S[g, \phi]$ is the \textbf{instantiated action}: a standard Lorentzian functional
defined over spacetime histories in the instantiated domain $\mathcal{G}$.  Its
stationary points $\delta S / \delta g^{\mu\nu} = 0$, $\delta S/\delta\phi = 0$
are equations of motion.  Noether's theorem, conservation laws, and all dynamical
structure belong here.  $S$ is a \emph{projected} object: it does not exist in the
precausal substrate; it is what $\mathcal{F}$ becomes under the action of $\Pi$, with
the Bernoulli measure $d\mu$ mapping to the covariant measure $\sqrt{-g}\,d^4x$ and
the gradient term $c|\nabla_{\mathcal{C}}\psi|^2$ projecting to the kinetic sector of
$\mathcal{L}$.

The relationship between $\mathcal{F}$ and $S$ is the PST instance of the standard
Euclidean-to-Lorentzian correspondence.  A Landau--Ginzburg free energy is formally a
Euclidean action: it is minimised rather than made stationary, integrated against a
positive-definite measure, with no timelike direction.  The standard route from Euclidean
to Lorentzian is Wick rotation: $t \to -i\tau$ reverses the sign of the time-kinetic
term and changes the signature from $(+,+,+,+)$ to $(-,+,+,+)$.  In PST, sublimation
\emph{is} the Wick rotation.  The moment the $(-,+,+,+)$ Lorentzian signature emerges
from the functor $\Phi$ is precisely the moment $\mathcal{F}$ ceases to be the relevant
functional and $S$ takes over.  This is not an embarrassing gap in the formalism.  It is
a structural prediction: the Lagrangian cannot exist before the threshold because the
time one would integrate it over does not yet exist.  That $S$ has no precausal
counterpart is a theorem of the framework, not a shortcoming.

\textit{F--S Dichotomy.}  $\mathcal{F}$ governs the precausal domain: equilibrium,
threshold, and vacuum structure, with no dynamics and no Noether charges.  $S$ governs
the instantiated domain: equations of motion, conservation laws, and all of quantum field
theory.  The passage from $\mathcal{F}$ to $S$ is sublimation; it is not reversible;
and nothing on the $S$-side can be traced back to the precausal level without crossing
the threshold.  Every subsequent use of Noether's theorem, every conserved charge, and
every field-theoretic result in this paper belongs exclusively to $S$.

\subsection{The projected action and the emergence of the Einstein--Hilbert term}

\emph{Newton's constant from the spectral action.}  Newton's constant is the
gravitational coefficient of the spectral action $\mathrm{Tr}\,f(D/\Lambda)$~\cite{ChamseddineConnes1997}.
Its heat-kernel asymptotics contribute a cosmological term at order
$\Lambda^4$, the Einstein--Hilbert term $\tfrac{1}{16\pi G}\int R\sqrt{-g}$ at
order $\Lambda^2$, and the Yang--Mills and Higgs terms at order $\Lambda^0$.
The Seeley--DeWitt heat-kernel computation carried out in this subsection and
the next is exactly that $\Lambda^2$ coefficient: $1/G \sim f_2\,\Lambda^2\,
\mathcal{N}$, with $\mathcal{N}$ the trace over the finite internal factor and
the cutoff $\Lambda \sim 1/d_0$ set by the Landau--Ginzburg coherence scale.
The scales $r_0$ and $d_0$ are the Landau--Ginzburg vacuum scales of the
order parameter, and $G$ emerges as a derived consistency relation among the
observables $(m_h, M_P, v)$ rather than a first-principles output.

The field equations~\eqref{eq:einstein} were presented above as a conservation statement, but their specific tensor structure: why the left-hand side is $G_{\mu\nu}$ and not $R_{\mu\nu}$, or $R\,g_{\mu\nu}$, or some other curvature combination: requires a derivation that has so far been deferred.  That derivation begins with the projected action.

The potential functional $\mathcal{F}(\psi, C)$ of equation~\eqref{eq:Fcomplex} contains three terms.  The first two, $a|\psi|^2$ and $b|\psi|^4$, determine the equilibrium structure of the order parameter.  The third, $c\,|\nabla_{\mathcal{C}}\psi|^2$, is the gradient term: it penalises variation of the order parameter across configuration space and controls the energetic cost of excitations above the vacuum.  Under the projection $\Phi$, this term acquires a precise geometric interpretation.

When the order parameter has settled to its vacuum value $\psi = r_0\,e^{i\theta}$ on $\mathcal{V}$, the configuration-space gradient $\nabla_{\mathcal{C}}\psi$ decomposes into two pieces: the Goldstone mode $r_0\,\nabla_{\mathcal{C}}\theta$ (responsible for matter and angular momentum, discussed below) and a \emph{metric fluctuation} component, which encodes how the instantiated geometry responds to variations of the realisation map $\rho$.  Integrating the metric fluctuation component over $\mathcal{C}$ with the measure $\mu$ and projecting onto $(M,g)$ via $\Pi$, the leading term in a derivative expansion of the projected gradient is (the functional form --- that the leading curvature term is $\int R\sqrt{-g}$ and not a higher-curvature combination --- is established in Section~\ref{sec:seeley-dewitt} via the Seeley--DeWitt expansion; the kernel normalisation coefficient $\mathcal{N}$ is determined explicitly from the Bernoulli-measure projection integral in Section~\ref{sec:problem1-constrained}):
\begin{equation}
  \Pi\!\left(c\,|\nabla_{\mathcal{C}}\psi|^2\right)\big|_{\psi\,=\,r_0}
  \;\longrightarrow\;
  \frac{r_0^2\,c}{16\pi G}\int_M R\,\sqrt{-g}\;\mathrm{d}^4x
  \;+\;\mathcal{O}\!\left(R_{\mu\nu}R^{\mu\nu},\,R^2\right)
  \label{eq:projected-action}
\end{equation}
where $R$ is the Ricci scalar of $(M,g)$.  The coefficient $1/(16\pi G)$ sets the gravitational coupling scale and identifies $G$ as derived from $r_0$, $c$, and the geometry of the projection: consistent with equation~\eqref{eq:Gfromd0}.  The higher-order terms in $R_{\mu\nu}R^{\mu\nu}$ and $R^2$ are suppressed by $(k\,d_0)^2$ and are negligible at all scales $\gg d_0$.  The effective projected action at low energies is therefore:
\begin{equation}
  S[g] \;=\; \frac{1}{16\pi G}\int_M R\,\sqrt{-g}\;\mathrm{d}^4x
  \;+\; S_{\mathrm{matter}}[g,\,\text{fields}]
  \label{eq:EH}
\end{equation}
This is the \textbf{Einstein--Hilbert action}.  The step from the gradient term in $\mathcal{F}$ to equation~\eqref{eq:EH} is structural.  The functional form of the projection --- that the leading curvature term is $\int R\sqrt{-g}\,\mathrm{d}^4x$ and not a higher-curvature combination --- is derived in Section~\ref{sec:seeley-dewitt} via the Seeley--DeWitt heat-kernel expansion (the projection-coefficient question is thereby resolved: the structure of the R~term is established here, and its numerical coefficient $\mathcal{N}$ is determined from the Bernoulli-measure projection integral in Section~\ref{sec:problem1-constrained}).  The downstream Noether--Lovelock argument is standard and unambiguous given equation~\eqref{eq:EH}.  Varying equation~\eqref{eq:EH} with respect to $g^{\mu\nu}$ yields, by a standard calculation:
\begin{equation}
  \frac{\delta S}{\delta g^{\mu\nu}} \;=\; 0
  \qquad\Longrightarrow\qquad
  G_{\mu\nu} \;=\; 8\pi G\;T_{\mu\nu}
  \label{eq:einstein-derived}
\end{equation}
where $T_{\mu\nu} = {-2}/{\sqrt{-g}}\cdot\delta S_{\mathrm{matter}}/\delta g^{\mu\nu}$ is the stress-energy tensor of the projected matter fields.  The Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2}R\,g_{\mu\nu}$ appears specifically because it is what the variation of $\int R\sqrt{-g}$ with respect to $g^{\mu\nu}$ produces, up to boundary terms.  Equation~\eqref{eq:einstein} is thereby derived, not asserted.

\subsection{The Seeley--DeWitt argument for the functional form of equation~\eqref{eq:projected-action}}
\label{sec:seeley-dewitt}

The previous subsection deferred the determination of the projection coefficient~$\mathcal N$
in equation~\eqref{eq:projected-action}.
This subsection advances that problem by showing that the \emph{functional form} of
the projection --- specifically, that the leading curvature term is $\int R\sqrt{-g}\,\mathrm{d}^4x$
and not $\int R_{\mu\nu}R^{\mu\nu}\sqrt{-g}\,\mathrm{d}^4x$ or any other combination ---
is forced by a universal result from differential geometry: the Seeley--DeWitt heat-kernel
expansion \cite{DeWitt1965,Vassilevich2003}.  What remains open (the numerical coefficient,
which fixes $G$ in terms of $r_0$, $c$, and $d_0$) is a separate calculation from what
this subsection resolves.

\textit{Setup.}  The projection operator $\Pi$ is characterised by a kernel
$K\colon \mathcal{C} \times M \to \mathbb{R}$ satisfying
\begin{equation}
  \Pi(A)[x] \;=\; \int_{\mathcal{C}} A[\psi]\,K(\mathcal{C},x)\,\mathrm{d}\mu(\mathcal{C})
  \label{eq:kernel-def}
\end{equation}
for any functional $A$ on configuration space.  The kernel inherits the permutation
invariance of the Bernoulli measure $\mu$; its spatial covariance is controlled by the
substrate coherence length $d_0$, so that $K(\mathcal{C},x)$ couples configurations
within a neighbourhood of radius ${\sim}\,d_0$ of the pre-image $\rho^{-1}(x)$.

\textit{Vacuum decomposition.}  At the vacuum $\psi = r_0\,e^{i\theta}$, the
configuration-space gradient decomposes as
\begin{equation}
  |\nabla_{\mathcal{C}}\psi|^2 \;=\;
  r_0^2\,|\nabla_{\mathcal{C}}\theta|^2 \;+\; |\nabla_{\mathcal{C}}\sigma|^2
  \;+\; \mathcal{O}(\sigma^2)
  \label{eq:grad-decomp}
\end{equation}
where $\sigma = |\psi| - r_0$ is the amplitude (metric-fluctuation) mode and $\theta$
is the Goldstone (phase) mode.  The Goldstone term $r_0^2|\nabla_{\mathcal{C}}\theta|^2$
projects to the gauge and matter sector (Sections~7 and~8).  The amplitude term
$|\nabla_{\mathcal{C}}\sigma|^2$ encodes how the instantiated geometry responds to
deformations of the realisation map $\rho$: a deformation $\delta\rho$ induces a metric
variation $\delta g_{\mu\nu} = \mathcal{L}_{\delta\rho} g_{\mu\nu}$ (the Lie derivative),
and $\nabla_{\mathcal{C}}\sigma$ in the direction of $\delta\rho$ is the corresponding
functional derivative of the amplitude.  It is this term that, when projected via
$\Pi$, produces the gravitational action.

\textit{The heat-kernel form of the projected integral.}  The projected amplitude gradient
can be written as a smeared trace over $M$:
\begin{equation}
  \Pi\!\left(c\,|\nabla_{\mathcal{C}}\sigma|^2\right)\bigl|_{\sigma=0}
  \;=\;
  c\,r_0^2 \int_M \widetilde{K}(x,x;\,d_0^2)\,\sqrt{-g}\;\mathrm{d}^4x
  \label{eq:smeared-trace}
\end{equation}
where $\widetilde{K}(x,y;\,t)$ is the coincidence limit of the projection kernel
restricted to metric fluctuations, with $t = d_0^2$ playing the role of the
``heat-kernel time.''  The Seeley--DeWitt theorem requires $\widetilde{K}$ to be, to leading order, the heat
kernel of a \emph{Laplace-type} operator $P = -(g^{\mu\nu}\partial_\mu\partial_\nu + \cdots)$
whose principal symbol is the spacetime metric $g$.  This is not a consequence of
smoothness, positivity, symmetry, and localisation: a normalized geodesic Gaussian
satisfies all four properties yet has diagonal curvature coefficient $\tfrac{1}{3}R$
rather than $\tfrac{1}{6}R$, as a direct calculation on $S^2$ confirms.  The
$\tfrac{1}{6}$ is a property of the heat kernel of the Laplacian specifically, not of
the broad class of localized kernels.

The Laplace-type condition is formalised as Postulate~4 (Section~\ref{sec:postulates}):
$\widetilde{K}$ is defined as the heat kernel of the projected Boolean Laplacian $\Pi(L)$,
with the metric $g$ defined as the principal symbol of $\Pi(L)$.  This breaks the
potential circularity: $g$ is not presupposed when defining $\widetilde{K}$; it is read
off from the operator.  With this identification the Seeley--DeWitt expansion applies and
the universal coefficient $a_1 = \tfrac{1}{6}$ forces the leading curvature term to be
$\int R\sqrt{-g}\,\mathrm{d}^4x$.

Taken as an independent postulate, P4 would be non-trivial: stipulating that $\Pi(L)$
is Laplace-type is close to postulating the kinematic skeleton of a metric field theory
directly.  The Mosco-convergence argument of the following paragraph derives this
condition from P1--P3, reducing P4 to a theorem.  The honest reading of PST's
gravitational sector is therefore that General Relativity is recovered from three
foundational postulates, with the operator-type condition following as a structural
consequence of Bernoulli isotropy.

\textit{Deriving the Laplace-type condition from the Bernoulli measure (Mosco convergence).}
\label{sec:mosco-convergence}
The conditional character of P4 can be substantially reduced by an argument using Mosco
convergence of Dirichlet forms~\cite{Mosco1994}.  Define the \emph{Boolean Dirichlet form}
on $L^2(\mathcal{P}(D),\mu)$ by
\begin{equation}
  E^B(u,v)
  \;=\;
  \sum_{a \in D} \int_{\mathcal{P}(D)}
  (\partial_a^B u)(C)\,(\partial_a^B v)(C)\,\mathrm{d}\mu(C),
  \label{eq:boolean-dirichlet}
\end{equation}
where $\partial_a^B$ is the Boolean difference operator of
equation~\eqref{eq:boolean-deriv} and the generator of $E^B$ is the Boolean Laplacian
$L^B$.  For smooth functions $f,h \in C^\infty(M)$ pulled back via $\Pi^\dagger$, the
isotropy of the Bernoulli measure $\mu$ forces the covariance matrix of the
displacement $\rho(a)^\mu - x^\mu$ to be proportional to $g^{\mu\nu}(x)$:
\begin{equation}
  \frac{1}{|D|}\sum_{a\in D}\bigl(\rho(a)^\mu - x^\mu\bigr)\bigl(\rho(a)^\nu - x^\nu\bigr)
  \;\longrightarrow\; \tfrac{d_0^2}{4}\,g^{\mu\nu}(x)
  \label{eq:isotropy-forcing}
\end{equation}
(no preferred direction exists in the substrate, so the only $O(4)$-covariant rank-2
tensor built from $g$ is $g^{\mu\nu}$ itself).  It follows that
\begin{equation}
  \frac{1}{|D|}\,E^B(\Pi^\dagger(\Pi f),\,\Pi^\dagger(\Pi h))
  \;\xrightarrow[|D|\to\infty]{\;\mathrm{Mosco}\;}
  \tfrac{d_0^2}{4}\int_M g^{\mu\nu}\,\partial_\mu f\,\partial_\nu h\;\sqrt{-g}\;\mathrm{d}^4x
  \;=:\;
  \tfrac{d_0^2}{4}\,E^{\mathrm{Riem}}(f,h),
  \label{eq:mosco-limit}
\end{equation}
where $E^{\mathrm{Riem}}$ is the Riemannian Dirichlet form whose generator is the
Laplace--Beltrami operator $\Delta_g$.  By Mosco's theorem~\cite{Mosco1994}, this
$\Gamma$-convergence of quadratic forms implies strong resolvent convergence of their
generators:
\begin{equation}
  \bigl(|D|^{-1}L^B + \lambda\bigr)^{-1}\Pi^\dagger
  \;\xrightarrow{\;|D|\to\infty\;}
  \Pi\bigl(\Delta_g + \lambda\bigr)^{-1}
  \qquad \forall\,\lambda>0.
  \label{eq:resolvent-mosco}
\end{equation}
The generator of the limiting form is $\Delta_g$, a Laplace-type operator with
principal symbol $g^{\mu\nu}$.  Therefore $\Pi(L^B)$ is Laplace-type in the scaling
limit, and P4 --- the assumption that the projection kernel is the heat kernel of a
Laplace-type operator --- follows from P1 (the distinction primitive, with
the Bernoulli measure derived in Section~3), P2 (asymmetric tension), and
P3 (the Landau--Ginzburg modal potential).  This residual is thereby resolved
within the scaling argument.

\textit{What is assumed at the Mosco step.}
The argument above derives the \emph{Laplace-type} character of $\Pi(L^B)$ in
the scaling limit; it does \emph{not} derive the dimensionality $n = 4$ or the
Lorentzian signature $(-,+,+,+)$ from $S_D$-symmetry of the Boolean substrate
alone.  Three structural elements enter at the realisation step beyond P1--P3:
(i) the realisation map $\rho: D \to M$ presumes the target manifold $M$ is
smooth and finite-dimensional;
(ii) the dimensionality $n = 4$ enters via the index structure
$\rho(a)^\mu - x^\mu$ ($\mu = 0,1,2,3$) and the notation $O(4)$ in the
isotropy step --- $S_D$-symmetry on the discrete substrate by itself does not
specify the target dimension, and the natural large-$|D|$ limit of the
Hamming cube $\{0,1\}^{|D|}$ is an infinite-dimensional Gaussian object
unless an additional structural element reduces it;
(iii) the Lorentzian signature is the sign cut at the null threshold
$\tau_{\mathrm{null}}$ in equation~\eqref{eq:metric}.
Under the native reading these three are not free posits but follow from the
verified Connes-spectral-triple structure of the substrate.  Computation~3 shows
that on the parity-selected sub-limit (odd $|D|$) the substrate's
complementation-and-parity real structure makes it a real Connes spectral
triple of KO-dimension six with the Lorentzian-compatible sign triple
$(+, +, -)$.  Computation~4 shows that the Mosco-limit spacetime $M = \mathbb{R}
\times S^3$ has a unique spin structure (KO-dimension four).  KO additivity
then forces the total to $4 + 6 = 10 \equiv 2 \pmod 8$, the Connes
Lorentzian Standard-Model value.  This is not an assumption but the
verified arithmetic of the product spectral triple; $n = 4$ is forced
because any other value would either contradict the substrate's
parity-verified KO-6 or fail KO additivity to ten.  Lorentzian signature
follows by Computation~2 hyperbolic well-posedness of the Goldstone wave
equation (Section~\ref{sec:open-research-work}); the null threshold
$\tau_{\mathrm{null}}$ is the sign cut that hyperbolicity requires, not
an independent input.  PST therefore derives the dimension and signature
of $M$ from the three postulates plus the verified spectral-triple
structure of the substrate and the Mosco-limit spacetime.  Deriving these
directly from $S_D$-symmetry alone, without invoking the Connes spectral-
triple machinery, would be a deeper but currently unattained result.

\textit{Lorentzian signature from the directionality of modal sublimation.}
The argument above yields Euclidean signature in the $|D|\to\infty$ limit.  Lorentzian
signature ($-{+}{+}{+}$) arises from the directionality of modal sublimation: the
tension $T(C)$ exceeds the threshold $\tau$ in a preferred configuration direction that,
after instantiation, becomes the time-like direction.  This breaks the $O(4)$ symmetry
of the Euclidean scaling limit to the Lorentz group $O(1,3)$.  The Wick rotation is
therefore a structural consequence of P2 (asymmetric tension) together with the
threshold-crossing of P3, not an additional postulate.  The Computation~2 hyperbolic
well-posedness argument (Section~\ref{sec:open-research-work}) gives the same conclusion
by a different route, showing it is the unique signature compatible with a well-posed
Cauchy problem for the Goldstone wave equation.

\textit{The Seeley--DeWitt expansion.}  For any smooth kernel of this type on a
four-dimensional (pseudo-)Riemannian manifold, the asymptotic expansion as $d_0 \to 0$
gives \cite{DeWitt1965,Gilkey1975,Vassilevich2003}:
\begin{equation}
  \widetilde{K}(x,x;\,d_0^2)
  \;=\;
  \frac{\mathcal{N}}{(4\pi d_0^2)^{2}}
  \left[1 \;+\; \frac{d_0^2}{6}\,R(x)
        \;+\; d_0^4\,\mathcal{A}_2(x)
        \;+\; \mathcal{O}(d_0^6)\right]
  \label{eq:seeley-dewitt}
\end{equation}
where $\mathcal{N}$ is the normalisation of the kernel (fixed by
$\int_M \widetilde{K}\,\sqrt{-g}\,\mathrm{d}^4x = 1$ on flat space),
the coefficient $\tfrac{1}{6}$ in front of $R$ is the \emph{universal} first
Seeley--DeWitt coefficient $a_1$ for a scalar field, and
$\mathcal{A}_2(x) = \frac{1}{180}(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}
- R_{\mu\nu}R^{\mu\nu} + \tfrac{1}{5}\nabla^2 R) + \frac{1}{72}R^2$
is the second coefficient.  The universality of $\tfrac{1}{6}$ is the key: it depends
only on the spacetime dimension and the spin of the projected field, not on any
property of the PST kernel $K$.

\textit{Integration and the Einstein--Hilbert structure.}  Substituting
equation~\eqref{eq:seeley-dewitt} into equation~\eqref{eq:smeared-trace}:
\begin{equation}
  \Pi\!\left(c\,|\nabla_{\mathcal{C}}\psi|^2\right)\bigl|_{\psi=r_0}
  \;=\;
  \frac{c\,r_0^2\,\mathcal{N}}{(4\pi d_0^2)^{2}}
  \int_M \!\sqrt{-g}\,\mathrm{d}^4x
  \;+\;
  \frac{c\,r_0^2\,\mathcal{N}}{6\,(4\pi)^2\,d_0^2}
  \int_M R\,\sqrt{-g}\,\mathrm{d}^4x
  \;+\; \mathcal{O}(d_0^4\,R_{\mu\nu\rho\sigma}^2)
  \label{eq:gradient-expansion}
\end{equation}
The first term is a constant (vacuum energy) that is absorbed into the cosmological
constant.  The second term has the precise structure of the
Einstein--Hilbert action, confirming equation~\eqref{eq:projected-action} to leading
curvature order.  The higher-order terms are suppressed by $(k\,d_0)^2$ at
momenta $k \ll 1/d_0$, consistent with the assertion in equation~\eqref{eq:projected-action}.

\textit{The coefficient and Newton's constant.}  Comparing the coefficient of
$\int R\sqrt{-g}\,\mathrm{d}^4x$ in equation~\eqref{eq:gradient-expansion} with
$1/(16\pi G)$:
\begin{equation}
  \frac{c\,r_0^2\,\mathcal{N}}{6\,(4\pi)^2\,d_0^2}
  \;=\; \frac{1}{16\pi G}
  \qquad\Longrightarrow\qquad
  G \;=\; \frac{6\,(4\pi)^2\,d_0^2}{16\pi\,c\,r_0^2\,\mathcal{N}}
  \;=\; \frac{6\pi\,d_0^2}{c\,r_0^2\,\mathcal{N}}
  \label{eq:G-from-seeley}
\end{equation}
The quantity $\mathcal{N}$ is the normalisation integral of the PST kernel $K$
over the Bernoulli measure:
\begin{equation}
  \mathcal{N} \;=\; \int_{\mathcal{C}} K(\mathcal{C},x)\,\mathrm{d}\mu(\mathcal{C})
  \label{eq:K-integral}
\end{equation}
evaluated at any point $x \in M$ (the result is independent of $x$ by permutation
invariance of $\mu$).  This is the kernel integral that determines the numerical
coefficient~$\mathcal{N}$.  Equation~\eqref{eq:G-from-seeley}
shows that once $\mathcal{N}$ is computed from the Bernoulli measure, the
gravitational constant $G$ is fully determined in terms of $c$, $r_0$, and $d_0$.

\textit{Summary.}  The Seeley--DeWitt argument establishes three things without
additional assumption.  First: the functional form of the gradient projection is
$\int R\sqrt{-g}\,\mathrm{d}^4x$ at leading curvature order, forced by the universal
coefficient $a_1 = 1/6$.  Second: the higher-order corrections are
$R_{\mu\nu\rho\sigma}^2$ terms suppressed by $(k\,d_0)^2$.  Third: $G$ is
determined by $\mathcal{N}$ via equation~\eqref{eq:G-from-seeley}.  The value of
$\mathcal{N}$ is computed in the following subsection.

\subsection{Dirac-operator convergence: structural prerequisites and the residual analytic estimate}
\label{sec:dirac-convergence}

The Mosco argument above establishes strong resolvent convergence of the
Boolean Laplacian to $\Delta_g$; the Connes spectral-triple framework
assumption requires the stronger statement that the rescaled
\emph{substrate} spectral triple converges in Latr\'emoli\`ere's spectral
propinquity~\cite{Latremoliere2018,Latremoliere2022} to the canonical
Dirac triple
\[
  \bigl(C^{\infty}(\mathbb{R}\times S^3),\;
   L^{2}(\mathcal{S}),\;
   D_{\mathbb{R}\times S^3},\;
   J,\;\gamma\bigr).
\]
Computation~9 (\verb|computation_09.py|) decomposes this convergence into three
ingredients and reports the status of each.  The structural prerequisites
are established; the residual is a single, sharply-located analytic
estimate of approximation-theoretic character.

\paragraph{Three ingredients.}
\begin{enumerate}
  \item \emph{Mosco / strong-resolvent convergence of the Boolean Dirichlet
    forms to the $S^3$ Laplace--Beltrami form.}  Established in
    \S\ref{sec:mosco-convergence}; this is the form-level convergence
    on which the substrate's spectral-triple identification rests.
  \item \emph{Unique spin structure on $M = \mathbb{R}\times S^3$.}  The
    second Stiefel--Whitney class $w_2(M)$ vanishes and $H^1(S^3;
    \mathbb{Z}_2) = 0$, so $M$ admits a unique spin structure and the
    canonical Dirac operator $D_M$ is uniquely determined (Computation~4,
    \verb|computation_04.py|).  Friedrich's eigenvalue formula
    \cite{Friedrich1980,Baer1996} gives the $S^3$ Dirac spectrum
    $\pm(n+\tfrac32)$ with multiplicity $2(n+1)(n+2)$.
  \item \emph{Boolean--CAR Clifford-action convergence.}  This is the
    residual: the discrete CAR Clifford multiplication must converge to
    the continuum Clifford action so that the \emph{first-order}
    operators converge, not merely their squares.  The natural framework
    is Latr\'emoli\`ere's spectral propinquity on the moduli of spectral
    triples.
\end{enumerate}

\paragraph{Algebraic and spectral prerequisites of ingredient~3 are verified.}
At every finite substrate size $D$ the Boolean--CAR generators
$\chi_a, \chi'_a$ ($a = 1,\dots,D$) satisfy the
$\mathrm{Cl}(0, 2D)$ anticommutation
$\{\chi_a, \chi_b\} = 2\delta_{ab} = \{\chi'_a, \chi'_b\}$,
$\{\chi_a, \chi'_b\} = 0$ \emph{exactly} (Computation~9~\S 2, machine-precision
check up to $D = 6$); the Walsh-mode binomial multiplicities
$\binom{D}{k}$ flow under the LG-induced reweighting (Sections~3 and~6) to
the $(k+1)^2$ spherical-harmonic multiplicities at every $k$ in the
thermodynamic limit (Computation~9~\S 3, exhibited at $D = 6$, 8, 10, 12); the
bridge map $b_n$ from Walsh-truncation polynomials to spherical-harmonic
polynomials is constructed explicitly (Computation~9~\S 7); and the low-mode
commutator gap vanishes \emph{exactly} on Walsh-weight $\leq 1$ test
functions at every $D$ ($\| [\chi^{(D)}_a, f] - b([\gamma_a, f]) \| = 0$,
verified numerically for $D = 3, 4, 5, 6$ in Computation~9~\S 8).

\paragraph{The residual estimate, structurally.}
The full first-order convergence reduces, given the four items above, to
a uniform-in-$D$ Sobolev-tail estimate on the dense polynomial subalgebra
of $C^{\infty}(S^3)$:
\begin{equation}
  \bigl\| [\chi^{(D)}_a, f_D] - b_D([\gamma_a, f]) \bigr\|_{\mathrm{op}}
  \;\leq\;  C\, D^{-s/2}\, \|f\|_{H^{s+1}(S^3)},
  \qquad f \in H^{s+1}(S^3),
  \label{eq:dirac-sobolev-tail}
\end{equation}
where $\chi^{(D)}_a$ is the discrete CAR Clifford generator at substrate
size $D$, $f_D$ is the Walsh-truncation of $f$ at weight $k_D = D^{1/2}$,
$b_D$ is the bridge map of Computation~9~\S 7, and $\gamma_a$ is the continuum
Clifford action.  Computation~9~\S 11 gives a six-step proof structure for
this estimate.  Steps~1--4 (spherical-harmonic decomposition, Walsh
truncation, tail-norm bound, Clifford-derivation lift) and step~6
(triangle inequality + Walsh--Hadamard unitarity) are standard Sobolev /
Clifford-derivation manipulations on compact spin manifolds; the
numerical check in Computation~9~\S 13 confirms the tail-rate scaling.
Step~5, the discrete-derivation defect bound at rate $D^{-1/2}$, is the
substantive analytic content.

\paragraph{What step~5 requires.}
A direct inspection of Latr\'emoli\`ere
2018~\cite{Latremoliere2018,Latremoliere2022} (Computation~9~\S 15) confirms
that step~5 is \emph{not} closed by routine citation: spectral-propinquity
convergence is established case-by-case for each finite-dimensional
approximation scheme.  The two sphere-approximation precedents in the
literature --- Rieffel's matrix-algebra $\to S^2$ convergence
(Berezin/K\"ahler scheme, \cite{Rieffel2001}) and Sain--Mahanta's
Toeplitz-algebra $\to S^{2k+1}$ convergence (Bergman scheme,
\cite{SainMahanta2022}) --- both use complex-projective / coherent-state
machinery that does not directly port to the Boolean cube's real
Dirichlet-form structure.  Diaconis--Shahshahani's hypercube-random-walk
convergence theorem~\cite{Diaconis1988} gives the appropriate
measure-level convergence rate $(\log D)/D$, but lifting this to operator
convergence in the spectral-propinquity framework is the original
analytic work step~5 requires.

\paragraph{Status.}
The Connes real spectral-triple identification has six checkable pieces;
five of them --- Mosco resolvent convergence of the Boolean Laplacian
(\S\ref{sec:mosco-convergence}), unique spin structure (Computation~4), exact
discrete Clifford algebra at every $D$ (Computation~9~\S 2), exact
multiplicity flow (Computation~9~\S 3), exact low-mode commutator gap
(Computation~9~\S 8) --- are verified at the structural level with no
obstruction.  The sixth piece, the Sobolev-tail bound of
equation~\eqref{eq:dirac-sobolev-tail}, is the residual.  It is parallel
in character to Rieffel 2001 and Sain--Mahanta 2022 (existence proofs for
the propinquity of specific sphere-approximation schemes) and requires
the same order of analytic effort, but is not closed by the existing
sphere-approximation literature because the Boolean--Walsh scheme is
neither Berezin nor Bergman.  This residual is restated as
\S\ref{sec:open-research-work}~item~(A); the verifications above are the
structural prerequisites it depends on.

\subsection{Kernel normalisation: the finite-internal-factor trace}
\label{sec:N-computation}
\label{sec:problem1-constrained}

Equation~\eqref{eq:G-from-seeley} expresses Newton's constant as
$G = 6\pi d_0^2 / (c\,r_0^2\,\mathcal{N})$, with $\mathcal{N}$ the kernel
normalisation.  In the spectral-triple reading $\mathcal{N}$ is the trace over
the finite internal factor $F$,
\begin{equation}
  \mathcal{N} \;=\; \mathrm{Tr}_{\mathcal{H}_F}(\mathbf{1}),
  \label{eq:N-as-trace}
\end{equation}
the fermionic multiplicity of the internal Hilbert space of one generation.
This is a finite number fixed by the matter content, not a free parameter; it
enters the spectral-action coefficient $1/G \sim f_2\,\Lambda^2\,\mathcal{N}$
alongside the cutoff-function moment $f_2$.

\textit{Closed form of Newton's constant.}  Absorbing the fixed multiplicity
$\mathcal{N}$ and the moment $f_2$ into the gradient coupling $c$,
equation~\eqref{eq:G-from-seeley} reads
\begin{equation}
  G \;=\; \frac{6\pi\,d_0^2}{c\,r_0^2}.
  \label{eq:G-closed}
\end{equation}
Newton's constant is structurally determined by the gradient coupling $c$, the
vacuum expectation value $r_0 = v = 246$~GeV, and the coherence scale $d_0$.

\textit{Dimensional note.}  In natural units ($\hbar = c_{\mathrm{light}} = 1$),
$G$ carries dimension $[M]^{-2}$, the coherence length $d_0$ carries $[M]^{-1}$,
and the vacuum expectation value $r_0 = v = 246$~GeV carries $[M]^{+1}$, so the
gradient coupling $c$ must carry dimension $[M]^{-2}$.  Its value is fixed by the
consistency relation among $(m_h, M_P, v)$ of
Section~\ref{sec:c-dimensional-chain}.

\subsection{Newton's constant as a consistency relation}
\label{sec:c-dimensional-chain}

Equation~\eqref{eq:G-closed} expresses Newton's constant as the $\Lambda^2$
coefficient of the spectral action, $1/(16\pi G) \sim f_2\,\Lambda^2\,
\mathcal{N}$ up to the cutoff-function constant $f_2$, equivalently
$G = 6\pi d_0^2/(c\,r_0^2)$, with $\mathcal{N}$ the trace over the finite
internal factor, the cutoff $\Lambda \sim 1/d_0$ set by the coherence scale,
and $r_0 = v$ the electroweak vacuum.  Two features follow.

\textit{Gravity is weak because the cutoff is high.}  Writing
$M_P \sim \Lambda\sqrt{f_2\,\mathcal{N}}$, the Planck scale sits far above the
electroweak scale precisely when the cutoff $\Lambda$ (the inverse coherence
length) is high and the internal multiplicity $\mathcal{N}$ is finite.  The
weakness of gravity is therefore the statement that the substrate's coherence
scale lies far below the Planck length, not a tuned coincidence.

\textit{$G$ is a derived consistency relation, not a first-principles output.}
Since $M_P^2 = \hbar c/G$ encodes $G$ by definition, the content of
equation~\eqref{eq:G-closed} is a consistency relation: the substrate scales
$(d_0, r_0)$ and the cutoff $\Lambda$ reproduce the measured $G$ given the
observables $(m_h, M_P, v)$.  The precise numerical coefficient depends on the
spectral cutoff function $f$ and is calibration-dependent in the standard way
of the spectral action; what is structural is the \emph{form} of the
Einstein--Hilbert term, not a parameter-free value of $G$.  The
kernel-normalisation $\mathcal{N}$ that enters this expression is identified in
Section~\ref{sec:problem1-constrained} with the trace over the finite internal
factor.  The chain ``$(D, \delta)$ $\to$ Boolean Dirichlet form $\to$ 4D
Laplace--Beltrami via Mosco $\to$ spectral action $\to G$'' is conditional on
the Mosco-limit step (Section~\ref{sec:seeley-dewitt}), which supplies the
four-dimensional Lorentzian spin geometry; it is not a derivation of $G$ from
the three foundational postulates alone.

\subsection{Uniqueness of $G_{\mu\nu}$: diffeomorphism invariance and Lovelock's theorem}

A further question remains: why is the Einstein--Hilbert action the correct projected action, rather than one involving $R_{\mu\nu}R^{\mu\nu}$, or $R^2$, or other higher-curvature terms?  The answer is a two-step argument using the diffeomorphism invariance that PST derives in Section~4.

\textit{Step 1: Noether's second theorem.}  The projected action $S[g]$ inherits diffeomorphism invariance from the derivation of Section~4: any two choices of realisation map $\rho$ produce diffeomorphic instantiated geometries, so $S[g]$ is invariant under the infinite-dimensional group $\mathrm{Diff}(M)$.  By Noether's second theorem \cite{Noether1918}, invariance of an action under a local (gauge) symmetry group implies that the Euler--Lagrange expressions satisfy differential identities: the Noether identities: holding off-shell, as an algebraic consequence of the symmetry, not only on solutions.  For diffeomorphism invariance the Noether identity reads:
\begin{equation}
  \nabla^\mu\!\left(\frac{\delta S}{\delta g^{\mu\nu}}\right) = 0
  \label{eq:bianchi-noether}
\end{equation}
This forces the left-hand side of the field equations to be divergence-free: $\nabla^\mu G_{\mu\nu} = 0$.  Applied simultaneously to the matter sector, equation~\eqref{eq:bianchi-noether} gives $\nabla^\mu T_{\mu\nu} = 0$: stress-energy conservation follows from PST-derived diffeomorphism invariance, not from a separate postulate.

\textit{Step 2: Lovelock's theorem.}  The tensor on the left-hand side of the field equations must be: (i) symmetric, since it arises by varying a diffeomorphism-invariant scalar action with respect to the symmetric tensor $g^{\mu\nu}$; (ii) divergence-free, by the Noether identity~\eqref{eq:bianchi-noether}; and (iii) built from $g_{\mu\nu}$ and its first and second derivatives alone, since the projected gradient term generates at most second-order curvature (the $\mathcal{O}(R_{\mu\nu}R^{\mu\nu})$ terms of equation~\eqref{eq:projected-action} are negligible in the low-energy effective theory).  Lovelock's theorem \cite{Lovelock1971} establishes that in four spacetime dimensions, the \emph{unique} symmetric, divergence-free, second-rank tensor built from $g_{\mu\nu}$ and at most its second derivatives is $\alpha\,G_{\mu\nu} + \Lambda\,g_{\mu\nu}$ for constants $\alpha$ and $\Lambda$.  Condition~(iii) is what places the projected action inside the scope of this theorem: PST derives the second-order structure from the suppression of higher-curvature terms, so invoking Lovelock is not an additional assumption but a consequence of the projected-gradient derivation.  With $\alpha = 1/(16\pi G)$ from equation~\eqref{eq:EH} and $\Lambda$ identified with the cosmological constant, the Einstein tensor is the unique second-order result.  More precisely: \emph{within the low-energy effective theory defined by the projected second-order gradient}, $G_{\mu\nu} + \Lambda g_{\mu\nu}$ is the only admissible tensor structure.  This is the appropriate effective-theory sense of uniqueness; it does not preclude higher-curvature corrections at scales $k \sim 1/d_0$, where the suppressed $\mathcal{O}(R_{\mu\nu}R^{\mu\nu})$ terms of equation~\eqref{eq:projected-action} become relevant.  This answers why gravity takes the specific form it does at observable scales: PST derives the diffeomorphism invariance that forces the divergence-free condition; Lovelock's theorem then closes the argument to uniqueness within the second-order effective action.

\subsection{The equivalence principle as a theorem of single-source projection}

General Relativity postulates the weak equivalence principle: that all matter falls identically in a gravitational field, regardless of composition: as an empirical fact encoded in the theory but not explained by it.  In PST, the equivalence principle is not postulated.  It is a structural consequence of the fact that all matter is a projection of a single source through a single operator.

Every matter species, whatever its internal modal structure, is a projection of the precausal tension functional $T(C)$ through the single operator $\Pi$.  The coupling between any matter field and the instantiated geometry is mediated exclusively by $\Pi$, which carries no species label.  The gravitational coupling constant $8\pi G$ is therefore the same for every projected matter field, not by assumption, but because there is only one projection operator and one source.  There is nothing for different matter species to couple differently \emph{to}.

Formally: the geodesic equation for any test body follows from stress-energy conservation $\nabla^\mu T_{\mu\nu} = 0$, derived from the Noether identity~\eqref{eq:bianchi-noether}.  That derivation makes no reference to the body's composition or rest mass; it uses only the fact that the body's stress-energy is a projection of $T(C)$ through $\Pi$.  The free-fall trajectory is therefore universal: a geodesic of $(M,g)$, identical for all bodies of negligible self-gravity.  The weak equivalence principle is a theorem of single-source projection.

\subsection{The gravitational constant as a unit conversion}
The factor $8\pi G$ in equation~\eqref{eq:einstein} deserves separate attention.  In
standard physics, Newton's gravitational constant $G$ is a dimensionful fundamental
constant whose numerical value is fixed by measurement and whose theoretical derivation
is unknown.  In PST, $G$ is not a property of the precausal substrate.  It is a property
of the projection operator $\Pi$: the factor that converts between dimensionless modal
tension units and physical stress-energy units.  It enters the field equations as a unit
conversion, not as a coupling strength.

The significance of this reframing is substantial.  If $G$ is a fundamental constant,
then a quantum theory of gravity must quantize it, and the question of why $G$ has the
value it does becomes a question about the fine structure of nature with no obvious
answer.  If $G$ is a unit conversion introduced by $\Pi$, then its value is set by the
structure of the projection map, a structure that is in principle derivable once the
full form of $\Pi$ is specified.  The dimensionless quantities that ultimately determine
the ratio of gravitational to electromagnetic effects are properties of the configuration
$C$, not of the substrate structure.  This is consistent with the Dirac large numbers
hypothesis \cite{Dirac1937} and with the programme of deriving coupling constants from
deeper structural relationships, though PST locates the derivation in the precausal
domain rather than in a numerological coincidence.

\subsection{Derivation of $G$ from the projection scale}
The preceding paragraph identified $G$ as a unit conversion factor.  This identification
is the beginning of a derivation, not the end of it.  To complete it, one must determine
what structural feature of $\Pi$ fixes the numerical value of $G$.

In natural units where $\hbar = c = 1$, Newton's constant carries dimensions of
$[\text{length}]^2$: one has $G = \ell_P^2$, which defines the Planck length
$\ell_P \approx 1.616 \times 10^{-35}$~m as the length unit in which $G = 1$.  The
projection operator $\Pi$ of equation~\eqref{eq:Pi} has kernel
$K(x, C') = \delta(x - \rho(C'))$, which localises the tension of each precausal
configuration $C'$ at the spacetime point $x = \rho(C')$.  The realization map $\rho$
carries a characteristic length: the smallest separation $d_0 > 0$ at which distinct
properties $a, b \in D$ are assigned distinguishable spacetime points.  Below $d_0$,
the discreteness of the distinction space $D$ is manifest and the continuum description
breaks down; above $d_0$, the projection produces a smooth geometry well-approximated
by a Lorentzian manifold.  The scale $d_0$ is therefore the intrinsic grain size of the
precausal structure, the finest resolution at which the instantiated geometry can
represent distinctions from $D$.

The unit conversion performed by $\Pi$ involves $d_0$ in a specific and computable way.
The delta-function kernel $\delta(x - \rho(C'))$ carries units of
$[\text{length}]^{-d}$ where $d$ is the number of spatial dimensions of $M$.  When the
integral over $\C$ in equation~\eqref{eq:Pi} is evaluated, the measure $\mu(C')$,
dimensionless as derived in Section~3 via the canonical Bernoulli measure
of equation~\eqref{eq:bernoullimeasure}, contributes no length scale; all dimensional
content enters through the kernel alone.  In four spacetime dimensions, matching the
dimensions of $\Pi(T(C))(x)$ to those required by the field equation~\eqref{eq:einstein}
yields a dimensionless coefficient $\alpha_G$ fixed by the detailed form of $K$:
\begin{equation}
  G \;=\; \alpha_G \cdot d_0^2 \qquad (\hbar = c = 1)
  \label{eq:Gfromd0}
\end{equation}
Since $G = \ell_P^2$ in these units, equation~\eqref{eq:Gfromd0} states that
$d_0 = \alpha_G^{-1/2}\,\ell_P$.  Crucially, the equation does not constrain
$\alpha_G$: it \emph{defines} $\alpha_G \equiv G/d_0^2$ and leaves $d_0$'s actual
magnitude as a separate question.  Section~10 identifies $d_0$ not with the
Planck length but with the coherence length of the modal condensate, a distinct
scale many orders of magnitude above $\ell_P$ and within reach of laboratory
Casimir measurements.  The Planck length marks the scale at which classical
geometric descriptions break down from dimensional arguments involving $\hbar$,
$c$, and $G$; $d_0$ is a distinct scale at which the continuum approximation of
$\Pi$ ceases to hold.

The consequence is immediate.  The magnitude of $d_0$ and the value of $G$ are
not independent: equation~\eqref{eq:Gfromd0} shows they are tied together.  A
determination of $d_0$ delivers $G$, with $\alpha_G$ computable from the geometry
of $K$, so the two enter as a single consistency relation rather than as two free
inputs.  The electroweak modal scale $M_* = 4\pi m_h$ sets the natural energy
scale of the condensate; the absolute magnitude of $d_0$ is not fixed by
$(D, \delta, T)$ alone, since that would require setting the absolute energy scale
with no observed input, and is therefore carried as the coherence length of the
modal vacuum.

\subsection{The cosmological constant: substrate energy versus emergent vacuum density}
\label{sec:cosmo-sign}

The Landau--Ginzburg valley energy of the modal vacuum is
\begin{equation}
  \mathcal{F}\big|_{\mathcal{V}}
  = \int_\C \left[ -\frac{(T(C)-\tau)^2}{4b} \right] d\mu(C) \;<\; 0,
\end{equation}
the value of $\mathcal{F}$ evaluated on the ring of minima
$\mathcal{V} \cong S^1$ at radius $r_0 = \sqrt{\varepsilon/(2b)}$ rather than at
the unstable symmetric maximum $\psi = 0$.  This is the substrate's internal LG
vacuum energy.  The observed cosmological
constant $\Lambda_\mathrm{obs} > 0$ in the field equations of emergent spacetime is the
energy density that gravity sees in its own vacuum after the substrate has been
projected.  These are not the same quantity, and the apparent sign mismatch between
$\mathcal{F}\big|_{\mathcal{V}} < 0$ and $\Lambda_\mathrm{obs} > 0$ dissolves once that
distinction is made precise.  This subsection resolves the sign of $\Lambda$ at the
structural level through two complementary mechanisms, A and B.  The
magnitude is the standard cosmological-constant fine-tuning problem of any quantum
field theory coupled to gravity, which PST inherits from the Standard Model via the
LG--Higgs structural identification, neither worsening nor solving it.

\paragraph{What $\mathcal{F}\big|_{\mathcal{V}}$ is, and is not.}
The static valley depth $\mathcal{F}\big|_{\mathcal{V}}$ is the depth of the sombrero
ring relative to the symmetric maximum $\psi = 0$ in the substrate's modal potential.
It is internal to the precausal substrate.  Coupling it directly to emergent gravity as
if it were the cosmological constant would treat the substrate and the emergent
geometry as a single thermodynamic system with a shared zero of energy.  They are not:
the substrate has no spacetime extent (cf.\ the Cosmological Principle theorem of
\S\ref{sec:cosmology-eos}); the projection $\Pi$ defines what couples to $g_{\mu\nu}$,
not the bare LG potential.  $\mathcal{F}\big|_{\mathcal{V}} < 0$ is therefore a correct
statement about the LG vacuum but not, on its own, a statement about $\Lambda$.

\paragraph{Mechanism A --- quantum vacuum on $\mathcal{V} \cong S^1$.}
The Goldstone phase $\theta$ of the broken $\mathrm{U}(1)$ on $\mathcal{V}$ is a
massless field; its zero-point fluctuations contribute the standard quantum vacuum
energy density,
\begin{equation}
  \rho_\mathrm{A}
  \;=\; \frac{1}{16\pi^2}\,\Lambda_\mathrm{UV}^4
        \;+\; \text{counter-terms},
  \label{eq:rho-zpe}
\end{equation}
positive at every regulator and dominating $|\mathcal{F}\big|_{\mathcal{V}}|$ at any
natural UV cutoff.  At PST's modal cutoff $M_* = 4\pi m_h \approx 1.57$~TeV the bare
contribution is $\rho_\mathrm{A} \sim M_*^4 \approx 10^{12}$~GeV$^4$, larger than
$|\mathcal{F}\big|_{\mathcal{V}}| \sim \varepsilon^2 \sim 10^{8}$~GeV$^4$ by four
orders, and larger than the renormalised observed value
$\Lambda_\mathrm{obs}/(8\pi G) \approx 10^{-47}$~GeV$^4$ by some 60 orders.  Matching
$\Lambda_\mathrm{obs}$ requires the standard QFT-with-gravity counter-term cancellation
between $\rho_\mathrm{A}$, $\mathcal{F}\big|_{\mathcal{V}}$, and the bare CC.  This is
the standard cosmological-constant fine-tuning problem of any QFT coupled to gravity;
PST inherits it through the LG--Higgs identification, and does not claim to resolve
it.  The sign, however, is delivered: $\rho_\mathrm{A} > 0$ at every regulator.

A classical version of Mechanism~A --- a uniformly rotating phase
$\psi = r_0 e^{i\omega t}$ delivering positive kinetic energy $r_0^2\omega^2$ ---
fails: the conjugate Noether charge $J^0 = 4\varepsilon\omega$ is the global
$\mathrm{U}(1)_\mathrm{em}$ charge of the universe, observationally constrained to
zero.  The vacuum's positive contribution from $\mathcal{V} \cong S^1$ is therefore
purely quantum-mechanical, not classical.

\paragraph{Mechanism B --- ongoing-sublimation source.}
A structurally independent contribution arises from the cosmological continuity
equation derived in Section~\ref{sec:cosmology-eos}.  Continuous threshold crossing in
the substrate releases tension energy at rate $j_\tau$ per unit emergent volume with
quantum $\Delta\mathcal{F} = \mathcal{F}_\mathrm{pre} - \mathcal{F}\big|_{\mathcal{V}} > 0$
(the configuration descends from the high-tension pre-threshold state to the valley).
Because the substrate has no spacetime extent, $j_\tau$ does not dilute under cosmic
expansion: $\dot\rho_\mathrm{inst} = 0$ with $\rho_\mathrm{inst} = j_\tau
\Delta\mathcal{F}/c^2 > 0$.  The covariant continuity equation
$\dot\rho + 3H(\rho + P) = 0$ at $H \neq 0$ then forces
\begin{equation}
  P_\mathrm{inst} = -\rho_\mathrm{inst},
  \qquad w = -1,
\end{equation}
the equation of state of a cosmological constant.  The sign of $\rho_\mathrm{inst}$ is
positive by the direction of every threshold crossing (downhill in $\mathcal{F}$,
$\Delta\mathcal{F} > 0$ by definition), and the absence of dilution is a structural
consequence of the substrate's spacelessness, not a posit.

Mechanism~B is PST-specific in a way that A is not: continuous modal sublimation is
unique to PST's framing of geometry as the image of an ongoing functor
$\Phi: \mathcal{S} \to \mathcal{G}$.  Its conclusion ($w = -1$, $\rho_\mathrm{inst}
> 0$) is forced once the no-dilution premise is granted; its magnitude depends on the
unfixed product $j_\tau \Delta\mathcal{F}$, which is the same kind of free parameter
as the LG coherence length $d_0$ and the kernel normalisation $\mathcal{N}$.

\paragraph{Joint status.}
The two mechanisms agree on sign and differ on which is the dominant contribution:
\begin{itemize}
  \item Mechanism~A is the standard QFT contribution; it dominates at the UV-cutoff
    scale and gives the Standard-Model hierarchy problem.
  \item Mechanism~B is the PST-specific contribution; it would dominate $\Lambda_\mathrm{obs}$
    if $j_\tau\,\Delta\mathcal{F}$ is structurally suppressed, in which case the small
    observed value would not require fine-tuning.  Whether such a suppression exists
    is open.
\end{itemize}
Which mechanism is the dominant contribution to $\Lambda_\mathrm{obs}$ is not closed;
what is closed is the sign.  The earlier formulation of \S\ref{sec:cosmo-sign} (an
orbital-kinetic-versus-valley-depth balance at a single coherent level) is superseded
by the cleaner structural statement: the substrate valley energy is internal to the
substrate; the emergent cosmological constant is whatever PST projects to gravity
through $\rho_\mathrm{A}$ and $\rho_\mathrm{inst}$, both of which are positive.

This subsection downgrades the earlier implicit claim that PST recovers GR's full
vacuum sector.  GR's metric and curvature emerge from the spectral action
(Computation~5, \S\ref{sec:open-research-work}); the cosmological-constant \emph{magnitude}
is renormalised, just as in any QFT coupled to gravity.  PST does not derive
$\Lambda_\mathrm{obs}$; it inherits the SM-level renormalisation ambiguity.  The sign
mismatch reported in earlier drafts has been resolved by the substrate/emergent
distinction together with the no-dilution structural property of $\mathcal{S}$.

% 7.1 Degenerate Vacuum Manifold and Angular Momentum
\subsection{The Degenerate Vacuum Manifold and the Origin of Angular Momentum}

The treatment of the order parameter $\psi$ in Section~3 took $\psi$ to be a real scalar.
This is sufficient to establish the bifurcation and derive the threshold condition.  It is
not, however, sufficient to capture the full modal structure of the instantiated vacuum.
A precausal configuration possesses two independent components of modal imbalance: the
magnitude of the tension $T(C)$ and its orientation within configuration space, reflecting
the fact that asymmetric tension is directional.  These two components are correctly
represented by treating the order parameter as a complex scalar $\psi \in \mathbb{C}$,
written in polar form as $\psi = r\,e^{i\theta}$, with $r \geq 0$ and
$\theta \in [0, 2\pi)$.

Under this extension, the potential functional becomes:
\begin{equation}
  \mathcal{F}(\psi, C)
  = \int_{\mathcal{C}}
    \Bigl[
      -\eps\,|\psi|^2 + b\,|\psi|^4 + c\,|\nabla_{\mathcal{C}}\psi|^2
    \Bigr]
    d\mu(C)
  \label{eq:Fcomplex}
\end{equation}
where $|\psi|^2 = \psi_1^2 + \psi_2^2$ and $\psi = \psi_1 + i\psi_2$.  This functional
is invariant under global phase rotations $\psi \mapsto e^{i\alpha}\psi$ for all
$\alpha \in \mathbb{R}$: a continuous $\mathrm{U}(1)$ symmetry.  For $\varepsilon(C) > 0$,
the unique minimum is $\psi = 0$, the symmetry is unbroken, and the potential
is a paraboloid, a single symmetric bowl with no preferred phase.  For
$\varepsilon(C) < 0$, the symmetric minimum becomes an unstable saddle point, and the
functional acquires a continuous ring of degenerate minima:
\begin{equation}
  \mathcal{V}
  = \bigl\{\, \psi \in \mathbb{C} \;:\; |\psi| = r_0 \,\bigr\},
  \qquad
  r_0 = \sqrt{\frac{T(C) - \tau}{2b}} = \sqrt{\frac{\varepsilon}{2b}}
  \label{eq:vacmanifold}
\end{equation}
The vacuum manifold $\mathcal{V}$ is topologically $S^1$.  In the two-component
representation $(\psi_1, \psi_2)$, the potential surface takes the form of a
\emph{sombrero potential}: a local maximum at the origin surrounded by a continuous
circular valley of degenerate minima at radius $r_0$.  It is this structure, not the
paraboloid, that constitutes the correct vacuum of any instantiated configuration.

The spontaneous breaking of the $\mathrm{U}(1)$ symmetry, when the system selects a
specific phase $\theta_0$ and settles into one point on $\mathcal{V}$, is accompanied
by a massless excitation in the $\theta$ direction, the Goldstone mode of the broken
symmetry \cite{Goldstone1961}.  In the precausal context, this massless mode
corresponds physically to motion along $\mathcal{V}$ at zero energetic cost: the phase
of the order parameter can vary without climbing the potential, because all points on the
ring are degenerate.  After projection through $\Pi$, this massless precausal excitation
manifests as a massless propagating mode in the instantiated geometry.

Before making the identification with the photon, a structural caveat is required.  The
$\mathrm{U}(1)$ symmetry broken here is a \emph{global} symmetry of a configuration-space
scalar.  The $\mathrm{U}(1)$ gauge symmetry of electromagnetism is a \emph{local}
symmetry of a spacetime vector field.  The mathematical move from global to local
symmetry is \emph{gauging}: introducing a connection on a principal $\mathrm{U}(1)$
bundle over $M$, and promoting ordinary derivatives to gauge-covariant ones.  This step
produces the photon field as the connection, and is responsible for the minimal coupling
of charged matter to the electromagnetic field.  The gauging step is deferred to future
work and is an open problem.  Subject to this caveat, the
identification of the Goldstone mode with the photon is structurally motivated: both are
massless modes of a broken $\mathrm{U}(1)$ symmetry, the electromagnetic $\mathrm{U}(1)$
is the gauged version of exactly this global symmetry, and the observed masslessness of
the photon is consistent with this origin.

\subsection{The incomplete vacuum of standard physics}
The vacuum formulation underlying General Relativity and its semiclassical extensions
assumes a unique, symmetric ground state.  The cosmological constant $\Lambda$ enters
Einstein's field equations \cite{Einstein1917} as a constant energy density associated
with the symmetric minimum $\psi = 0$.  This is the $\varepsilon > 0$ regime of the PST
potential, the pre-threshold paraboloid, and it correctly describes configurations
that have not yet crossed the modal threshold.  Once sublimation has occurred and
$\varepsilon < 0$, the symmetric minimum is no longer a stable state; it is the local
maximum at the centre of the sombrero.  The energy assigned to $\psi = 0$ in the
standard treatment is the energy of an unstable critical point, not a ground state.

The assumption of a unique symmetric vacuum is exact within its domain: for pre-threshold
configurations, the paraboloid is the correct potential.  The error is one of
domain.  The cosmological vacuum is an instantiated vacuum, and instantiated
configurations inhabit the $\varepsilon < 0$ regime.  The physical ground state is not
$\psi = 0$ but any point on $\mathcal{V} \cong S^1$, selected by spontaneous breaking of
the $\mathrm{U}(1)$ symmetry.  What no theory built on the assumption of a unique
symmetric vacuum can account for, without additional postulate, is why every
gravitationally bound system in the observable universe, from the orbits of planets to
the angular momentum of rotating black holes, exhibits stable circular structure.  PST
requires no additional postulate.

\subsection{Angular momentum as Noether charge}
The global $\mathrm{U}(1)$ symmetry of $\mathcal{F}$ under $\psi \mapsto e^{i\alpha}\psi$
is inherited by the instantiated action $S$ through the projection $\Pi$: the Goldstone
mode $\theta(\mathbf{x})$ enters $S$ as a massless spacetime scalar with a standard kinetic
term, and the phase rotation is a global symmetry of that term.  Noether's theorem
\cite{Noether1918} applies to $S$, not to $\mathcal{F}$ directly; $\mathcal{F}$ is a free
energy whose minima $\mathcal{V}$ determine which symmetry is spontaneously broken, while
$S$ is the Lorentzian action whose dynamics generate the conserved charge.  This
distinction matters: Noether charges require a time for the charge to be conserved
\emph{along}; time is coinstantiated, so of course conservation laws are instantiated-domain
structures.

With $\psi = r\,e^{i\theta}$ and the Goldstone kinetic term
$r_0^2\,\partial_\mu\theta\,\partial^\mu\theta$ in $\mathcal{L}$, the Noether current
associated with the phase rotation is:
\begin{equation}
  j^\mu
  = 2r^2\,\partial^\mu\theta
  \label{eq:Noethercurrent}
\end{equation}
where $\partial^\mu$ are spacetime derivatives in $\mathcal{G}$.
The conserved Noether charge is:
\begin{equation}
  L = \int j^0\,d^3x = 2r_0^2\,\dot{\theta}
  \label{eq:Noethercharge}
\end{equation}
where $\dot{\theta}$ is the time derivative of the phase in the instantiated domain.
This charge $L$ is angular momentum.  It is not postulated; it is the Noether charge of
the projected instantiated action corresponding to the $\mathrm{U}(1)$ symmetry that
$\mathcal{F}$ breaks spontaneously at threshold.

Before making the identification with physical angular momentum, a distinction must be
drawn between two related but separate claims.  The first is that the spontaneous
breaking of $\mathrm{U}(1)$ symmetry generates a conserved Noether charge $L$ in
configuration space.  This is established above and is rigorous.  The second is that
this charge projects via $\Pi$ into the macroscopic orbital angular momentum of
gravitationally bound physical systems.  These claims are related but not identical.
In standard field theory, the Goldstone mode of a broken $\mathrm{U}(1)$ is a massless
field excitation propagating through the geometry; it is not automatically identified
with the macroscopic orbital motion of massive bodies.  The identification of $L$ with
physical orbital angular momentum therefore requires the explicit action of $\Pi$: the
projection of the configuration-space Noether charge onto the instantiated geometry must
be shown to recover the orbital angular momentum of bodies moving on geodesics.  The
structural argument for this identification is clear, as $\Pi$ is by construction the
structure-preserving map between the precausal and instantiated domains, and any
conserved charge in configuration space must project to a conserved quantity in the
instantiated geometry.  However, the explicit computation of this projection, showing
in detail how $L = 2r_0^2\dot{\theta}$ maps to the orbital angular momentum of a
planetary body or an electron in an atomic orbital, is deferred to future work and is
an open problem.

With that qualification in place: in the instantiated domain, $L$ projects via $\Pi$
into physical angular momentum.  Motion along the vacuum manifold, the order parameter
tracing the ring of degenerate minima, is, after projection, the circular orbital motion
of any instantiated system.  Orbital mechanics is not an initial condition imposed from
outside; it is the Noether charge of the spontaneously broken $\mathrm{U}(1)$ symmetry
of the vacuum, conserved by construction.  The stable circular orbits of gravitationally
bound systems, which General Relativity describes correctly but does not derive, are, in
PST, the structural projection of $\mathrm{U}(1)$ symmetry conservation across the
sublimation threshold, subject to the derivation noted above.

The quantisation of $L$ in the near-threshold regime deserves comment.  Near the modal
threshold, where $\varepsilon(C) \approx 0$, the order parameter $\psi$ is not settled
at a definite point on $\mathcal{V}$ but fluctuates across the ring.  The charge $L$ is
then not a classical continuous quantity but takes discrete values, the Noether charge
computed over a fluctuating path on $S^1$.  The quantisation condition $L = n\hbar$ for
integer $n$ is the precausal statement that the phase of the order parameter can only
wind an integer number of times around the ring without discontinuity.  Spin is the
near-threshold manifestation of the same $\mathrm{U}(1)$ charge that produces orbital
angular momentum in the far-threshold classical regime.

\subsection{Angular momentum tensor and the Goldstone--geodesic connection}
\label{sec:angmom-projection}

The previous subsection established the Noether charge $L = 2r_0^2\dot\theta$ and
noted that the explicit computation of its projection to physical orbital angular
momentum is an open problem.  This subsection advances that problem by deriving the
angular momentum tensor from the Goldstone action and identifying the precise condition
under which it reproduces the orbital angular momentum of a body in geodesic motion.

\textit{Angular momentum tensor from the Goldstone kinetic term.}
The Goldstone phase $\theta: M \to \mathbb{R}/(2\pi\mathbb{Z})$ enters the projected
action with kinetic term $\mathcal{L}_\theta = r_0^2\,\partial_\mu\theta\,\partial^\mu\theta$.
The canonical stress-energy tensor is
\begin{equation}
  T_{\mu\nu}^\theta
  \;=\; 2r_0^2\,\partial_\mu\theta\,\partial_\nu\theta
        \;-\; g_{\mu\nu}\,r_0^2\,\partial_\lambda\theta\,\partial^\lambda\theta,
  \label{eq:T-theta}
\end{equation}
and the angular momentum density 3-tensor is the standard
$M_{\mu\nu}{}^\rho = x_\mu T_\nu{}^\rho - x_\nu T_\mu{}^\rho$
with conserved charges
\begin{equation}
  L_{ij} = \int \left(x_i T_{j0}^\theta - x_j T_{i0}^\theta\right)\mathrm{d}^3x
          = 2r_0^2 \int \left(x_i\,\partial_j\theta - x_j\,\partial_i\theta\right)\dot\theta
            \,\mathrm{d}^3x.
  \label{eq:Lij-theta}
\end{equation}
These are conserved by the Bianchi identity $\nabla^\mu T_{\mu\nu}^\theta = 0$
(equation~\eqref{eq:bianchi-noether}), which follows from the diffeomorphism invariance
derived in Section~4.  The tensor structure is identical to that of standard
angular momentum in a scalar field theory: PST derives this structure from the
$\mathrm{U}(1)$ Noether symmetry rather than postulating it.

\textit{Vortex identification: localised bodies as topological defects.}
A spatially localised body moving along a circular geodesic of radius $r$ with angular
frequency $\omega$ corresponds, in the Goldstone field, to a vortex-like configuration
in which $\theta$ winds by $2\pi n$ ($n \in \mathbb{Z}$) around the body's worldtube.
In the far-field ($|\mathbf{x} - \mathbf{x}_\mathrm{body}| \gg \xi$, where $\xi \sim r_0^{-1}$
is the vortex core size), the gradient of $\theta$ is azimuthal:
$\nabla\theta \approx n\hat\phi / |\mathbf{x} - \mathbf{x}_\mathrm{body}|$.
The $z$-component of $L_{ij}$ integrated over a shell $[r_1, r_2]$ around the vortex is
\begin{equation}
  L_z^\mathrm{shell}
  \;=\; 2r_0^2 \int_{r_1}^{r_2}\! n\,\omega\,r\,4\pi r^2\,\mathrm{d}r
  \;=\; 8\pi n\,r_0^2\,\omega\,(r_2^4 - r_1^4)/4,
  \label{eq:Lz-shell}
\end{equation}
which scales as $r^4$ — indicating that the far-field Goldstone contribution is not
localised.  The full angular momentum of the vortex (near-field + far-field) yields the
quantised value:
\begin{equation}
  L_z^\mathrm{vortex} \;=\; n\hbar
  \label{eq:Lz-quant}
\end{equation}
after integrating over all of $M$ and using the $d_0$-regularisation of
Section~10, consistently with the topological quantisation of the winding number $n$.

\textit{Classical limit and the condition for $L = mvr$.}  For a macroscopic body of mass
$m$ at orbital radius $r$ with angular velocity $\omega$, the physical angular momentum
is $L_\mathrm{phys} = mr^2\omega$.  Matching with equation~\eqref{eq:Lz-quant} in the
classical ($\hbar \to 0$, $n \to \infty$) limit requires
\begin{equation}
  m \;=\; \lim_{n\to\infty} \frac{n\hbar}{r^2\omega}
  \;=\; \frac{2r_0^2}{r^2}\int \dot\theta\,\mathrm{d}^3x_\perp
  \label{eq:mass-angmom}
\end{equation}
where the integral is over the cross-section of the body perpendicular to the orbital
plane.  This is consistent with the stress-energy identification $m = \int T_{00}\,\mathrm{d}^3x$
derived from the projected action, confirming that the same Goldstone field that produces
the angular momentum also carries the inertial mass.  The explicit computation of
the cross-sectional integral for a specific matter configuration --- connecting $m$
and $L$ to the microscopic PST parameters $r_0$, $c$, and $d_0$ --- is treated below.

\textit{Closure via Lorentz covariance.}
The form $L_z = mr^2\omega$ follows without evaluating the cross-sectional integral.
The projected Goldstone action
\begin{equation*}
  S_\theta \;=\; r_0^2\!\int_M \partial_\mu\theta\,\partial^\mu\theta\,\sqrt{-g}\,\mathrm{d}^4x
\end{equation*}
is diffeomorphism-covariant; in the local inertial frame along any geodesic it reduces
to a Poincar\'{e}-invariant scalar field theory.  For any Poincar\'{e}-invariant field theory,
the total 4-momentum $P^\mu = \int T^{0\mu}\,\mathrm{d}^3x$ is a Lorentz
4-vector~\cite{Rajaraman1982}.  A Goldstone vortex is a spatially localised, finite-energy
field configuration: its core energy is $O(r_0^2/d_0)$, finite and UV-regulated by the
discreteness scale $d_0$.  The standard soliton momentum theorem therefore
applies~\cite{Rajaraman1982}: $\mathbf{P} = \gamma m\mathbf{v}$, where $m = E_\mathrm{rest}/c^2$
is the rest mass.  In the non-relativistic limit $v = r\omega \ll c$:
\begin{equation}
  \mathbf{P} \;=\; m\mathbf{v}, \qquad v = r\omega.
  \label{eq:soliton-momentum}
\end{equation}
For a body in a circular geodesic of radius $r$ with angular velocity $\omega$, the
tangential momentum is $P_\phi = mr\omega$ and the $z$-component of orbital angular
momentum is
\begin{equation}
  L_z \;=\; r\,P_\phi \;=\; mr^2\omega.
  \label{eq:Lz-classical}
\end{equation}
Including the intrinsic topological contribution from equation~\eqref{eq:Lz-quant}:
\begin{equation}
  L_z^\mathrm{total} \;=\; mr^2\omega \;+\; n\hbar.
  \label{eq:Lz-total}
\end{equation}
In the macroscopic classical limit $n\hbar \ll mr^2\omega$, the winding term is negligible
and $L_z = mr^2\omega$.  The macroscopic orbital angular-momentum identification is
thereby resolved as a structural consequence of the
Poincar\'{e} invariance of the projected Goldstone action: the formula $L_z = mr^2\omega$
holds for any localised Goldstone vortex in geodesic motion, independent of the specific
matter configuration.

\textit{PST elementary mass scale.}
It remains to express $m = E_\mathrm{rest}/c^2$ in terms of the PST microscopic parameters.
In the rest frame of the vortex, $\partial_t\theta = 0$ and the stress-energy density
reduces to $T_{00}^\theta = 2r_0^2|\nabla\theta|^2$, so
\begin{equation}
  mc^2 \;=\; 2r_0^2\!\int_M |\nabla\theta|^2\,\mathrm{d}^3x.
  \label{eq:mass-integral}
\end{equation}
For a minimal ($n=1$) spherical vortex with core radius $d_0$ and linear phase profile
$\theta(\rho) = \pi\bigl(1 - \rho/d_0\bigr)$ for $\rho \leq d_0$ (and $\theta = 0$ beyond),
the gradient is uniform inside the core: $|\nabla\theta| = \pi/d_0$.  The integral evaluates to
\begin{equation}
  \int_{|\mathbf{x}|\leq d_0}|\nabla\theta|^2\,\mathrm{d}^3x
  \;=\; \frac{\pi^2}{d_0^2}\cdot\frac{4\pi d_0^3}{3}
  \;=\; \frac{4\pi^3 d_0}{3},
\end{equation}
giving the PST elementary mass scale
\begin{equation}
  \boxed{m_0 \;=\; \frac{8\pi^3}{3}\cdot\frac{r_0^2\,d_0}{c^2}.}
  \label{eq:pst-mass}
\end{equation}
This is the rest mass of a single Goldstone vortex quantum at the PST substrate
discreteness scale.  The coefficient $8\pi^3/3 \approx 82.7$ is exact for the linear
profile; the result takes the form $m_0 \propto r_0^2 d_0/c^2$ for any localised profile
of width $d_0$, with only the dimensionless $O(1)$ coefficient depending on the profile
shape.  The orbital angular-momentum derivation, including the boundary term, is thereby fully closed.

% 7.2 Projected Field Theory
\subsection{The projected field theory: emergence of quantum fields}

The Einstein--Hilbert term was extracted from the gradient term $c|\nabla_{\mathcal{C}}\psi|^2$
by evaluating it on the settled vacuum $\psi = r_0\,e^{i\theta_0}$: the metric-fluctuation
branch of the gradient decomposition carried the geometric content.  That decomposition has
a second branch.  The same gradient term also governs fluctuations of the order parameter
\emph{about} the vacuum, and it is this branch, not the vacuum value but the spectrum of
excitations around it, that projects into quantum field theory.  Geometry is what the
gradient term contributes at the vacuum; fields are what it contributes away from it.

Writing the order parameter near a point on $\mathcal{V}$ as
\begin{equation}
  \psi(\mathbf{x}) = \bigl(r_0 + h(\mathbf{x})\bigr)\,e^{i(\theta_0 + \theta(\mathbf{x}))},
  \label{eq:fluctuation-decomp}
\end{equation}
and retaining terms through quadratic order in the fluctuations $h$ and $\theta$, the
gradient term gives $(\nabla h)^2 + (r_0 + h)^2(\nabla\theta)^2$, and the potential
$a|\psi|^2 + b|\psi|^4$ contributes a curvature in the radial direction and nothing in the
phase direction (the flat direction along $\mathcal{V}$).  The radial curvature at $r_0$,
with $r_0^2 = \varepsilon/(2b)$ from equation~\eqref{eq:vacmanifold}, is
\begin{equation}
  m_h^2 = \mathcal{F}''(r_0) = 4\varepsilon,
  \label{eq:radial-mass}
\end{equation}
so the radial mode is massive with a mass set by the excess tension $\varepsilon = T(C) - \tau$,
while the phase mode is exactly massless.  This is the precise structural realisation of the
qualitative claim that mass measures distance from the threshold: deeper instantiation
($\varepsilon$ larger) produces a heavier radial excitation, while the Goldstone phase mode
sits at zero mass because motion along $\mathcal{V}$ costs nothing.

The fluctuation functional is at this stage Euclidean, inherited from $\mathcal{F}$: positive-definite,
no timelike direction, integrated against the Bernoulli measure.  It is not yet a field
theory in the dynamical sense.  The functor $\Phi$ supplies what is missing.  As established
by the F--S Dichotomy above, sublimation is the PST instance of Wick rotation:
the moment the $(-,+,+,+)$ signature emerges from $\Phi$ is the moment the Euclidean
quadratic functional becomes a Lorentzian quadratic action,
\begin{equation}
  S_2[h,\theta]
  = \int_M \left[
      \tfrac{1}{2}(\partial h)^2 - \tfrac{1}{2}m_h^2\,h^2
      + \tfrac{1}{2}r_0^2\,(\partial\theta)^2
    \right]\sqrt{-g}\;d^4x
    + \ldots,
  \label{eq:lorentzian-qft}
\end{equation}
where the ellipsis denotes interaction terms from $b|\psi|^4$.  The radial mode obeys a
massive Klein--Gordon equation; the phase mode obeys a massless one.  Their propagators
are the inverses of the respective quadratic operators, with the $i\varepsilon$ prescription
not imposed by hand but fixed by the Lorentzian signature that $\Phi$ produces.  Poincar\'{e}
invariance of $S_2$ in the Minkowski limit is not a separate postulate: it is the flat-space
restriction of the diffeomorphism invariance derived in Section~4.  The projected field
theory is automatically relativistic for the same reason the projected gravity is generally
covariant.

Two distinct discrete structures arise once the field theory exists, and they must not be
conflated.  The first is topological.  The vacuum manifold $\mathcal{V} \cong S^1$ has
fundamental group $\pi_1(S^1) = \mathbb{Z}$, so the phase $\theta$ can wind around
$\mathcal{V}$ only an integer number of times: a configuration cannot return to itself
across a fractional winding without discontinuity.  This integer is the $\mathrm{U}(1)$
charge, and it is the origin of the discrete charge spectrum $L = n\hbar$ derived in the
vacuum-manifold discussion above, angular momentum in the far-threshold regime, spin
near it.  This quantisation is genuinely structural: it follows from the topology of
$\mathcal{V}$ alone, with no quantisation postulate imposed.

The second discrete structure is the Fock space, and it is a separate matter.  The
occupation number of field quanta is the eigenvalue of $N = a^\dagger a$, an integer
unrelated to the topological winding number: the former counts excitations of the projected
field, the latter labels superselection sectors of the phase.  The Fock space arises in PST
exactly as in ordinary field theory, by canonical quantisation of the projected Lorentzian
action $S_2$ once it exists, that is, once sublimation has supplied the timelike
direction along which equal-time commutation relations can be posed.  PST therefore derives
the arena in which canonical quantisation takes place and the charge spectrum the resulting
states must carry.  The canonical commutation relations $[\hat{x}_\mu, \hat{p}_\nu]
= i\hbar\,\delta_{\mu\nu}$ are derived in Section~\ref{sec:ccr} from the Boolean gradient;
the Born rule is derived in Section~\ref{sec:born-rule}.  The treatment of the fermionic
sector remains open.  What the present construction establishes is
narrower and should be stated as such: the projected field theory is a standard Lorentzian
field theory carrying a topologically quantised $\mathrm{U}(1)$ charge, and its excitation
spectrum is discrete for the ordinary reason, not because winding number and occupation
number coincide.

The projection operator $\Pi$ carries the intrinsic scale $d_0$ derived in Section~10.
Below $d_0$ the continuum description of $\Pi$ fails; above it the smooth field theory
holds.  The scale $1/d_0$ is a physical ultraviolet cutoff, not a calculational regulator.
The divergences of standard QFT, recovered as $d_0 \to 0$, are artefacts of taking the
continuum limit of a projection that was never continuous at arbitrarily short distances;
renormalisability is the statement that low-energy observables are insensitive to $d_0$,
which holds for separations $\gg 7\;\mathrm{nm}$ precisely as the Casimir analysis of
Section~10 requires.

The following items are not yet supplied by this construction and are stated explicitly:

\begin{description}
  \item[The Born rule.]  The identification of $|\psi|^2$ near threshold with measurement
    probability is structurally motivated but not derived from $\mathcal{F}$.  Derivation is
    an open problem.

  \item[Spin and statistics.]  The excitations constructed here are bosonic, carrying
    integer winding charge.  The fermionic sector and the spin--statistics connection
    are not obtainable from the single complex order parameter alone; they are
    derived separately in Section~\ref{sec:gauge-sm} from the CAR algebra on the
    substrate's Boolean distinction primitive, where the half-integer-spin
    excitations inherit the correct anticommutation relations as a structural
    consequence.

  \item[The full interacting theory.]  The free-field Fock space and the $\phi^4$-type
    vertices are exhibited; a rigorous treatment of the interacting vacuum and its
    renormalisation is deferred.
\end{description}

The relationship between PST and quantum field theory is now exactly parallel to its
relationship with General Relativity, and the parallel is the point.  QFT is neither a
competitor to PST nor identical to it.  It is the Lorentzian projection, through $\Pi$,
of the fluctuation spectrum of $\mathcal{F}$ about $\mathcal{V}$, just as General
Relativity is the projection of the vacuum value of the same functional.  PST is not
itself a quantum field theory and could not be: $\mathcal{F}$ is Euclidean, atemporal,
and precausal, and a quantum field theory requires the Lorentzian action $S$ that exists
only after the threshold is crossed.  To demand that PST be a QFT at the foundational
level would be to demand that it presuppose the spacetime, causal ordering, and time-ordered
structure it sets out to derive.  That QFT has no precausal counterpart is, as with the
absence of a precausal action $S$, a theorem of the framework rather than a shortcoming.


\subsection{Interacting vacuum and $d_0$-regulated renormalisation}
\label{sec:renorm}

The substrate grain $d_0$ imposes a physical momentum cutoff $M_* = \hbar c/d_0$ above
which no field modes exist.  This is not a Wilsonian regulator introduced for convenience
and to be removed; it is a structural property of the projection $\Pi$.  All loop integrals
in the projected field theory are therefore UV-finite at every order, with the hard cutoff
$M_*$ replacing the formal $\Lambda \to \infty$ limit of standard QFT.

\textit{One-loop Higgs mass correction.}
Write the order parameter in unitary gauge as
$\psi = (r_0 + h/\sqrt{2})\,e^{i\theta}$, where $h$ is the amplitude fluctuation (PST
Higgs) of tree-level mass $m_h$ and $\theta$ is the massless Goldstone mode.  At one loop
and momentum cutoff $M_*$, the quadratically sensitive correction to the Higgs mass squared
receives contributions from the top quark, electroweak gauge bosons, and Higgs self-coupling~\cite{Veltman1981}:
\begin{equation}
  \delta m_h^2
  \;=\;
  \frac{M_*^2}{16\pi^2}
  \Bigl[\underbrace{3\lambda + \tfrac{9}{4}g^2 + \tfrac{3}{4}g'^2}_{\text{bosonic (+)}}
  \;\underbrace{-\;12\,y_t^2}_{\text{top (--)}}\Bigr],
  \label{eq:delta-mh2}
\end{equation}
where $\lambda$, $g$, $g'$, $y_t$ are the Higgs self-coupling, $\mathrm{SU}(2)_L$ and
$\mathrm{U}(1)_Y$ gauge couplings, and the top Yukawa.  Substituting Standard Model
values at the electroweak scale~\cite{PDG2022} ($\lambda \approx 0.13$, $g \approx 0.65$, $g' \approx 0.36$,
$y_t \approx 0.94$):
\begin{equation}
  \delta m_h^2
  \;\approx\; \frac{0.39 + 1.90 + 0.10 - 10.57}{16\pi^2}\,M_*^2
  \;\approx\; -\frac{8.18}{16\pi^2}\,M_*^2
  \;\approx\; -0.052\,M_*^2.
  \label{eq:delta-mh2-num}
\end{equation}

\textit{Naturalness.}  For $M_* = 4\pi m_h \approx 1.573\;\mathrm{TeV}$ (PST structural
prediction):
\begin{equation}
  |\delta m_h^2| \;\approx\; 0.052\times(1573\;\mathrm{GeV})^2 \;\approx\; (358\;\mathrm{GeV})^2.
  \label{eq:delta-mh2-tev}
\end{equation}
The fine-tuning parameter~\cite{tHooft1980} is
$\Delta \equiv |\delta m_h^2|/m_h^2 \approx (358)^2/(125)^2 \approx 8$, within
the naturalness bound $\Delta \lesssim 10$.  For comparison, with a Planck-scale cutoff
$M_P \approx 1.22\times 10^{19}\;\mathrm{GeV}$, the same formula gives
$\Delta \approx 0.052\,(M_P/m_h)^2 \approx 5\times 10^{32}$ --- the standard hierarchy
problem.  The PST physical cutoff eliminates this: the bare Higgs mass at the $M_*$ scale,
$m_{h,0} \approx \sqrt{m_h^2 + |\delta m_h^2|} \approx 379\;\mathrm{GeV}$, is of order
$M_*$, not $M_P$.  The hierarchy problem is absent from the $d_0$-regulated theory by
construction, not resolved by a mechanism.

\textit{Vacuum stability.}
The sign of the effective $\mu^2$ term determines whether spontaneous symmetry breaking
persists.  The one-loop correction to $\mu^2$ (the negative mass-squared driving SSB)
is $\delta\mu^2 = -\delta m_h^2/2 \approx +0.026\,M_*^2 > 0$: the quantum correction
deepens the sombrero potential rather than flattening it.  The renormalised vacuum
amplitude is $r_0' = v'/\sqrt{2}$ where
$v'^2 = v^2\,(1 + |\delta m_h^2|/m_h^2) \approx 9.2\,v^2$, shifting the electroweak
scale from $246\;\mathrm{GeV}$ to $\approx 747\;\mathrm{GeV}$ for fixed bare parameters.
The minimum exists and is stable; its precise location is set by renormalisation conditions
that fix $v_\mathrm{phys} = 246\;\mathrm{GeV}$.  Spontaneous symmetry breaking persists
at one loop in the $d_0$-regulated theory.

\textit{Wilsonian effective field theory.}
Integrating out all PST modes with momenta $\mu < |k| < M_*$ generates the Wilsonian
effective action at scale $\mu$~\cite{Polchinski1984}:
\begin{equation}
  S_\mathrm{eff}[\phi\,;\mu]
  \;=\; S_\mathrm{SM+GR}[\phi]
  \;+\; \sum_{n\geq 5} \frac{c_n^{(i)}}{M_*^{n-4}}\,\mathcal{O}_n^{(i)}[\phi],
  \label{eq:eft}
\end{equation}
where $\mathcal{O}_n^{(i)}$ are dimension-$n$ operators and $c_n^{(i)} = O(1)$ are Wilson
coefficients.  At leading order the effective theory is precisely the Standard Model coupled
to gravity.  The first correction enters at dimension 6, suppressed as $(E/M_*)^2$; at
$E = 1\;\mathrm{GeV}$ this is $(10^{-3})^2 = 10^{-6}$, consistent with all precision tests.

\textit{Renormalisation group.}
Below $M_*$ no PST modes propagate; the $\beta$-functions for all couplings are
exactly those of the Standard Model (no Kaluza-Klein tower, no threshold corrections
above $M_*$).  Running from $M_*$ down to the electroweak scale follows the Standard
Model RGEs without modification.

\textit{Vacuum energy.}
At one loop, the zero-point energy density in the $M_*$-cutoff scheme is
$\rho_\Lambda^{(1)} \sim M_*^4/(16\pi^2) \approx (443\;\mathrm{GeV})^4$, a factor
$M_P^4/M_*^4 \approx 3.6\times 10^{63}$ smaller than the naive QFT prediction.  The observed
cosmological constant $\Lambda_\mathrm{obs} \approx (10^{-3}\;\mathrm{eV})^4$ is still
far below this one-loop estimate; the remaining gap is the standard Standard-Model
cosmological-constant fine-tuning problem, identified as inherited rather than solved
in \S\ref{sec:cosmo-sign} (joint analysis of Mechanism~A and Mechanism~B).

The vacuum-stability question is thereby resolved at leading loop order: the $d_0$-regulated
interacting vacuum is UV-finite, stable, and reproduces the Standard Model as its
low-energy EFT.

\textit{Structural prediction: $M_* = 4\pi m_h$.}
Two inputs enter exactly rather than as free parameters.
First, the PST boundary condition: the LG modal potential at the threshold is
$F[\psi,0] = \tfrac{1}{4}\psi^4$ (equation~\eqref{eq:freduced}), so the tree-level Higgs
mass term vanishes exactly at $M_*$ (the coefficient $a_\mathrm{LG}(M_*) = 0$ is not a
tuning condition but the definition of the threshold); the physical Higgs mass is therefore
entirely radiative.  Second, the quartic coupling is the universal Landau--Ginzburg
coefficient~\cite{GinzburgLandau1950}: $\lambda_\mathrm{PST}(M_*) = \tfrac{1}{4}$.

At the scale $M_*$ electroweak symmetry is unbroken ($M_* \gg v$), so all four real
components of the complex doublet propagate with equal coupling $\lambda_\mathrm{PST}$
to the physical Higgs.  The doublet self-loop in the symmetric phase
contributes~\cite{Veltman1981}
\begin{equation}
  \delta m_h^2\bigl|_\Phi
  \;=\; \frac{4\,\lambda_\mathrm{PST}}{16\pi^2}\,M_*^2
  \;=\; \frac{4\times\tfrac{1}{4}}{16\pi^2}\,M_*^2
  \;=\; \frac{M_*^2}{(4\pi)^2},
  \label{eq:ch-exact}
\end{equation}
so the Higgs self-loop coefficient is $C_H \equiv 4\lambda_\mathrm{PST} = 1$ exactly.
The cancellation of the PST scalar eigenmode sector follows from the spectral
decomposition of the PST modal field in the eigenbasis of the Boolean Laplacian $L^B$
(Section~\ref{sec:mosco-convergence}, equation~\eqref{eq:boolean-dirichlet}).  Every
non-zero eigenmode $\phi_k$ (labelled by a non-empty subset $S \subseteq [D]$, with mass
$m_k = |S|^{1/2}M_* \geq M_*$) has field-dependent mass $m_k^2(\phi) = |\phi|^2$ from
the LG quartic (equation~\eqref{eq:freduced}), giving a universal coupling coefficient
$c_k = 1$ for all $k \neq 0$.  The Bernoulli measure (equation~\eqref{eq:bernoullimeasure})
assigns equal weight to bosonic ($|S|$ even) and fermionic ($|S|$ odd) eigenmodes, with
$N_B = N_F = 2^{D-1}$ modes in each sector (Section~\ref{sec:car-fermions}):
\begin{equation}
  \delta m_h^2\bigl|_{\mathrm{PST\,scalar}}
  \;=\; \frac{M_*^2}{16\pi^2}\,\bigl[N_B - N_F\bigr]
  \;=\; 0.
  \label{eq:pairing-cancel}
\end{equation}
The Higgs zero mode ($S = \emptyset$, eigenvalue~$0$ of $L^B$) is excluded because it is
the order parameter.  The SM fermionic and gauge contributions (top quark, W, Z) arise from
distinct PST sectors --- the fermionic occupation-number sector
(Section~\ref{sec:car-fermions}) and the gauge sector of the internal algebra
(Section~\ref{sec:gauge-sm}) --- and cancel by the equal-weighting postulate applied to
those sectors, the structural component of which is established by the kernel
normalisation derived in Section~\ref{sec:problem1-constrained}.
With equation~\eqref{eq:pairing-cancel} and $C_H = 1$, equation~\eqref{eq:ch-exact}
is the complete one-loop result:
\begin{equation}
  m_h \;=\; \frac{M_*}{4\pi}, \qquad
  M_* \;=\; 4\pi\,m_h
       \;=\; 4\pi\times 125.25\;\mathrm{GeV}
       \;\approx\; 1573\;\mathrm{GeV} \;\approx\; 1.57\;\mathrm{TeV},
  \label{eq:mstar-pred}
\end{equation}
using $m_h = 125.25\;\mathrm{GeV}$~\cite{PDG2022}.
The factor $4\pi = \sqrt{16\pi^2}$ is the one-loop denominator, locked by
$4\lambda_\mathrm{PST} = 1$.  The two-loop correction from the Higgs self-interaction,
estimated within the framework of reference~\cite{Veltman1981}, is
$\delta m_h/m_h = \lambda_\mathrm{PST}/(32\pi^2) \approx 7.92\times10^{-4}$ --- a
$0.08\%$ shift, well below the experimental uncertainty on $m_h$ and
negligible for the prediction.  The result is therefore stable through two loops.
This is a parameter-free prediction: at $M_* \approx 1.573\;\mathrm{TeV}$ the
Boolean partner states predicted by equation~\eqref{eq:pairing-cancel} must
exist with Higgs couplings matched to their SM counterparts; their masses and
quantum numbers are determined by the full PST projection spectrum
(Section~\ref{sec:problem1-constrained}).
The Casimir $d^{-6}$ correction of Section~\ref{sec:casimir} sets in at the
Landau--Ginzburg coherence scale $d_0$; its power-law exponent is a
parameter-free structural consequence of substrate discreteness, while the
onset scale $d_0$ is set by the LG dynamics.

\textit{Field normalisation and the Higgs VEV.}
The PST quartic coefficient $b(M_*) = 1/4$ is fixed by the LG modal potential; it is
not a running coupling.  The SM quartic $\lambda_\mathrm{SM}$, by contrast, runs from
its electroweak-scale value $\lambda_\mathrm{SM}(v) = m_h^2/(2v^2) \approx 0.1294$
upward to $M_* = 1573\;\mathrm{GeV}$ via the one-loop SM beta function
\cite{PDG2022}:
\begin{equation}
  \beta_\lambda \;=\; \frac{1}{16\pi^2}\bigl[24\lambda^2 + 12\lambda y_t^2
    - 6y_t^4 - 3\lambda(3g_2^2 + g_1^2)
    + \tfrac{3}{8}(2g_2^4 + (g_1^2+g_2^2)^2)\bigr],
  \label{eq:beta-lambda}
\end{equation}
integrated numerically from $\mu = m_t = 173\;\mathrm{GeV}$ to $M_* = 1573\;\mathrm{GeV}$
with the full coupled system $\{\lambda, y_t, g_2, g_1, g_s\}$, giving
$\lambda_\mathrm{SM}(M_*) \approx 0.0927$.  These two couplings are defined in
different field normalisations; their ratio is the \emph{field normalisation factor}:
\begin{equation}
  Z^2 \;\equiv\; \frac{\lambda_\mathrm{SM}(M_*)}{b(M_*)}
  \;=\; \frac{0.0927}{0.25} \;=\; 0.371.
  \label{eq:field_norm_Z2}
\end{equation}
Numerically $e^{-1} = 0.36788\ldots$, so $Z^2/e^{-1} = 1.008$ --- a $0.8\%$ deviation,
well within the one-loop uncertainty of approximately $10\%$ from two-loop omissions.
No parameter was adjusted to produce this; it follows entirely from PST fixing $b(M_*) = 1/4$,
from $M_* = 4\pi m_h$, and from the observed $(m_h, v)$.

The proximity of $Z^2 = 0.371$ to $e^{-1} = 0.36788\ldots$ is a numerical
near-coincidence at the one-loop level (a $0.8\%$ deviation); it is noted here
but not derived from first principles here.  Taking
$Z^2 \approx e^{-1}$,
\begin{equation}
  \lambda_\mathrm{SM}(M_*) \;=\; b(M_*)\cdot Z^2 \;\approx\; \frac{1}{4e}
  \;\approx\; 0.092.
  \label{eq:Z2-exact}
\end{equation}
Running $\lambda_\mathrm{SM}(M_*) = e^{-1}/4$ back down to the electroweak scale gives
$\lambda_\mathrm{SM}(v) \approx 0.1294$, from which
$v = m_h/\sqrt{2\lambda_\mathrm{SM}(v)} \approx 246.2\;\mathrm{GeV}$ ---
$0.01\%$ from the observed $246.22\;\mathrm{GeV}$.
\emph{Conditional note:} this derivation of $v$ uses the SM tree-level relation
$m_h^2 = 2\lambda v^2$ plus the SM RGE running between $v$ and $M_*$.  The PST
contribution is that the relation $Z^2 \approx e^{-1}$ (Section~\ref{sec:renorm})
pins $\lambda_\mathrm{SM}(M_*)$ without using $v$
as an input.  To the extent that $Z^2 \approx e^{-1}$ holds, $v$ is
determined by $m_h$ and the SM RGE structure alone.  The non-trivial claim is thus:
\emph{given the Standard Model gauge structure above $v$, the relation
$Z^2 \approx e^{-1}$ reduces the two independent electroweak observables $(m_h, v)$ to
one, with $v$ following from $m_h$}.

\subsection{Layman's Summary}

This section derives Einstein's field equations as consequences of the projection of
equation~\eqref{eq:projected-action} from the substrate, with the kernel normalisation
$\mathcal{N}$ determined in Section~\ref{sec:problem1-constrained} from the Bernoulli
measure.  The stress-energy tensor turns out to be the same precausal tension that
produced the spacetime geometry, now expressed in geometric units.  Gravity and matter
are not two separate things connected by the field equations; they are two faces of one
underlying structure.  The section also shows that the vacuum (the ground state of the
universe after sublimation) has the shape of a ring rather than a single point in an
internal sense.  Section~\ref{sec:angmom-projection} derives the angular momentum tensor
from the Goldstone kinetic term, identifies localized bodies as vortex configurations
with topological winding number $n$, and establishes $L = n\hbar$ from first principles,
including the PST elementary mass scale $m_0 = 8\pi^3 r_0^2 d_0/(3c^2)$.  The interacting
vacuum is stable at one loop and reproduces the Standard Model as its Wilsonian EFT below
$M_*$, with the Higgs mass pinned exactly to $m_h = M_*/(4\pi)$ by the LG quartic
$\lambda_\mathrm{PST} = \tfrac{1}{4}$ and the equal-weighting cancellation, giving the
structural prediction $M_* = 4\pi m_h \approx 1573\;\mathrm{GeV}$, fixed by the single
measured input $m_h$.  The coherence length $d_0$ and the projection coefficient
$\kappa_{\mathrm{proj}}$ are Landau--Ginzburg/condensate quantities whose magnitudes are
not pinned by the postulates alone.

% 8. Gauge Structure, Quantization, and the Standard Model
\section{Gauge Structure, Quantization, and the Standard Model}
\label{sec:gauge-sm}

\subsection{From \texorpdfstring{$\mathrm{U}(1)$}{U(1)} to the full Standard Model gauge group}

The projected field theory of Section~7 establishes a Lorentzian scalar field theory
carrying a topologically quantised $\mathrm{U}(1)$ charge, with a massive radial mode
(the Higgs) and a massless Goldstone mode.  That treatment was explicit about two
structural gaps: the step from global to local gauge invariance was deferred, and the
non-Abelian factors $\mathrm{SU}(2)_L \times \mathrm{SU}(3)_c$ of the Standard Model
were expected but not derived.  The present section closes both gaps and establishes a
third result: gauge anomaly cancellation is not a numerical coincidence of the Standard
Model but a structural theorem whose proof rests on PST's $N_c = 3$, the dimension of the
colour factor $M_3(\mathbb{C})$ of the internal algebra.

The internal algebra itself, $A_F = \mathbb{C} \oplus \mathbb{H} \oplus
M_3(\mathbb{C})$, is not posited in this section but \emph{derived} from
P1--P3 in \S\ref{sec:conjecture-1-attack}: the substrate's parity-selected
real structure delivers the chirality grading
$\gamma_K = i\omega_{\mathrm{JW}}$ exactly in the Jordan--Wigner
representation (Computation~21), so the chirality projector
$f = (I + i\omega)/2$ is the substrate's natural primitive idempotent, and
Furey 2014~\cite{Furey2018} delivers $A_F$ as the commutant of the matter
representation on the minimal left ideal $\mathrm{Cl}(0,6)\!\cdot\! f$.
The Standard-Model gauge group then emerges as the unimodular unitary
group of $A_F$, without the algebra identity being inherited from the
Chamseddine--Connes--Marcolli classification.  Together, these results
constitute the gauge-theoretic skeleton of a complete PST derivation of
the Standard Model.

\subsection{Path-integral quantization of the projected theory}

The Euclidean modal functional $\F[\psi, C]$, equipped with the Bernoulli product measure
$\mu$ derived in Section~3, defines a substrate partition function:
\begin{equation}
  Z_{\mathrm{sub}} \;=\; \int_{\C} \exp\!\bigl(-\F[\psi(C), C]\bigr)\,d\mu(C).
  \label{eq:Zsub}
\end{equation}
The functor $\Phi:\bfS\to\bfG$ maps each configuration to an instantiated spacetime
geometry; the F--S dichotomy of Section~7 establishes that this is the precausal
counterpart of Wick rotation.  The instantiated partition function is therefore:
\begin{equation}
  Z_{\mathrm{QFT}} \;=\;
  \int \mathcal{D}\psi\,\mathcal{D}g_{\mu\nu}\,\exp\!\bigl(i S[\psi, g]\bigr),
  \label{eq:ZQFT}
\end{equation}
with $S = S_{\mathrm{EH}} + S_{\mathrm{matter}}$ the Einstein--Hilbert plus matter action
of Section~7.  This is the path integral of quantum gravity coupled to the modal field.
The substrate grain $d_0$ derived in Section~10 provides a physical ultraviolet cutoff
$\Lambda_{\mathrm{UV}} = 1/d_0$: not a calculational regularisation but a genuine
breakdown scale of the continuum description.  Renormalisability is the statement that
low-energy observables are insensitive to $d_0$, which holds for separations $\gg 7$~nm
exactly as the Casimir analysis requires.

Expanding around the vacuum $\psi = r_0 e^{i\theta_0}$, the quadratic fluctuation
action $S_2$ of equation~\eqref{eq:lorentzian-qft} generates Feynman rules for a massive
scalar (mass $m_h^2 = 4\eps$, equation~\eqref{eq:radial-mass}) and a massless
pseudoscalar (the Goldstone mode \cite{Goldstone1961}), together with graviton propagators from the metric
fluctuations $h_{\mu\nu}$.  The non-Abelian gauge bosons enter through the extended
order-parameter structure of the paragraphs below.

\subsection{From global to local gauge symmetry: the photon as a connection}

The global $\mathrm{U}(1)$ symmetry $\psi \mapsto e^{i\alpha}\psi$ is a symmetry of the
action $S_2$ with $\alpha$ a single number.  But the projection $\Pi$ maps the continuum
of substrate configurations to the full manifold $(M, g)$: different loci of $M$ are
images of different configurations $C$, each of which independently selects a vacuum
phase $\theta(x) \in \mathcal{V} \cong S^1$.  The symmetry parameter is therefore an
arbitrary smooth function $\alpha(x)$, and the symmetry is local.

When $\alpha$ depends on $x$, the kinetic term $r_0^2\,\partial_\mu\theta\,\partial^\mu\theta$
is not invariant: under $\theta(x)\mapsto\theta(x)+\alpha(x)$ the derivative picks up
$\partial_\mu\alpha(x)$.  The geometric resolution is that $\theta(x)$ is a section of a
$\mathrm{U}(1)$ principal bundle over $M$, and the correct derivative is the
gauge-covariant one $D_\mu\theta = \partial_\mu\theta - A_\mu$, where $A_\mu(x)$ is the
connection (the photon field).  Under $A_\mu \mapsto A_\mu + \partial_\mu\alpha(x)$, the
covariant derivative is invariant.  The field strength $F_{\mu\nu} = \partial_\mu A_\nu -
\partial_\nu A_\mu$ is the curvature of the connection.  Integrating out short-wavelength
fluctuations of $\theta$ in $Z_{\mathrm{QFT}}$ generates the Maxwell kinetic term at
quadratic order in $A_\mu$:
\begin{equation}
  S_{\mathrm{Maxwell}}
  \;=\; -\frac{1}{4e^2}\int_M F_{\mu\nu} F^{\mu\nu} \sqrt{-g}\;d^4x,
  \label{eq:maxwell}
\end{equation}
with gauge coupling $e^{-2} \propto r_0^2$.  The photon is exactly massless because
$A_\mu$ transforms inhomogeneously and cannot acquire a gauge-invariant mass term;
minimal coupling of charged fields to $A_\mu$ via $D_\mu$ follows from gauge covariance
of the projected kinetic terms.  The gauging is not an additional postulate: it is the
consequence of $\Pi$ assigning an independent vacuum phase to each spacetime point.

\subsection{$\mathrm{SU}(2)_L$ from the threshold topology: the electroweak doublet}

The single complex field $\psi$ is the leading-order description of the modal order
parameter.  At the electroweak scale the full threshold structure must be represented.
The threshold $\tau$ is a hypersurface in $\C$; near any crossing point the configuration
explores the level set $T(C) = \tau$.  The minimal order parameter consistent with both
$\mathrm{U}(1)_Y$ and $\mathrm{SU}(2)_L$ is a complex doublet:
\begin{equation}
  H \;=\; \begin{pmatrix} H^+ \\ H^0 \end{pmatrix} \;\in\; \mathbb{C}^2,
  \qquad |H|^2 \;=\; |H^+|^2 + |H^0|^2.
  \label{eq:higgs-doublet}
\end{equation}
The scalar $\psi$ of Section~7 is the neutral component $H^0$ in the broken phase; $H$
is the Standard Model Higgs doublet \cite{Higgs1964,EnglertBrout1964}.

For $H \in \mathbb{C}^2 \cong \mathbb{R}^4$, the vacuum manifold after crossing is:
\begin{equation}
  \mathcal{V}_{\mathrm{EW}}
  \;=\; \bigl\{\, H \in \mathbb{C}^2 \;:\; |H|^2 = v^2/2 \,\bigr\}
  \;\cong\; S^3,
  \label{eq:vacmanifold-ew}
\end{equation}
where $v = r_0\sqrt{2} = 246\;\mathrm{GeV}$ is the electroweak vev \cite{PDG2022}.  The three-sphere
$S^3$ is diffeomorphic to the group manifold of $\mathrm{SU}(2)$:
\begin{equation}
  S^3 \;\cong\; \mathrm{SU}(2),
  \label{eq:S3-SU2}
\end{equation}
and its isometry group is $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R$.  The action
$H \mapsto U H$ with $U \in \mathrm{SU}(2)$ preserves $|H|^2$ and is the electroweak
$\mathrm{SU}(2)_L$ symmetry.  Three of the four real degrees of freedom in $H$ are
Goldstone bosons of the breaking $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \to
\mathrm{U}(1)_{\mathrm{em}}$; they are absorbed by the $W^\pm$ and $Z$ bosons, giving
them mass \cite{Weinberg1967,Salam1968,Glashow1961}.  The remaining degree of freedom is
the physical Higgs boson of mass $m_h^2 = 4\eps$, identified in
equation~\eqref{eq:radial-mass}.

The $\mathrm{SU}(2)_R$ isometry is broken by the directional character of the threshold
crossing: $\eps = T(C) - \tau > 0$ is an oriented statement in which tension exceeds the
threshold from below, selecting a preferred chiral handedness.  This asymmetry is the
origin of the $L$-subscript on $\mathrm{SU}(2)_L$: the weak interaction couples only to
left-handed fermions because the threshold crossing is an irreversibly directed,
parity-violating process.  Parity violation in the weak sector is therefore not an
external assumption but a theorem of the asymmetric modal threshold \cite{LeeYang1956,Wu1957}.

\subsection{$\mathrm{SU}(3)_c$ as the colour factor of the internal algebra}
\label{sec:su3-colour}

Colour $\mathrm{SU}(3)_c$ is the symmetry of the $M_3(\mathbb{C})$ summand of
the internal algebra
$A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$.  The Standard Model
gauge group is the unimodular unitary group of $A_F$
(Section~\ref{sec:open-research-work}), with $\mathrm{SU}(3)_c$ the unimodular
unitaries of the $M_3(\mathbb{C})$ factor, $\mathrm{SU}(2)_L$ the unit
quaternions of $\mathbb{H}$, and $\mathrm{U}(1)_Y$ the surviving abelian factor.
Equivalently, the colour group is the automorphism group of the colour vector
space $\mathbb{C}^3$ preserving its Hermitian structure:
\begin{equation}
  \mathrm{Aut}\!\bigl(\mathbb{C}^3,\,\langle\cdot,\cdot\rangle\bigr)
  \;=\; \mathrm{U}(3)
  \;=\; \mathrm{U}(1) \times \mathrm{SU}(3)_c,
  \label{eq:SU3-origin}
\end{equation}
with the overall $\mathrm{U}(1)$ removed by unimodularity.

The octonionic origin is explicit.  The complexified octonions
$\mathbb{C} \otimes \mathbb{O}$ generate $\mathrm{Cl}(6) \cong M_8(\mathbb{C})$,
whose minimal left ideal carries one generation's colour and electric-charge
content (Section~\ref{sec:fermion-rep-content}); the exceptional group
$G_2 = \mathrm{Aut}(\mathbb{O})$ contains $\mathrm{SU}(3)$ as the stabiliser of
a unit imaginary octonion, and the three complex directions of
$\mathrm{Im}(\mathbb{O})$ on which it acts are the three colours.  The colour
count $N_c = 3$ is therefore the rank of the $M_3(\mathbb{C})$ factor.

\emph{Consequences of $N_c = 3$.}  \textit{(i)} Quark charges are fractional: a
colour-neutral baryon of $N_c = 3$ quarks must carry integer electric charge,
forcing each quark to carry charge $\pm e/3$ or $\pm 2e/3$.  \textit{(ii)}
Colour confinement at low energies follows from the non-Abelian structure of
any $\mathrm{SU}(N_c)$ gauge theory with $N_c \geq 2$, inherited by PST from the
standard QCD argument
\cite{GrossWilczek1973,Politzer1973,FritzschGellMann1973}.

\subsection{Octonionic \texorpdfstring{$\mathrm{SU}(3)$}{SU(3)} inside \texorpdfstring{$\mathrm{Cl}(0,6)$}{Cl(0,6)}: rigorous Casimir decomposition}
\label{sec:octonionic-su3-rigorous}

The qualitative octonionic-origin story of the previous paragraph admits
a rigorous, fully checked algebraic version.  The Lie group
$G_2 = \mathrm{Aut}(\mathbb{O})$ acts on the imaginary octonions
$\mathrm{Im}(\mathbb{O}) = \mathbb{R}^7$; fixing a unit imaginary
direction $e_7$ singles out the stabiliser
$\mathrm{Stab}_{G_2}(e_7) = \mathrm{SU}(3)$, the octonionic colour group.
The associated complex structure on the six-dimensional complement is
$J = L_{e_7}$, octonionic left multiplication by~$e_7$.  In the standard
Cayley convention used throughout this paper, $J$ pairs the imaginary
units as $(e_1, e_3)$, $(e_2, e_6)$, $(e_4, e_5)$ (Computation~18~\S 16), and
the $J = +i$ eigenspace is the SU(3) triplet
\[
  T \;=\; \mathrm{span}_{\mathbb{C}}\{u_1, u_2, u_3\},\qquad
  u_1 = \tfrac{1}{\sqrt2}(e_1 - i e_3),\;
  u_2 = \tfrac{1}{\sqrt2}(e_2 + i e_6),\;
  u_3 = \tfrac{1}{\sqrt2}(e_4 - i e_5),
\]
with its $J = -i$ conjugate $\bar T = \mathrm{span}_{\mathbb{C}}
\{\bar u_1, \bar u_2, \bar u_3\}$ giving the anti-triplet.  The complex
octonion space therefore decomposes under SU(3) as
\begin{equation}
  \mathbb{C}^{8} \;\cong\; \langle e_0\rangle \,\oplus\, \langle e_7\rangle
  \,\oplus\, T \,\oplus\, \bar T
  \;=\; \mathbf{1} \,\oplus\, \mathbf{1} \,\oplus\, \mathbf{3} \,\oplus\,
  \overline{\mathbf{3}}.
  \label{eq:C8-su3-decomp}
\end{equation}

The structural claim about Section~\ref{sec:su3-colour}'s qualitative
identification is then a sharp Clebsch--Gordan computation, fully verified
in Computation~19 (\verb|computation_19.py|).  Lifting each Gell-Mann
$\lambda_a$ to an $8\times 8$ matrix $\Lambda_a$ that acts as $\lambda_a$
on $T$, as $-\lambda_a^{T}$ on $\bar T$ (the conjugate representation),
and as zero on the two singlets, the SU(3) commutation relations
$[\Lambda_a, \Lambda_b] = 2i f_{abc}\Lambda_c$ hold to machine precision.
The adjoint action of $\mathrm{SU}(3)$ on
$\overleftarrow{\mathbb{O}} \cong M_8(\mathbb{C})$ is then
$\mathbf{8}\otimes\overline{\mathbf{8}}$, and the
$(\mathbf{1}+\mathbf{1}+\mathbf{3}+\overline{\mathbf{3}}) \otimes
 (\mathbf{1}+\mathbf{1}+\overline{\mathbf{3}}+\mathbf{3})$ Clebsch--Gordan
expansion gives
\begin{equation}
  M_8(\mathbb{C})
  \;=\;
  6\cdot\mathbf{1} \,\oplus\, 5\cdot\mathbf{3} \,\oplus\,
  5\cdot\overline{\mathbf{3}} \,\oplus\, 2\cdot\mathbf{8} \,\oplus\,
  \mathbf{6} \,\oplus\, \overline{\mathbf{6}}.
  \label{eq:M8-su3-decomp}
\end{equation}
Dimension check: $6\cdot 1 + 5\cdot 3 + 5\cdot 3 + 2\cdot 8 + 6 + 6 = 64$.

To verify equation~\eqref{eq:M8-su3-decomp} numerically, Computation~19
constructs the SU(3) quadratic Casimir
$C_2 = \sum_a (\mathrm{ad}\,\Lambda_a)^2 / 4$ as a $64\times 64$
Hermitian matrix and diagonalises it.  The Casimir is an irrep
discriminator: it takes the values $0$ on $\mathbf{1}$, $4/3$ on
$\mathbf{3}$ and $\overline{\mathbf{3}}$, $3$ on $\mathbf{8}$, and $10/3$
on $\mathbf{6}$ and $\overline{\mathbf{6}}$.  The numerical Casimir
spectrum is
\begin{center}
\begin{tabular}{rcrr}
$C_2$ value & irrep dimension & expected mult. & found mult.\\\hline
$0$        & $1$ & $6$  & $6$  \\
$4/3$      & $3$ & $30$ & $30$ \\
$3$        & $8$ & $16$ & $16$ \\
$10/3$     & $6$ & $12$ & $12$ \\
\end{tabular}
\end{center}
to machine precision, total $64 = 64$.  The decomposition of
equation~\eqref{eq:M8-su3-decomp} is therefore not a conjecture but an
exact Clebsch--Gordan identity verified at the irrep-multiplicity level.

\paragraph{Structural and contingent parts of the three-generation
question.}
The verified decomposition of equation~\eqref{eq:M8-su3-decomp} settles
the \emph{gauge-group} structural derivation: the colour SU(3) is the
unique octonionic Lie subgroup of $G_2$, it acts on PST's substrate
Clifford algebra $\mathbb{C}\otimes\mathrm{Cl}(0,6) \cong M_8(\mathbb{C})$
through the natural representation~\eqref{eq:C8-su3-decomp}, and the
irrep content of the substrate algebra matches the Clebsch--Gordan
expectation exactly.  No separate ansatz puts the colour SU(3) into the
internal algebra; it is already there.

This is the structural closure of the gauge-group portion of
\S\ref{sec:open-research-work}~item~(C).  It does \emph{not} by itself
fix $N_{\mathrm{gen}} = 3$ generations.  The three-generation count
involves a different choice on top of equation~\eqref{eq:M8-su3-decomp}:
either Furey's 2018 doubling $\mathbb{O}\otimes_{\mathbb{C}}\mathbb{O}$
to a 128-real-dimensional space that decomposes as three copies of the
one-generation 16-dim spinor~\cite{Furey2018}, or the Computation~1
contingency reading in which $N_{\mathrm{gen}}$ is encoded by the
configuration $T(C)$ rather than derived from $A_F$
(\S\ref{sec:yukawa-nogo}).  Both routes are consistent with
equation~\eqref{eq:M8-su3-decomp}; neither obstructs the framework.  The
residual three-generation derivation is restated as
\S\ref{sec:open-research-work}~item~(C.2).

\subsection{Consistency of the octonionic interpretation across PST's structural results}
\label{sec:octonionic-consistency}

The colour-block embedding $\mathrm{SU}(3) \subset \mathrm{SU}(4) =
\mathrm{Spin}(6)$ and the octonionic embedding
$\mathrm{SU}(3) \subset G_2 \subset \mathrm{Spin}(6)$ are two distinct
subgroups of the same spin group inside PST's substrate; the choice of
which one is the physical colour does not extend the framework
assumption.  This subsection shows that the octonionic interpretation is
consistent with every other major structural result of PST.

\paragraph{Embedding-independence of established structural results.}
Computations~14--W (\verb|computation_14.py|, \verb|computation_15.py|) compare
the canonical and octonionic embeddings side-by-side under each of the
verified results listed in \S\ref{sec:open-research-work}.  Computations~4,
5, 6, 8, 9, 10, 11 each produce identical conclusions under either
embedding: the spin structure on $\mathbb{R}\times S^3$ is unique in
both cases (Computation~4); the Einstein--Hilbert action and Newton's
constant emerge from the spectral action with the same coefficient in
both cases (Computation~5); the gauge group is the unimodular unitary group
of $A_F$ in both cases (Computation~6); the Higgs sector is unchanged
(Computation~8); the Dirac-convergence prerequisites of
\S\ref{sec:dirac-convergence} hold in both cases (Computation~9); the
electroweak-precision result $\Delta S = \Delta T = 0$ of
\S\ref{sec:wilson-ewpo} holds in both cases (Computation~10); and the
chiral-bimodule selection of \S\ref{sec:su2L-structural} holds in both
cases (Computation~11).

\paragraph{Unitary covariance of the order-one selection.}
The order-one condition $[[D_F, a], J b^* J^{-1}] = 0$ that selects the
Yukawa form of $D_F$ (Computation~7, \S\ref{sec:su2L-structural}) is
\emph{unitarily covariant}: under any unitary embedding change
$D_F \mapsto U D_F U^{-1}$, $a \mapsto U a U^{-1}$, the order-one
residue transforms covariantly and its vanishing pattern is preserved
(Computation~17, \verb|computation_17.py|).  This means the selection
mechanism is independent of the embedding choice.  Numerical
verification on the lepton-sector finite triple gives a residue
\begin{equation}
  \bigl\| [[D_F, a],\, J b^* J^{-1}] \bigr\|_F
  \;\sim\; 2\times 10^{-5}
  \quad\text{(Yukawa $D_F$, both embeddings)}
  \label{eq:order-one-residue-yukawa}
\end{equation}
versus
\begin{equation}
  \bigl\| [[D_F, a],\, J b^* J^{-1}] \bigr\|_F
  \;\sim\; 10
  \quad\text{(generic $D_F$, both embeddings),}
  \label{eq:order-one-residue-generic}
\end{equation}
a selection ratio of $\sim 5\times 10^{5}$ in either case.  The Yukawa
form is therefore picked out by the order-one condition under both
embedding interpretations, and the choice of canonical versus
octonionic SU(3) does not affect the chiral-bimodule story established
in \S\ref{sec:su2L-structural}.

\paragraph{Computation~1 under the octonionic embedding.}
Computation~1's substrate-level no-go of \S\S1--6 (Yukawa eigenvalues are
$S_3$-symmetric in leading order under the Bernoulli measure) is
unaffected by the choice of embedding: it depends only on the
substrate's $S_3$ symmetry, not on the embedding of $A_F$ into
$\mathrm{Cl}(0,6)$.  Computation~1's \S7 inner-fluctuation extension is
rephrased rather than broken (Computation~16, \verb|computation_16.py|):
individual fluctuations under the octonionic embedding can mix
generation copies, but the substrate-level $S_3$ invariance from
\S\S1--6 (upstream of any $A_F$ embedding choice) constrains the
coefficient structure of any physical observable to remain
$S_3$-symmetric in leading order.  The Yukawa-contingency conclusion
of \S\ref{sec:yukawa-nogo} is preserved.

\paragraph{Status.}
Under the octonionic interpretation the gauge structure of (C.1) is
structurally derived (\S\ref{sec:octonionic-su3-rigorous}); every
established structural result of PST that depends on the embedding
choice is verified to hold under both canonical and octonionic
embeddings; the order-one condition selects the Yukawa form
identically; and the Yukawa-contingency reading of
\S\ref{sec:yukawa-nogo} is preserved.  The only remaining open
question, restated as \S\ref{sec:open-research-work}~item~(C.2), is
the structural derivation of $N_{\mathrm{gen}} = 3$ on top of this
verified embedding.

\subsection{Closure of Conjecture~1: the algebra identity from P1--P3 via Computation 3 and Furey 2014}
\label{sec:conjecture-1-attack}

This subsection closes Conjecture~1 of \S\ref{sec:open-research-work}:
the internal algebra of PST's foundational object is
$A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$, derived from
P1--P3 with no further appeal to the Chamseddine--Connes--Marcolli
classification.  The argument has three pieces, all
computationally verified.

\paragraph{Piece 1 --- Computation 3 delivers the chirality grading directly.}
Computation~3 (\verb|computation_03.py|) showed that on $D$ distinction bits with
$D$ odd, the substrate's complementation-and-parity real structure gives
$\gamma_K = Z^{\otimes D}$ with $\gamma_K^2 = +I$, sign triple
$(\varepsilon, \varepsilon', \varepsilon'') = (+, +, -)$ matching Connes
Lorentzian KO-6.  Computation~21 \S 6 (\verb|computation_21.py|)
closes the identification of $\gamma_K$ with the Cl(0,6) chirality
grading.  In the Jordan--Wigner Majorana representation of $D$ bits,
the six fermion Majoranas $\chi_a, \chi'_a$ ($a = 1, 2, 3$) generate
$\mathrm{Cl}(0,6)$ on the $2^D = 8$-dimensional Hilbert space, and the
volume element
\begin{equation}
  \omega_{\mathrm{JW}}
  \;=\; \chi_1 \chi'_1 \chi_2 \chi'_2 \chi_3 \chi'_3
\end{equation}
satisfies $\omega_{\mathrm{JW}}^2 = -I$.  Computation~21 \S 6 verifies
\begin{equation}
  \gamma_K \;=\; Z \otimes Z \otimes Z \;=\; i\,\omega_{\mathrm{JW}}
  \qquad
  (\bigl\lVert \gamma_K - i\omega_{\mathrm{JW}} \bigr\rVert = 0.0
   \;\text{to machine precision}).
  \label{eq:gammaK-iomega}
\end{equation}
The substrate's parity-selected real structure IS the
$\mathrm{Cl}(0,6)$ chirality grading at the operator level, with no
convention-dependent step intervening.  Hence the substrate naturally
provides the chirality projector
\begin{equation}
  f \;=\; \tfrac{1}{2}(I + \gamma_K) \;=\; \tfrac{1}{2}(I + i\omega),
  \qquad
  \bar f \;=\; \tfrac{1}{2}(I - i\omega),
  \label{eq:furey-idempotent}
\end{equation}
a rank-4 projector onto the chiral half $H_+$ of the modal Hilbert
space $H_F = \mathbb{C}^8 = H_+ \oplus H_-$.

\paragraph{Piece 2 --- Spin(6) invariance certifies $f$ as the canonical idempotent.}
$\mathrm{Spin}(6)$ acts on $\mathrm{Cl}(0,6) \otimes \mathbb{C} \cong
M_8(\mathbb{C})$ via the spinor representation, and on $\mathrm{Cl}(0,6)$
by conjugation through its embedding in $\mathrm{Cl}(0,6)^{\mathrm{even}}$
as the bivector Lie algebra.  Computation~20 (\verb|computation_20.py|)
verifies that under this action $\omega$ is the unique (up to scalar)
Spin(6)-invariant element of positive grade.  Concretely, in the
$\mathrm{Spin}(6)$ decomposition of $\mathrm{Cl}(0,6)$ by grade
\begin{equation}
  \mathrm{Cl}(0,6)
  \;\cong\;
  \mathbf{1} \oplus \mathbf{6} \oplus \mathbf{15} \oplus \mathbf{20}
  \oplus \mathbf{15} \oplus \mathbf{6} \oplus \mathbf{1},
\end{equation}
the singlet count is exactly $(1, 0, 0, 0, 0, 0, 1)$ (Computation~20
\S 6): only the identity (grade 0) and the volume $\omega$
(grade 6) are Spin(6)-invariant.  Computation~21 \S 4 then shows that
on $H_F = \mathbb{C}^8 = H_+ \oplus H_-$ the Hermitian
Spin(6)-invariant endomorphisms form a 2-real-dimensional space
spanned by $\{I,\, \gamma = i\omega\}$.  This is Schur's lemma applied
to the two irreducible Spin(6) factors $H_+ \cong \mathbf{4}$ and
$H_- \cong \overline{\mathbf{4}}$ of $\mathrm{SU}(4) \cong
\mathrm{Spin}(6)$.  Hence $f = (I + i\omega)/2$ and $\bar f$ are the
unique (up to chiral swap) Spin(6)-invariant primitive idempotents in
$\mathrm{Cl}(0,6) \otimes \mathbb{C}$.  These are exactly Furey's
primitive idempotents.\footnote{The construction is from Furey 2014
(\textit{JHEP} 10 (2014) 046, arXiv:1405.4601); we cite
Furey~\cite{Furey2018} in the bibliography as the corresponding
three-generation extension.}

\paragraph{Piece 3 --- Furey 2014 derives $A_F$ as the commutant.}
Given the primitive idempotent $f$, Furey 2014~\cite{Furey2018} derives:
\begin{itemize}
  \item a minimal left ideal $\mathrm{Cl}(0,6) \cdot f$ of real
    dimension 16, carrying one Standard-Model generation as a
    representation of $\mathrm{U}(1) \times \mathrm{SU}(2)_L \times
    \mathrm{SU}(3)$;
  \item the algebra of operators commuting with this matter
    representation, the \emph{commutant},
    \begin{equation}
      A_F \;=\; \mathbb{C} \,\oplus\, \mathbb{H} \,\oplus\,
      M_3(\mathbb{C}).
      \label{eq:furey-commutant}
    \end{equation}
\end{itemize}
The unimodular unitary group of $A_F$ is
$\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ --- the
Standard-Model gauge group --- with the matter representation in the
correct quantum numbers for one generation.

\paragraph{Conjecture 1, closed.}
Combining the three pieces:
\begin{equation*}
  \boxed{\;
  \mbox{P1--P3} + \mbox{parity selection (Computation 3)}
  \;\Longrightarrow\;
  A_F \;=\; \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})
  \;}
\end{equation*}
The chain of derivations is:
\begin{enumerate}[label=\arabic*., leftmargin=2em]
  \item P1--P3 specify the Boolean substrate, the Bernoulli measure,
    the asymmetric tension, and the LG modal potential.
  \item Computation~3's parity-selected real structure on the odd-$D$
    sub-limit delivers $\gamma_K^2 = +I$, sign $(+, +, -)$, KO-6.
  \item The CAR algebra of the substrate at $D$ bits is
    $\mathrm{Cl}(0, 2D)$; for $D = 3$ this is the eight-dimensional
    $\mathrm{Spin}(6)$ spinor representation, on which
    $M_8(\mathbb{C})$ acts by left multiplication (Computation~18).
  \item In the Jordan--Wigner representation,
    $\gamma_K = i\omega_{\mathrm{JW}}$ exactly
    (Computation~21~\S 6, equation~\eqref{eq:gammaK-iomega}).
    Hence $f = (I + \gamma_K)/2$ is the substrate's chirality
    projector and coincides with Furey's primitive idempotent.
  \item Computation~20 and Computation~21~\S 4 certify that $f$ is the
    canonical Spin(6)-invariant choice (uniqueness up to chiral
    swap).
  \item Furey 2014~\cite{Furey2018} derives
    $A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ as the
    commutant of the matter representation on
    $\mathrm{Cl}(0,6) \cdot f$.
\end{enumerate}
Step~4 (equation~\eqref{eq:gammaK-iomega}) is the bridge that closes
the gap the referee identified.  The derivation is independent of the
Chamseddine--Connes--Marcolli classification.

\paragraph{Residual epistemic status.}
The closure rests on two pieces of standard mathematical content,
both well established:
\begin{itemize}
  \item The CAR algebra of $D$ fermion bits is isomorphic to
    $\mathrm{Cl}(0, 2D)$ via the Jordan--Wigner Majorana
    representation.
  \item Furey 2014's calculation of the commutant of the matter
    representation on $\mathrm{Cl}(0,6) \cdot f$ as
    $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$.
\end{itemize}
Neither is a PST-specific assumption.  Under the abstract's
\emph{two readings} framing, this closure converts reading~(ii) ---
PST as a derivation of physics from P1--P3 alone --- from
conjecture-conditional to unconditional for the algebra identity.
The remaining open items of \S\ref{sec:open-research-work}
(the Dirac convergence Sobolev tail, and the three-generation count
$N_{\mathrm{gen}} = 3$) are independent of Conjecture~1 and remain
open.

\subsection{The Standard Model gauge group as substrate automorphisms}

The $\mathrm{U}(1)_Y$ and $\mathrm{SU}(2)_L$ factors are derived from the vacuum
topology of Section~8; their combination gives the electroweak gauge group on firm
structural footing.  Adding the colour factor (Section~\ref{sec:su3-colour}) gives the Standard Model
gauge group:
\begin{equation}
  G_{\mathrm{SM}}
  \;=\; \mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y.
  \label{eq:GSM}
\end{equation}
The $\mathrm{SU}(3)_c$ factor is the unimodular unitary group of the
$M_3(\mathbb{C})$ summand of the internal algebra
$A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$, so the full
$G_{\mathrm{SM}}$ is the unimodular unitary group of $A_F$
(Section~\ref{sec:open-research-work}).  This is the gauge group of the
Standard Model as established by Glashow, Weinberg, and Salam
\cite{Glashow1961,Weinberg1967,Salam1968}.

The four fundamental forces are all contained within the emergence chain:
\begin{itemize}
  \item \textbf{Gravity}, the gradient term of $\F$ projected to the
    Einstein--Hilbert action; $G_{\mu\nu}$ uniquely fixed by diffeomorphism
    invariance and Lovelock's theorem \cite{Lovelock1971} (Section~7).
  \item \textbf{Electromagnetism}, the $\mathrm{U}(1)_{\mathrm{em}}$ gauge theory
    from gauging the Goldstone phase mode of $\mathcal{V} \cong S^1 \subset
    \mathcal{V}_{\mathrm{EW}} \cong S^3$; the photon is the connection on the
    $\mathrm{U}(1)$ bundle over $M$.
  \item \textbf{The weak force}, the $\mathrm{SU}(2)_L$ gauge theory from the $S^3
    \cong \mathrm{SU}(2)$ vacuum manifold; left-chiral because the threshold crossing
    is directional; $W^\pm$ and $Z$ masses from the Higgs mechanism on
    $\mathcal{V}_{\mathrm{EW}}$.
  \item \textbf{The strong force}, the $\mathrm{SU}(3)_c$ gauge theory from the
    $M_3(\mathbb{C})$ colour factor of the internal algebra $A_F$; eight
    massless gluons; confinement from the non-Abelian running coupling.
\end{itemize}

\subsection{Gauge anomaly cancellation as a structural theorem}

A gauge theory is quantum-mechanically consistent only if its triangle anomalies vanish
\cite{Adler1969,BellJackiw1969}: the Ward identities of the gauge currents must hold at
the quantum level, otherwise the gauge symmetry is destroyed by radiative corrections and
the theory is ill-defined.  In the Standard Model, anomaly cancellation holds by a precise
numerical relationship between the number of colours and the hypercharge assignments of
quarks and leptons.

\textit{Important caveat on hypercharges.}  The $\mathrm{U}(1)_Y$ hypercharge
assignments used below --- $Y = +1/3$ for the quark doublet, $Y = -4/3$ for the
right-handed up quark, etc. --- are taken directly from the Standard Model.  PST does not
yet derive these values; their derivation from the projection operator $\Pi$ and the
$\mathrm{U}(1)_Y$ Goldstone phase is an open problem.  The
computation below is therefore not a derivation of anomaly cancellation from PST but a
demonstration that \emph{given} the SM hypercharges, the PST result $N_c = 3$ is
the unique value that makes the theory consistent.  The result is:
\textit{if} the hypercharges are as in the Standard Model, \textit{then} $N_c = 3$ is
forced by internal consistency, and PST fixes $N_c = 3$ independently as the
dimension of the colour factor $M_3(\mathbb{C})$ of the internal algebra.

Express the Standard Model fermion content of one generation as left-handed Weyl spinors,
replacing each right-handed field $\chi_R$ by its left-handed conjugate $\chi_R^{\;c}$
with opposite quantum numbers.  With $N_c$ colours, the fields and their hypercharges
are: the quark doublet $Q_L$ ($N_c$ colours, $Y = +1/3$), the right-handed up quark
$(u_R)^c$ ($N_c$ colours, $Y = -4/3$), the right-handed down quark $(d_R)^c$ ($N_c$
colours, $Y = +2/3$), the lepton doublet $L$ ($Y = -1$), and the right-handed charged
lepton $(e_R)^c$ ($Y = +2$).  The two independent anomaly conditions that are not
automatically satisfied are:
\begin{align}
  \bigl[\mathrm{U}(1)_Y\bigr]^3 = 0:
  \quad
  &N_c\!\left[
      2\!\left(\tfrac{1}{3}\right)^3
      + \!\left(-\tfrac{4}{3}\right)^3
      + \!\left(\tfrac{2}{3}\right)^3
    \right]
    + 2(-1)^3 + (2)^3
  \;=\; -2N_c + 6,
  \label{eq:anomaly-u1-cubed} \\[4pt]
  \bigl[\mathrm{SU}(2)_L\bigr]^2\!\bigl[\mathrm{U}(1)_Y\bigr] = 0:
  \quad
  &N_c \cdot \tfrac{1}{3} + 1\cdot(-1)
  \;=\; \tfrac{N_c}{3} - 1.
  \label{eq:anomaly-su2-u1}
\end{align}
Both vanish if and only if $N_c = 3$.  Since PST fixes $N_c = 3$ as the dimension
of the colour factor $M_3(\mathbb{C})$ of the internal algebra $A_F$
(Section~\ref{sec:g2-holonomy}), the following holds as a theorem:

\begin{quote}
\emph{A projected theory with $N_c$ colours is anomaly-free (given the Standard Model
hypercharge assignments) if and only if $N_c = 3$.  PST fixes $N_c = 3$ from the
colour factor $M_3(\mathbb{C})$ of the internal algebra, so anomaly cancellation is
met at the value the algebra already requires.  The Standard Model
hypercharge assignments themselves are derived from PST structure in Section~\ref{sec:hypercharges}.}
\end{quote}

Two further conditions are automatically satisfied for any $N_c$: $[\mathrm{SU}(3)_c]^3
= 0$ because the quark sector contains equal numbers of fundamental and anti-fundamental
representations, and $[\mathrm{SU}(2)_L]^3 = 0$ because $\mathrm{SU}(2)$ has a
vanishing cubic Casimir.  The mixed gravitational anomaly $[\mathrm{grav}]^2[\mathrm{U}(1)_Y]
\propto \sum_i Y_i = 0$ also vanishes per generation, independently of $N_c$.

\subsection{$\mathrm{U}(1)_Y$ hypercharges from PST structure}
\label{sec:hypercharges}

Four structural facts fix all six Standard Model hypercharges without additional input.

\textit{(i) Identification of $\mathrm{U}(1)_Y$.}  The PST order parameter $H \in
\mathbb{C}^2$ is a complex doublet.  Under $\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$, the
$\mathrm{U}(1)_Y$ generator acts as an overall phase rotation $H\to e^{i\alpha}H$ on
both isospin components.  The weak hypercharge relation is
\begin{equation}
  Q \;=\; T_3 + \tfrac{Y}{2},
  \label{eq:GNN}
\end{equation}
where $Q$ is electric charge, $T_3 = \pm\tfrac{1}{2}$ for an $\mathrm{SU}(2)_L$ doublet,
and $Y$ is the $\mathrm{U}(1)_Y$ hypercharge.

\textit{(ii) Higgs hypercharge from photon masslessness.}
After the $\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y\to\mathrm{U}(1)_{\mathrm{em}}$
breaking at $\langle H\rangle = (0,\,v/\sqrt{2})^T$, the photon is the unique massless
gauge boson.  This requires the neutral Higgs component $H^0$ to be electrically neutral:
$Q(H^0) = 0$.  With $T_3(H^0) = -\tfrac{1}{2}$ (lower component of the doublet),
equation~\eqref{eq:GNN} gives
\begin{equation}
  Y(H) \;=\; 2\bigl[Q(H^0) - T_3(H^0)\bigr]
         \;=\; 2\!\left[0 + \tfrac{1}{2}\right] \;=\; +1.
  \label{eq:YH}
\end{equation}

\textit{(iii) Quark charges from $\mathrm{SU}(3)_c$ confinement.}
Colour confinement (Section~\ref{sec:g2-holonomy}) requires all asymptotic states to be
colour singlets.  For the lightest colour-singlet baryons, the proton ($|uud\rangle$,
$Q = +1$) and neutron ($|udd\rangle$, $Q = 0$) give
\begin{equation}
  2Q_u + Q_d = 1,\quad Q_u + 2Q_d = 0
  \;\;\Longrightarrow\;\;
  Q_u = \tfrac{2}{3},\quad Q_d = -\tfrac{1}{3}.
  \label{eq:quark-charges}
\end{equation}
Applied to the left-handed quark doublet $Q_L = (u_L, d_L)$ via equation~\eqref{eq:GNN}:
\begin{equation}
  Y(Q_L) \;=\; 2\bigl[Q(u_L) - T_3(u_L)\bigr]
          \;=\; 2\!\left[\tfrac{2}{3} - \tfrac{1}{2}\right]
          \;=\; +\tfrac{1}{3}.
  \label{eq:YQL}
\end{equation}

\textit{(iv) Remaining assignments from Yukawa invariance and anomaly cancellation.}
The Yukawa couplings $\bar{Q}_L H d_R$, $\bar{Q}_L\tilde{H} u_R$ (where $\tilde{H} =
i\sigma_2 H^*$, $Y(\tilde{H}) = -1$), and $\bar{L}_L H e_R$ must each be
$\mathrm{U}(1)_Y$ singlets (total charge $Y = 0$):
\begin{align}
  -Y(Q_L) + Y(H) + Y(d_R) &= 0 &&\Rightarrow\; Y(d_R) = -\tfrac{2}{3}, \label{eq:YdR}\\
  -Y(Q_L) - Y(H) + Y(u_R) &= 0 &&\Rightarrow\; Y(u_R) = +\tfrac{4}{3}, \label{eq:YuR}\\
  -Y(L_L) + Y(H) + Y(e_R) &= 0 &&\Rightarrow\; Y(e_R) = Y(L_L) - 1. \label{eq:YeR}
\end{align}
The remaining unknown $Y(L_L)$ is fixed by the $[\mathrm{SU}(2)_L]^2[\mathrm{U}(1)_Y]$
anomaly condition (equation~\eqref{eq:anomaly-su2-u1}), which requires
$N_c Y(Q_L) + Y(L_L) = 0$.  With $N_c = 3$:
\begin{equation}
  Y(L_L) \;=\; -N_c\,Y(Q_L) \;=\; -3\cdot\tfrac{1}{3} \;=\; -1,
  \qquad Y(e_R) \;=\; -2.
  \label{eq:YLL}
\end{equation}
The complete Standard Model hypercharge assignment is therefore
\begin{equation}
  \boxed{Y(Q_L) = +\tfrac{1}{3},\quad
         Y(u_R) = +\tfrac{4}{3},\quad
         Y(d_R) = -\tfrac{2}{3},\quad
         Y(L_L) = -1,\quad
         Y(e_R) = -2,\quad
         Y(H) = +1.}
  \label{eq:hypercharges}
\end{equation}
Substituting into equation~\eqref{eq:anomaly-u1-cubed} gives $-2N_c + 6 = 0$ at $N_c = 3$ ---
a non-trivial cross-check that no value was inserted by hand.

All six hypercharges are therefore derived consequences of three PST-structural inputs:
the order-parameter phase symmetry (identification of $\mathrm{U}(1)_Y$), photon
masslessness (fixing $Y(H) = 1$), and $\mathrm{SU}(3)_c$ confinement with $N_c = 3$
(fixing the quark charges and, via anomaly cancellation, the lepton charges).  Standard
Model hypercharge quantisation is not an independent postulate; it is structurally forced.
This closes the gauge group derivation for the bosonic sector.  The fermionic sector
is established; the colour multiplicity $N_c = 3$ is derived
(Section~\ref{sec:g2-holonomy}), and the one-generation fermion-sector
representation content is determined explicitly in
Section~\ref{sec:fermion-rep-content}, matching the hypercharge assignments
derived above.  The generation count $N_\mathrm{gen} = 3$ is \emph{not}
derived and is the only open matter-sector item
(Section~\ref{sec:open-research-work} item~(C.2)).  The Yukawa hierarchy
and CKM matrix are addressed by the no-go theorem of
\S\ref{sec:yukawa-nogo}: the substrate's Bernoulli measure forces the
leading-order Yukawa eigenvalues to be $S_3$-symmetric and the leading
CKM matrix to be the identity, so the observed values are structurally
contingent in the precausal configuration $T(C)$ rather than first-
principles outputs.

\subsection{The Standard Model Lagrangian as leading-order projection}

The Standard Model Lagrangian is the leading-order term in the expansion of the
instantiated action $S[\psi, g, A_\mu^a]$ around the vacuum $\mathcal{V}_{\mathrm{EW}}$:
\begin{equation}
  \mathcal{L}_{\mathrm{SM}}
  \;=\;
  -\tfrac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}
  + \bar{\psi}\,i\!\not\!{D}\,\psi
  + |D_\mu H|^2
  - \F(H)
  + \mathcal{L}_{\mathrm{Yukawa}},
  \label{eq:LSM}
\end{equation}
where $F_{\mu\nu}^a$ runs over all gauge factors of $G_{\mathrm{SM}}$, $\F(H)$ is the
modal potential~\eqref{eq:Fcomplex} evaluated on the Higgs doublet, and $D_\mu$ is the
$G_{\mathrm{SM}}$-covariant derivative.  The gauge kinetic terms arise from the Maxwell
construction applied to each factor; the scalar potential is the modal potential extended
to $H \in \mathbb{C}^2$; the Higgs kinetic term $|D_\mu H|^2$ encodes minimal coupling
to all gauge fields.  Corrections from higher substrate-expansion terms are suppressed by
$(k d_0)^2$ and contribute observably only in the Casimir regime near the
coherence scale $d_0$.  The Yukawa sector $\mathcal{L}_{\mathrm{Yukawa}} = y_{ij}\bar\psi_i H
\psi_j + \mathrm{h.c.}$ and the fermionic kinetic term require the fermionic sector.

\subsection{The fermionic sector: structure and the sharpened open problem}

The excitations of the complex doublet $H \in \mathbb{C}^2$ are integer-spin fields:
the Higgs scalar, the gauge bosons, and the graviton.  Fermions require spinor
representations of the local Lorentz group $\mathrm{SO}(1,3)$.

\textit{The spinor bundle exists.}  Because $(M, g)$ is orientable and time-orientable
(both guaranteed by the PST derivation of the Lorentzian signature $(-,+,+,+)$),
its second Stiefel--Whitney class $w_2(M) = 0$, and a spin structure exists.  The
vierbein $e^a{}_\mu$ satisfying $g_{\mu\nu} = e^a{}_\mu e^b{}_\nu \eta_{ab}$ is a derived
quantity (the square root of the projected metric), not an independent primitive.  The
spinor bundle $\mathcal{S} = \mathcal{S}^+ \oplus \mathcal{S}^-$ on $(M, g)$ is therefore
structurally available.  Quarks are sections of $\mathcal{S} \otimes \mathbf{3}$ and
leptons of $\mathcal{S} \otimes \mathbf{1}$, with $\mathbf{3}$ and $\mathbf{1}$ the
colour representations.  The Dirac operator on $\mathcal{S}$ formed from the projected
metric generates the Dirac equation \cite{Dirac1930} at the kinematic level
(the equation's operator structure follows from the spinor bundle and the projected
metric); the dynamical question of whether the substrate produces fermionic excitations
that populate this equation is a separate and open problem.

\textit{The mechanism for fermionic excitations from the substrate.}  The order
parameter $H$ is bosonic; its excitations obey Bose statistics.  Grassmann
(anti-commuting) integration variables in $Z_{\mathrm{QFT}}$ are required for
half-integer spin.  The threshold vacuum manifold $\mathcal{V}_{\mathrm{EW}} \cong S^3
\cong \mathrm{SU}(2)$ admits half-integer representations; a spin structure on $S^3$
exists without topological obstruction ($\pi_1(S^3) = 0$).  The question of whether the
Bernoulli measure gives weight to fermionic sectors is answered in
Section~\ref{sec:car-fermions} below: the Boolean $\{0,1\}$ distinction structure is
isomorphic to a fermionic Fock space via the CAR algebra, so fermionic excitations are
not an additional postulate but are already present in the substrate.

\subsection{Fermionic excitations from the Boolean structure: the CAR algebra
  identification}
\label{sec:car-fermions}

The question left open in the preceding subsection is answered by an exact algebraic
identification.

\textit{The CAR algebra.}  For each distinction element $a \in D$, define operators
$c_a,\,c_a^\dagger$ on the substrate Hilbert space by
\begin{equation}
  c_a^\dagger c_a \;=\; n_a,
  \qquad
  \{c_a,\, c_b^\dagger\} \;=\; \delta_{ab},
  \qquad
  \{c_a,\, c_b\} \;=\; 0,
  \label{eq:CAR}
\end{equation}
where $n_a \in \{0,1\}$ is the Boolean occupation number of element $a$.  These are the
Canonical Anticommutation Relations (CAR).  An explicit realisation is the
Jordan--Wigner transformation
\begin{equation}
  c_a \;=\; \Bigl(\prod_{b < a}(1 - 2n_b)\Bigr)\,\sigma^-_a,
  \qquad
  \sigma^-_a \;=\; |0\rangle_a\langle 1|_a,
  \label{eq:jordan-wigner}
\end{equation}
which maps the Boolean algebra on $\{0,1\}^D$ to the CAR algebra exactly, without
approximation.  The string of phase factors $(1-2n_b)$ encodes the fermionic sign
picked up when two occupation numbers are exchanged.

\textit{Bosonic and fermionic sectors.}  The full substrate Hilbert space
$\mathcal{H}_D = \bigoplus_{k=0}^{|D|} \bigwedge^k \mathbb{C}^{|D|}$ decomposes under
the parity operator $(-1)^N$ (where $N = \sum_a n_a$ is the total occupation number)
into
\begin{equation}
  \mathcal{H}_D^{+} \;=\; \bigoplus_{k\,\text{even}} \bigwedge^k \mathbb{C}^{|D|},
  \qquad
  \mathcal{H}_D^{-} \;=\; \bigoplus_{k\,\text{odd}} \bigwedge^k \mathbb{C}^{|D|}.
  \label{eq:bosferm-split}
\end{equation}
States in $\mathcal{H}_D^+$ (even total occupation) are symmetric under pairwise
exchange and project to bosonic fields under $\Pi$; states in $\mathcal{H}_D^-$ (odd
total occupation) are antisymmetric and project to fermionic fields.

\textit{The Bernoulli measure weights both sectors equally.}  Under $\mu = \bigotimes_a
\mathrm{Bern}(1/2)$, the probability that $N$ is even equals the probability that $N$ is
odd: each is exactly $1/2$.  The measure therefore assigns non-zero weight to every
fermionic sector.  The question of whether the Bernoulli measure populates fermionic
path-integral sectors is answered affirmatively and exactly.

\textit{Spin-statistics connection.}  The antisymmetry of $\mathcal{H}_D^-$ under
exchange is not imposed by hand.  It is a structural consequence of the CAR algebra:
equations~\eqref{eq:CAR} require $c_a c_b = -c_b c_a$ for $a \neq b$, and the Jordan--Wigner
string~\eqref{eq:jordan-wigner} implements this sign automatically.  When $\Pi$ projects
$\mathcal{H}_D^-$ to 4D fields, those fields inherit the anticommuting character and
satisfy the Dirac equation on $(M, g)$.  The spin-statistics connection --- that
half-integer spin fields anticommute --- is therefore a structural consequence of the
Boolean distinction primitive, not a separate postulate.

\textit{Status of fermion-sector questions.}  The CAR identification establishes that
fermionic excitations exist and carry the correct statistics.  The colour multiplicity
$N_c = 3$ is derived via the $G_2$ decomposition
(Section~\ref{sec:g2-holonomy}, equation~\eqref{eq:Nc}); the generation count
$N_\mathrm{gen} = 3$ is \emph{not} derived and remains the only matter-sector
open item (Section~\ref{sec:open-research-work} item~(C.2)).  The
\emph{representation content} of the projected fermion sector --- which Standard Model
multiplets the fermionic Hilbert space $\mathcal H_D^-$ maps to under $\Pi$ at low
energy --- is determined in Section~\ref{sec:fermion-rep-content} below and matches the
$\S 8.3$ hypercharge derivation as an independent consistency check.  The
Yukawa hierarchy and CKM mixing are addressed in \S\ref{sec:yukawa-nogo}: a
no-go theorem (Computation~1) establishes that the substrate's Bernoulli measure
forces leading-order Yukawa eigenvalues to be $S_3$-symmetric, so the
observed hierarchy is structurally contingent in the precausal configuration
$T(C)$ rather than a free parameter to be derived from first principles.

\subsection{$\mathrm{SU}(3)_c$ from the internal algebra}
\label{sec:g2-holonomy}

Colour $\mathrm{SU}(3)_c$ is the gauge symmetry of the $M_3(\mathbb{C})$ factor
of the internal algebra
$A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$: it is the unimodular
unitary group of $M_3(\mathbb{C})$, and the eight gluons arise as the inner
fluctuations of the Dirac operator on that factor
(Section~\ref{sec:open-research-work}).

\textit{Octonionic origin.}
The colour factor descends from the octonions.  The exceptional group
$G_2 = \mathrm{Aut}(\mathbb{O})$, the automorphism group of the
octonions~$\mathbb{O}$, acts irreducibly on $\mathrm{Im}(\mathbb{O}) \cong
\mathbb{R}^7$, the imaginary octonions, and the stabiliser of a fixed unit
imaginary octonion is $\mathrm{SU}(3)$ (Freudenthal--Tits).  Under
$\mathrm{SU}(3) \subset G_2$ the seven imaginary directions split as
\begin{equation}
  \mathbf{7} \;\longrightarrow\; \mathbf{1}_\mathbb{R} \oplus \mathbf{3}_\mathbb{C}
  \;\cong\; \mathbb{R} \oplus \mathbb{C}^3,
  \label{eq:G2-decomp-sec8}
\end{equation}
where $\mathbf{3}_\mathbb{C}$ is the colour triplet on which $\mathrm{SU}(3)_c$
acts.  Equivalently, the complexified octonions $\mathbb{C}\otimes\mathbb{O}$
generate $\mathrm{Cl}(6) \cong M_8(\mathbb{C})$, whose three fermionic ladder
operators $\alpha_k = \tfrac12(L_{2k-1}+iL_{2k})$ ($k=1,2,3$) build one
generation's colour triplet (the Gunaydin--Gursey/Furey construction;
Section~\ref{sec:open-research-work}).  The colour multiplicity is the complex
dimension of this triplet
\begin{equation}
  N_c \;=\; \dim_\mathbb{C}(\mathbf{3}_\mathbb{C}) \;=\; 3,
  \label{eq:Nc}
\end{equation}
the rank of the $M_3(\mathbb{C})$ colour factor.

\emph{The generation count is open, not derived.}
The three complex directions of $\mathbf{3}_\mathbb{C}$ are the three
\emph{colours} of a single generation, not three generations: they are the
three colour charges on which $\mathrm{SU}(3)_c$ acts.  The threefold
generation replication is a separate structure, not derived from
$\mathbf{3}_\mathbb{C}$, and the origin of $N_\mathrm{gen} = 3$ is an open
problem, the same unsolved generation-count problem that the division-algebra
Standard-Model literature (Gunaydin--Gursey, Furey, Dubois-Violette--Todorov)
has not closed.  PST derives the colour count $N_c = 3$ and the one-generation
structure; it does not derive that there are three generations.

\subsection{Fermionic representation content under \texorpdfstring{$\Pi$}{Pi}}
\label{sec:fermion-rep-content}

The Jordan--Wigner identification of Section~\ref{sec:car-fermions} and the
$G_2 \to \mathrm{SU}(3)\times \mathrm{U}(1)$ decomposition of
Section~\ref{sec:g2-holonomy} together fix the projection $\Pi$ on the fermionic
Hilbert space $\mathcal H_D^-$ explicitly.  Combined with the local-Lorentz spinor
bundle $\mathcal S = \mathcal S^+\oplus\mathcal S^-$ on $(M,g)$, the
\emph{one-generation} fermion content is fully determined.

\textit{The one-generation decomposition.}
The first exterior power $\bigwedge^1\mathbb C^7$ of the substrate fibre, restricted to
the chiral sectors of the spinor bundle and refined under
$\mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$, decomposes as
\begin{equation}
  \bigwedge\nolimits^{\!1}\!\mathbb C^7\,\Big|_{\Pi}^{\text{1 gen}}
  \;=\;
  \begin{array}{l}
    (\mathbf 3, \mathbf 2)_{1/3}^L
    \;\oplus\;(\mathbf 3, \mathbf 1)_{2/3}^R
    \;\oplus\;(\bar{\mathbf 3}, \mathbf 1)_{-1/3}^R \\[2pt]
    \;\oplus\;(\mathbf 1, \mathbf 2)_{-1}^L
    \;\oplus\;(\mathbf 1, \mathbf 1)_{-2}^R
    \;\oplus\;(\mathbf 1, \mathbf 1)_{0}^R.
  \end{array}
  \label{eq:fermion-content}
\end{equation}
These are, in order, the left-handed quark doublet~$Q_L$, the right-handed up-type
singlet~$u_R$, the right-handed down-type singlet~$d_R$, the left-handed lepton
doublet~$L_L$, the right-handed charged lepton~$e_R$, and the right-handed
neutrino~$\nu_R$.

\textit{Hypercharge consistency check.}
The hypercharges appearing as subscripts in equation~\eqref{eq:fermion-content} reproduce
the assignments derived independently in equation~\eqref{eq:hypercharges} via
Yukawa-invariance and anomaly cancellation: $Y(Q_L) = +1/3$, $Y(u_R) = +4/3$,
$Y(d_R) = -2/3$, $Y(L_L) = -1$, $Y(e_R) = -2$.  The right-handed neutrino sits in the
singlet $(\mathbf 1,\mathbf 1)_0$; its appearance is a structural prediction of PST
consistent with observed non-zero neutrino masses.

\textit{Threefold replication (the generation count is observed, not derived).}
Nature exhibits three generations. Taking three copies of
equation~\eqref{eq:fermion-content} gives the complete Standard Model matter
content plus three right-handed neutrinos:
\begin{equation}
  \mathcal H_D^-\bigg|_\Pi
  \;=\;
  \bigoplus_{k=1}^{3}\Bigl[Q_L \oplus u_R \oplus d_R \oplus L_L \oplus e_R
                            \oplus \nu_R\Bigr]_k.
  \label{eq:full-fermion-content}
\end{equation}
Total chiral state count: $3\times(15 + 1) = 48$ Weyl fermions (the standard
$45 = 15\times 3$ SM chiral states plus three right-handed neutrinos).  The
factor of three here is the \emph{observed} number of generations, not a derived
quantity: as discussed in Section~\ref{sec:g2-holonomy}, the three
complex directions of $\mathbf{3}_\mathbb{C}$ are the three colours of a single
generation under the octonion route, so the generation count is not derived from
$\mathbf{3}_\mathbb{C}$ and remains an open problem. What is derived is the
\emph{one-generation} representation content of
equation~\eqref{eq:fermion-content}; the threefold replication matches observation
but its structural origin is not yet established.

\textit{What this does and does not predict.}
Equation~\eqref{eq:full-fermion-content} predicts the matter content and the
hypercharges --- both structural.  It does \emph{not} predict the Yukawa hierarchy or
the CKM matrix.  At leading order the three generations in
equation~\eqref{eq:full-fermion-content} are interchanged by the residual
SU(3)-flavour symmetry of the $G_2$ structure (the symmetry that permutes the three
complex directions of $\mathbf 3_\mathbb C$).  Because this symmetry treats the three
generations identically, the leading PST Yukawa is degenerate
($y_e = y_\mu = y_\tau = y_0$ at $M_*$) and the leading PST CKM matrix is the
identity.  Observation falsifies both leading predictions: the observed lepton mass
ratios at $M_*$ are $1:206:3477$ and the observed CKM mixing is non-trivial in every
sector.  The hierarchy and the mixing are therefore contingent in the precausal
configuration~$C$, not derivable from PST's structural ingredients.

\textit{Generation-symmetry no-go.}  The Bernoulli measure $\mu$ on $\{0,1\}^D$
is the unique probability measure invariant under the full automorphism group
$S_D \times \mathbb Z_2$ (bit permutations + complementation; paper §3).
Generation permutation~$S_3$ embeds as a specific subgroup of $S_D$: it permutes
the three pairs of bits $(e_{2k-1}, e_{2k})$ assigned to the three generations
via the Jordan--Wigner identification.  Therefore every observable computed from
$(D, \delta, \mu)$ is automatically $S_3$-invariant across generations.  This
applies in particular to every candidate sub-leading mechanism --- Jordan--Wigner
phases (vanishing in expectation), multi-instanton sectors (acting on the gauge
structure, which is generation-blind by definition), $1/D$ corrections to Mosco
convergence (inheriting the full $S_D \times \mathbb Z_2$ symmetry), and
radion-mediated splittings (acting on the compact fibre direction, shared across
generations).  Each preserves the SU(3)-flavour symmetry of the leading structure.

The leading PST Yukawa matrix is therefore necessarily diagonal with equal
eigenvalues, and the leading PST CKM matrix is the identity, with no
structural mechanism to lift this degeneracy.  The observed values must arise
from the precausal configuration $C$ itself --- they are not features of the
universal structural form that PST identifies.

\textit{SM-sector $\delta m_h^2$ cancellation (status sharpened).}
The structural equal-weighting $N_B = N_F$ of $\mathcal H_D^+$ and $\mathcal H_D^-$
under the Bernoulli measure (Section~\ref{sec:car-fermions}) is a closed result.
However, the numerical Veltman bracket
$6\lambda + \tfrac32 g_2^2 + \tfrac12 g_1^2 - 6 y_t^2$ at $M_* = 1573\,\mathrm{GeV}$
evaluates to $-3.04$ at one loop, an ${\sim}8\sigma$ deviation from zero on
two-loop precision.  PST's structural claim and the numerical Veltman vanishing
are therefore not equivalent statements.  The structural claim stands; clarifying
the precise relation between Bernoulli equal-weighting and the SM-coupling
combination is recorded as a residual.

\subsection{Yukawa hierarchy and CKM mixing as contingent in \texorpdfstring{$T(C)$}{T(C)}: a structural no-go}
\label{sec:yukawa-nogo}

The observed Yukawa hierarchy (charged-lepton masses in the ratio
$1 : 206 : 3477$ at $M_*$) and the non-trivial CKM mixing matrix are
\emph{not} structural predictions of PST.  They are contingent on the
precausal configuration $C$ — specifically, on the asymmetric-tension
distribution $T(C)$ that produced our universe.  This is a precise
statement of PST's scope, established as a theorem in Computation~1
(\verb|computation_01.py|).

\emph{No-go theorem (substrate level).}  The Bernoulli product measure
$\mu = \bigotimes_a \mathrm{Bern}(1/2)$ on $\mathcal{P}(D)$ is invariant
under the full permutation group $S_D \times \mathbb{Z}_2$ acting on the
$D$ substrate bits.  Generation permutation $S_3 \subset S_D$ (permuting
the bit-triples assigned to the three generations) is therefore a
symmetry of any observable computed from $(D, \delta, \mu)$ alone.
Computation~1 verifies this numerically for the leading proposed
generation-asymmetric mechanisms: the Yukawa overlap integral, the
multi-instanton Atiyah--Singer index per representation, and the
Jordan--Wigner expectation in the even-parity sector are all
$S_3$-invariant.  Therefore the leading PST Yukawa coupling is diagonal
with equal eigenvalues, the leading CKM matrix is the identity, and the
observed hierarchy and mixing must enter from outside the structural
ingredients.  The contingency is located precisely in $T(C)$.

\emph{Spectral-triple extension.}  The no-go survives the Connes--Chamseddine
spectral-triple framework via inner-fluctuation analysis.  Since $A_F$
acts on $H_F$ as $a^{(1)} \otimes I_\mathrm{gen}$ with no generation
labels, an inner fluctuation $\omega = \sum a_i\,[D, b_i]$ inherits the
generation structure of $D_F$ unchanged; it cannot lift a degeneracy
absent from $D_F$ nor remove one present.  The generation pattern is
locked in $D_F$, which is a contingent input.  This extension is a
direct corollary of the bimodule structure (Computation~1~\S7).

\emph{Quantitative contingency.}  The observed flavour parameters
(approximately twenty for Dirac neutrinos: nine Yukawa eigenvalues,
four CKM parameters, seven PMNS) consume about $31\%$ of the
$2^D = 64$ real degrees of freedom available in $T(C)$ at $D = 6$
substrate bits (Computation~1~\S8).  PST is therefore neither over-determined
nor under-equipped to encode the observed flavour pattern: there is
ample structural room for the contingency, with a definite count of how
much is consumed.

This is the honest scope-statement: PST does not derive contingent
parameter values; it derives the structural form within which those
parameters live.  In good company, this matches the noncommutative-
geometric Standard Model of Connes--Chamseddine, in which the Yukawa
eigenvalues are inputs to $D_F$ rather than outputs.

\subsection{Chirality of \texorpdfstring{$\mathrm{SU}(2)_L$}{SU(2)L} as a structural consequence}
\label{sec:su2L-structural}

The chirality of the weak gauge group --- that $\mathrm{SU}(2)_L$ acts on
left-handed doublets only --- is a structural consequence of PST's
substrate combined with the directed modal threshold of paper~\S3.
This is the chiral-bimodule support property of the Connes--Chamseddine
finite triple, established for PST in Computations~7, 8, and~11.

The substrate's three-sphere factor $S^3$ in $M = \mathbb{R} \times S^3$
carries an isometry algebra $\mathrm{so}(4) = \mathrm{su}(2)_L \oplus
\mathrm{su}(2)_R$, equivalently the algebra of quaternion left and right
multiplications on $\mathbb{H}$.  This furnishes \emph{two} candidate
$\mathrm{su}(2)$ representations on the internal Hilbert space $H_F$, one
acting on each chirality block of $\gamma_F$.  Both have clean block
support and the two are exact parity conjugates of each other ($P \cdot
H_A \cdot P^{-1} = H_B$ verified numerically; Computation~11,
\verb|computation_11.py|).  Together with Computation~7's verification that
the order-one condition selects the Yukawa form given the chiral
bimodule, the chirality reduces to a single physical question: which of
the two parity-conjugate $\mathrm{su}(2)$ embeddings is realised in
nature?

The answer is the parity-violation mechanism PST already commits to in
\S3: the directed modal threshold $\varepsilon > 0$ orients the $S^3$
volume form, fixing one factor as the physical $\mathrm{SU}(2)_L$.  The
same mechanism that produces parity violation selects the chiral
bimodule.  Chirality of $\mathrm{SU}(2)_L$ is therefore structural,
conditional only on the threshold-direction parity argument PST commits
to elsewhere.

\subsection{Layman's Summary}

This section bridges PST to the particle physics of the Standard Model.  It shows how quantum field theory's path integral arises from the substrate's partition function, and derives the gauge structure from the substrate's internal algebra $A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$.  The algebra itself is derived from P1--P3 in \S\ref{sec:conjecture-1-attack} (Computations~3, 21, and Furey 2014), not posited; the gauge group $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ is then the unimodular unitary group of $A_F$, with gauge bosons and the Higgs arising as inner fluctuations of the Dirac operator.  The colour group $\mathrm{SU}(3)_c$ is the $M_3(\mathbb{C})$ factor of $A_F$, and the octonionic origin of the internal algebra (via the imaginary octonions $\mathrm{Im}(\mathbb{O})$ on which $G_2$ acts) supplies one generation's colour and electric-charge content.  A structural consistency requirement --- gauge anomaly cancellation, which demands that the number of quark colour charges be exactly three --- is not an accident but a theorem conditional on $N_c = 3$: the same $N_c = 3$ that the algebra delivers also makes the quantum theory self-consistent.  The generation count $N_\mathrm{gen} = 3$ is \emph{not} derived: under the octonion route the three complex directions are spent on colour, so the threefold generation replication is a separate open question (item~(C.2) of \S\ref{sec:open-research-work}).  The relation $Z^2 \approx e^{-1}$ between the PST field-normalisation factor and the Standard Model coupling enters as a one-loop numerical near-coincidence, noted but not derived from first principles.

% 9. Physical Predictions
\section{Physical Predictions}

\subsection{Structural and specific predictions}

A foundational theory makes predictions at two levels.  Structural predictions follow
from the form of the theory alone, independently of the specific content of any
particular precausal configuration $C$.  Specific predictions require knowledge of $C$
and concern the particular values of physical parameters in our universe.  This section
focuses on structural predictions: consequences that hold for any instantiated geometry,
regardless of the detailed tension distribution that produced it.  These are the
predictions that a future derivation of the Standard Model from PST would inherit as
constraints; they are also the predictions through which PST can in principle be
distinguished from competing frameworks experimentally.

A second point about the character of predictions is worth stating explicitly.  PST is
not primarily a replacement for existing physical theories at the level at which those
theories are predictive.  General Relativity and Quantum Mechanics already give correct
answers to the questions they ask.  PST operates at the level beneath those questions,
at the level of why those theories work, why the spacetime arena exists, why matter
inhabits it, and why the symmetries they assume hold.  The predictions in this section
are therefore predictions about the domain where existing theories break down, approach
their limits, or require assumptions that PST renders derivable.

\subsection{The Born rule from the Bernoulli measure}
\label{sec:born-rule}

A foundational question that Section~8 noted explicitly as open is the derivation of the
Born rule: the identification of $|\psi|^2$ with measurement probability.  This
subsection provides a structural derivation rooted in the Bernoulli measure and the
Hilbert-space structure established in Section~3.

\textit{The PST Hilbert space.}  The gradient structure derivation of Section~3 established
that the space $L^2(\mathcal{P}(D), \mu)$ of square-integrable complex functionals on
configuration space, with the Bernoulli measure $\mu = \bigotimes_a \mathrm{Bern}(1/2)$,
is a separable Hilbert space.  The inner product is
\begin{equation}
  \langle \phi, \psi \rangle
  \;=\; \int_{\mathcal{P}(D)} \phi^*(C)\,\psi(C)\,\mathrm{d}\mu(C),
  \label{eq:L2-inner-prod}
\end{equation}
and the order parameter $\psi \in L^2(\mathcal{P}(D), \mu)$ is a unit vector (after
normalisation).  This is the quantum state space of PST, derived from the Bernoulli
measure rather than postulated.

\textit{Physical observables as projectors.}  A physical observable $\hat{O}$ with
outcomes $\{o_i\}$ and corresponding eigenspaces $\{\mathcal{H}_i\}$ defines an
orthogonal decomposition $L^2(\mathcal{P}(D),\mu) = \bigoplus_i \mathcal{H}_i$.  The
corresponding projection operators $P_i: L^2 \to \mathcal{H}_i$ satisfy $P_i P_j =
\delta_{ij} P_i$ and $\sum_i P_i = \mathbf{1}$.

\textit{The $|\psi|^2$ probability from the Bernoulli measure.}  The Bernoulli measure
$\mu$ assigns probability $\mu(S)$ to each measurable subset $S \subseteq \mathcal{P}(D)$.
Near the threshold, the physical probability of obtaining outcome $o_i$ is the rate at
which configurations in the eigenspace $\mathcal{H}_i$ cross the modal threshold $\tau$.
The threshold-crossing rate for a configuration $C$ is proportional to the modal
tension excess $|a_{\scriptscriptstyle LG}||\psi(C)|^2$ (from the linear stability analysis
of Section~3): configurations where $\psi$ has large amplitude cross the threshold more
frequently, and the crossing rate is proportional to $|\psi|^2$, not to $|\psi|$ or
$|\psi|^3$.  The probability weight associated with configuration $C$ crossing at outcome
$o_i$ is therefore $|\psi(C)|^2 \cdot \mathbf{1}_{\mathcal{H}_i}(C)$, giving:
\begin{equation}
  P(\text{outcome } o_i \mid \psi)
  \;=\;
  \frac{\displaystyle\int_{C:\,\psi(C)\in \mathcal{H}_i} |\psi(C)|^2\,\mathrm{d}\mu(C)}
       {\displaystyle\int_{\mathcal{P}(D)} |\psi(C)|^2\,\mathrm{d}\mu(C)}
  \;=\;
  \frac{\|P_i\psi\|^2}{\|\psi\|^2}
  \;=\; \|P_i\psi\|^2
  \label{eq:born-rule}
\end{equation}
for unit-normalised $\psi$.  This is the \textbf{Born rule}.

\textit{Gleason's theorem closes the argument.}  The probability assignment
$P \mapsto \|P\psi\|^2$ satisfies all probability axioms and, crucially, it is
\emph{additive on orthogonal projectors}: $P(P_i \oplus P_j) = \|P_i\psi\|^2 + \|P_j\psi\|^2$
for orthogonal $P_i$, $P_j$.  This additivity is guaranteed by the linearity of the
Bernoulli integral:
$\int_{S_i \cup S_j} |\psi|^2\,\mathrm{d}\mu = \int_{S_i}|\psi|^2\,\mathrm{d}\mu +
\int_{S_j}|\psi|^2\,\mathrm{d}\mu$ for disjoint $S_i$, $S_j$.
By Gleason's theorem \cite{Gleason1957} applied to the separable Hilbert space
$L^2(\mathcal{P}(D),\mu)$ (dimension $\geq 3$ for $|D| \geq 2$), the \emph{unique}
non-contextual probability measure on the orthogonal projectors of a Hilbert space is
$P \mapsto \|P\psi\|^2$. The Born rule is therefore not an additional postulate of
PST but a consequence of the Bernoulli measure and the Hilbert-space structure of
$L^2(\mathcal{P}(D),\mu)$.

\textit{What this establishes and what remains open.}  The derivation above establishes
the Born rule for measurements that decompose into orthogonal eigenspaces in
$L^2(\mathcal{P}(D),\mu)$.  What it does not yet supply is the full measurement
postulate (collapse of the wave function) and the treatment of the fermionic
sector.  The canonical commutation relations are recovered in the following subsection, given the identification of the projected Boolean gradient with $\partial_\mu$.

\subsection{Canonical commutation relations from the Boolean gradient}
\label{sec:ccr}

The projected Hilbert space $\mathcal{H} = L^2(M, d^4x)$ carries a natural position
operator:
\begin{equation}
  (\hat{x}_\mu\,\psi)(x) \;=\; x_\mu\,\psi(x),
  \label{eq:position-op}
\end{equation}
defined as multiplication by the spacetime coordinate $x_\mu$.  The momentum operator
is the image of the Boolean gradient under the projection $\Pi$:
\begin{equation}
  \hat{p}_\mu \;=\; -i\hbar\;\Pi\!\bigl(\partial_\mu^{\mathrm{B}}\bigr),
  \label{eq:momentum-op}
\end{equation}
where $\partial_\mu^{\mathrm{B}}$ is the Boolean (Efron--Stein) gradient
(equation~\eqref{eq:boolean-deriv}) in the $\mu$-direction.

\textit{Continuum limit of the Boolean gradient.}  For a smooth function $f \in L^2(M)$
pulled back to $L^2(\mathcal{P}(D),\mu)$ via $\Pi$, the Boolean central difference
acting on a single substrate element $a$ at position $x_a$ is:
\begin{equation}
  (\partial_a^{\mathrm{B}}\psi)(C)
  \;=\; \psi(C\cup\{a\}) - \psi(C\setminus\{a\})
  \;\xrightarrow{\;|dx_a|\to 0\;}
  \;2\,\partial_{x_\mu} f(x)\cdot\hat{e}_\mu,
  \label{eq:bool-to-sobolev}
\end{equation}
where $\hat{e}_\mu$ is the unit vector in the $\mu$-direction and the factor 2 arises
because adding and removing $a$ displace the projected centre $x(C)$ by $+dx_a$ and
$-dx_a$ respectively.  Including the factor $\tfrac{1}{2}$ implicit in the projection
gives $\Pi(\partial_\mu^{\mathrm{B}}) = \partial_\mu$, so that
$\hat{p}_\mu = -i\hbar\,\partial_\mu$ acts on $L^2(M, d^4x)$ as the standard
momentum operator.

\textit{Derivation.}  Acting on any $\psi \in \mathcal{H}$:
\begin{align}
  [\hat{x}_\mu,\,\hat{p}_\nu]\,\psi
  &\;=\; -i\hbar\bigl(x_\mu\,\partial_\nu\psi \;-\;
    \partial_\nu(x_\mu\psi)\bigr)  \notag\\
  &\;=\; -i\hbar\bigl(x_\mu\,\partial_\nu\psi \;-\;
    \delta_{\mu\nu}\psi \;-\; x_\mu\,\partial_\nu\psi\bigr)  \notag\\
  &\;=\; i\hbar\,\delta_{\mu\nu}\,\psi.
  \label{eq:ccr-derived}
\end{align}
The canonical commutation relations $[\hat{x}_\mu, \hat{p}_\nu] = i\hbar\,\delta_{\mu\nu}$
follow once $\hat{p}_\mu = -i\hbar\,\partial_\mu$ is identified on $L^2(M, d^4x)$.  The
product rule step (second line) is then trivial.  The substantive step is the continuum
limit~\eqref{eq:bool-to-sobolev}: the identification of the Boolean gradient with the
standard momentum operator.  The heuristic argument above is made rigorous in the following paragraph via strong
resolvent convergence.

\textit{Rigorous continuum limit via strong resolvent convergence.}
The Mosco-convergence result of Section~\ref{sec:seeley-dewitt}
(equation~\eqref{eq:resolvent-mosco}) makes the limit~\eqref{eq:bool-to-sobolev} rigorous.
Strong resolvent convergence of the rescaled Boolean Laplacian to $\Delta_g$ implies that
for any $f \in H^1(M)$ and $\lambda > 0$, the Boolean gradient
$|D|^{-1/2}\nabla^B(\Pi^\dagger f)$ converges in $L^2(\mathcal{P}(D),\mu)$ to the
Riemannian gradient $\nabla_g f$.  Under $\Pi$, this maps
$\partial_a^B \mapsto \partial_\mu$ in $L^2(M, \mathrm{d}^4x)$, so
$\hat{p}_\mu = -i\hbar\,\partial_\mu$ is the strong limit of the projected Boolean
gradient.  The operator $-i\hbar\,\partial_\mu$ is essentially self-adjoint on
$C_c^\infty(M) \subset L^2(M, \mathrm{d}^4x)$ by standard elliptic theory~\cite{Taylor1996};
its closure is the unique self-adjoint extension on the Sobolev domain $H^1(M)$.  The
continuum-limit identification $\Pi(\partial_\mu^B) = \partial_\mu$ is therefore not merely
heuristic: the Boolean gradient converges to the standard momentum operator in the strong
resolvent topology, and the resulting $\hat{p}_\mu$ is self-adjoint and densely defined.
The canonical commutation relations are thereby established.

\subsection{Quantum Fluctuations}

Near the modal threshold, where $\varepsilon(C) = T(C) - \tau \rightarrow 0$, the order
parameter $\psi$ is in the critical region: the two minima of the modal potential are
about to separate but have not yet done so.  The potential near $\psi = 0$
is nearly flat in the direction of bifurcation, meaning that the order parameter requires
almost no energy to shift significantly.  Small fluctuations in the tension $\delta T$
produce large fluctuations in the instantiated metric:
\begin{equation}
  \delta g_{\mu\nu}
  \;\sim\;
  \eta_{\mu\nu} \cdot \frac{\delta T}{2\,|T - \tau_{\mathrm{null}}|^{1/2}}\bigg|_{T \to \tau_{\mathrm{null}}}
  \label{eq:qfluc}
\end{equation}
This is the PST interpretation of quantum fluctuations.  They are not a fundamental
feature of the ontology, not irreducible randomness built into the fabric of reality.
They are the geometric signature of tension configurations that are close to the modal
threshold: configurations whose instantiation is marginal, whose order parameter has not
settled firmly into one of the two degenerate minima.

The Heisenberg uncertainty principle, $\Delta x \, \Delta p \geq \hbar/2$, acquires a
natural interpretation in this picture.  Position is registered by the realisation map
$\rho$, which assigns to each property a point in the instantiated manifold.  Momentum
is the projection of the directional tension gradient via $\Pi$.  Near the threshold,
$\rho$ is sensitive to small variations in $T(C)$ because the denominator in
equation~\eqref{eq:metric} is small near $\tau_\mathrm{null}$.  This sensitivity means
that determining position precisely, which amounts to resolving $\rho$ to high accuracy,
requires probing configurations at separations close to $\tau_\mathrm{null}$, which in
turn amplifies the uncertainty in the tension gradient that projects as momentum.  The
uncertainty relation is a lower bound on the product of these two projection
sensitivities; it is a structural property of $\Pi$ near the threshold, not an empirical
law imposed from outside.

Wave-particle duality has the same precausal interpretation.  A configuration that is
deep in the instantiated regime, far from the threshold with $\varepsilon(C) \gg 0$, has
a sharply defined order parameter and projects through $\Pi$ as a localised density: a
particle.  A configuration hovering near the threshold has a diffuse, oscillating order
parameter and projects as a spatially extended metric fluctuation: a wave.  The
transition between wave behaviour and particle behaviour is not a collapse caused by
measurement; it is the settling of the order parameter from the near-threshold critical
region into one of the two stable minima.  Measurement in PST is not a physical
interaction that disturbs a pre-existing state; it is the process by which a near-threshold
configuration is brought sufficiently far from $\tau$ that its projection becomes
localised.

\subsection{Spacetime Curvature}

At macroscopic scales, where $\varepsilon(C) \gg 0$ and the order parameter has settled
firmly into the instantiated regime, the dominant effect of the tension distribution is
not fluctuation but curvature.  A region of the instantiated manifold is curved to the
extent that the tension function $T(C)$ varies nonuniformly across the precausal
configurations that project into it.  From the tension-to-metric map~\eqref{eq:metric}
and the definition of the Ricci scalar:
\begin{equation}
  R\bigl(\Phi(C)\bigr) \;\propto\; \Pi\!\left(\nabla_{\C}^2\, T(C)\right)
  \label{eq:curvature}
\end{equation}
Curvature is the projection of the configuration-space Laplacian of asymmetric tension
onto the instantiated manifold.  The Laplacian $\nabla_{\C}^2$ acts on the discrete
power set $\mathcal{P}(D)$; its continuum realisation via $\Pi$ is an open problem.
A perfectly
uniform tension distribution, in which $T(C)$ is the same for all configurations,
projects as flat Minkowski space.  Every departure from flatness in the instantiated
geometry corresponds to a nonuniformity in $T(C)$.  Massive objects are therefore not
the source of curvature in a causal sense; they are the instantiated projection of
regions where the precausal tension distribution was particularly nonuniform, and the
curvature they produce is simply the geometric expression of that nonuniformity.

Gravitational waves follow immediately from this picture.  A propagating disturbance in
$T(C)$ in the precausal configuration space projects as a propagating disturbance in
$g_{\mu\nu}$ in the instantiated manifold.  Gravitational waves are not ripples in a
pre-existing fabric; they are oscillating tension gradients in configuration space
manifesting as metric perturbations.  Their propagation at the speed of light follows
from the fact that the null threshold $\tau_\mathrm{null}$ is an isotropic property of
the projection: disturbances propagate at the speed set by the boundary between timelike
and spacelike separations, which is fixed by $\tau_\mathrm{null}$ alone.  The recent
detection of gravitational waves by LIGO \cite{Abbott2016} is fully consistent with
this picture; the waves are a direct experimental observation of propagating tension
gradients in the precausal substrate.

Black holes are the extreme limit of this gradient picture: they correspond to
configurations in which $\nabla_\C^2 T(C) \rightarrow \infty$ at a point in
configuration space.  The event horizon is the projection of the set of configurations
at which the tension gradient reaches the null threshold condition, beyond which no
precausal configuration can project outward.  The Hawking temperature of a black hole \cite{Bekenstein1973,Hawking1975}
acquires a precausal interpretation as the thermal fluctuation of near-threshold
configurations at the horizon, consistent with the thermodynamic derivation of
Jacobson \cite{Jacobson1995} but now grounded in the precausal structure of $\Pi$
rather than in a phenomenological entropy argument.

\subsection{The weak-field limit and the recovery of Newtonian gravity}

Equation~\eqref{eq:einstein-derived} is the Einstein field equation.  For the claim that PST derives GR to be substantive, rather than merely renaming GR's content, the field equation must reduce to Newtonian gravity in the non-relativistic, weak-field limit.  The reduction is standard in GR; it is reproduced here in PST notation to confirm that no additional structure interferes.

In the weak-field, slow-motion regime, write $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ with $|h_{\mu\nu}| \ll 1$.  The projected tension reduces to a non-relativistic matter density: $\Pi(T(C))(x) \approx \rho(x)$, so that $T_{00} = \rho$ and all spatial components are negligible.  Working in harmonic gauge with $\bar{h}_{\mu\nu} = h_{\mu\nu} - \tfrac{1}{2}\eta_{\mu\nu}h$, the linearised field equations become:
\begin{equation}
  \nabla^2\bar{h}_{00} \;=\; -16\pi G\,\rho
  \label{eq:weakfield}
\end{equation}
Setting $\bar{h}_{00} = -4\phi$, where $\phi$ is the Newtonian gravitational potential ($g_{00} \approx -(1 + 2\phi)$), equation~\eqref{eq:weakfield} reduces to:
\begin{equation}
  \nabla^2\phi \;=\; 4\pi G\,\rho
  \label{eq:poisson}
\end{equation}
Poisson's equation for Newtonian gravity is recovered exactly, with $G$ playing the role of the unit-conversion factor between the projected matter density $\rho = \Pi(T(C))$ and the metric perturbation $\phi$.  At the next post-Newtonian order the same field equations reproduce perihelion precession ($\gamma = \beta = 1$) and gravitational light bending at the value GR predicts, since equation~\eqref{eq:einstein-derived} is the Einstein equation; PST introduces no additional fields or couplings that would shift the post-Newtonian parameters away from their GR values.  The sole physical differences from standard GR appear at scales near $d_0$ (the Casimir correction of Section~10) and in the ontological provenance of $G$ and $\Lambda$.

\subsection{Stress-Energy as Projected Tension}

The unification of geometry and matter as coprojections of the same precausal tension
resolves a longstanding conceptual problem in theoretical physics.  In General Relativity,
the statement ``matter curves spacetime'' is taken as a foundational description: matter
and spacetime are separate ontological kinds, and one acts on the other.  In Quantum
Field Theory, fields and their excitations are defined on a background spacetime, which
is treated as a fixed classical arena.  The two descriptions are mutually inconsistent at
a deep level: the quantum fields of QFT back-react on the spacetime metric, but the
resulting semiclassical equations are not derivable from a unified action, and the
renormalisation of the stress-energy tensor introduces ambiguities that have no physical
resolution within the framework.

PST dissolves this problem by removing the distinction between the two categories.
Geometry and stress-energy are coprojections: both are outputs of $\Phi$ and $\Pi$
acting on the same input $T(C)$.  The phrase ``matter curves spacetime'' is a useful
shorthand within the instantiated domain, but its deeper meaning is that the same
precausal tension structure that projects as matter also projects as curvature, at every
point simultaneously.  There is no back-reaction because there are not two things
acting on each other; there is one projection producing two aspects of a single
geometric object.

Mass, in this picture, is a measure of how far a configuration sits from the modal
threshold.  A configuration deep in the instantiated regime, with $\varepsilon(C) \gg 0$,
has a large order parameter $r_0 \propto \sqrt{\varepsilon}$ and projects as a dense,
heavy field configuration.  A configuration hovering near the threshold has a small
order parameter and projects as a light or massless excitation.  Massless fields,
including the photon, correspond to configurations that project through $\Pi$ at
exactly the null threshold $\tau_\mathrm{null}$: they occupy the boundary between
timelike and spacelike separation, which is precisely the condition for propagation at
the speed of light.  The masslessness of light is not a postulate; it is the projection
of the null threshold condition into the instantiated geometry.

\subsection{Vacuum Phenomena and the Casimir Effect}

Standard quantum field theory predicts a Casimir pressure between two parallel perfectly
conducting plates at separation $d$ of \cite{Casimir1948}:
\begin{equation}
  P_{\mathrm{standard}}(d) = -\frac{\pi^2 \hbar c}{240\, d^4}
  \label{eq:casimirstandard}
\end{equation}
At $d = 100\;\mathrm{nm}$ this evaluates to approximately $-13\;\mathrm{Pa}$, measurable
experimentally \cite{Lamoreaux1997,Bressi2002}.  The effect has been measured to
sub-percent accuracy at separations $160$--$750\;\mathrm{nm}$ \cite{Decca2003,Bordag2001}.

PST provides a precausal account of this effect.  The conducting plates impose boundary
conditions on the projection operator $\Pi$, restricting which sub-configurations $C'$
contribute to the kernel integral at points $x$ between the plates.  In standard QFT,
equivalent boundary conditions are imposed on field modes, and the two descriptions agree
at leading order.  The departure arises because the kernel $K(x, C') = \delta(x -
\rho(C'))$ has a spatial resolution structure set by the characteristic scale $d_0$ at
which the realization map $\rho: D \rightarrow M$ becomes sensitive to boundary geometry.
At separations $d \gg d_0$ the two descriptions are equivalent; at $d \sim d_0$ they
diverge.

By dimensional analysis, and the parity constraint that the correction must be even in
$d_0/d$ (odd powers would require a preferred orientation inconsistent with the isotropy
of the precausal substrate), the leading PST correction to the Casimir pressure is:
\begin{equation}
  P_{\mathrm{PST}}(d)
  = P_{\mathrm{standard}}(d)
    \left[1 + \xi\!\left(\frac{d_0}{d}\right)^{\!2}
    + \mathcal{O}\!\left(\!\left(\frac{d_0}{d}\right)^{\!4}\right)\right]
  \label{eq:psttot}
\end{equation}
where $\xi$ is a dimensionless coefficient of order unity set by the structure of $K$.
The correction term scales as $d^{-6}$:
\begin{equation}
  \delta P_{\mathrm{PST}}(d) = -\frac{\xi\,\pi^2 \hbar c\, d_0^2}{240\, d^6}
  \label{eq:casimir}
\end{equation}
The $d^{-6}$ scaling is a necessary consequence of any single-length-scale, isotropic
short-distance modification of the vacuum: the leading even power of $d_0/d$ multiplied
by the standard $d^{-4}$ Casimir term gives $d^{-6}$ regardless of the specific theory.
Absence of a $d^{-6}$ correction would falsify PST; presence of a $d^{-6}$ correction is
consistent with PST but does not select it from other minimal-length or EFT-cutoff models.
The quantity specific to PST is the \emph{amplitude} $\xi d_0^2$: $\xi = 90/\pi^2$
is fixed exactly by the kernel $K$ of the projection operator (below), while the
scale $d_0$ is the Landau--Ginzburg coherence length, set by the LG dynamics.

\subsection{The parameter-free exponent and the structure of falsifiability}

A careful reader will observe that equation~\eqref{eq:casimir} contains two quantities
whose precise values determine the amplitude: the coefficient $\xi$ and the scale $d_0$.
The situation is as follows.

The discreteness scale $d_0$ is the Landau--Ginzburg coherence length; its value is
set by the LG vacuum structure and is not pinned by the three foundational
postulates alone.  The coefficient $\xi$, by contrast, is determined exactly by the
Gaussian moments of the PST projection kernel $K$:
\begin{equation}
  \xi \;=\; \frac{15}{\pi^2}\,
    \frac{\displaystyle\int_0^\infty e^{-x^2/(4d_0^2)}\,x^4\,dx}
         {\displaystyle d_0^2\int_0^\infty e^{-x^2/(4d_0^2)}\,x^2\,dx}
  \;=\; \frac{15}{\pi^2}\times 6 \;=\; \frac{90}{\pi^2} \;\approx\; 9.12,
  \label{eq:xi-exact}
\end{equation}
\emph{Dimensional verification:} $\xi$ is a pure number.  Substituting $u = x/(2d_0)$
evaluates the integrals exactly:
\begin{equation}
  \int_0^\infty e^{-x^2/(4d_0^2)}\,x^4\,dx \;=\; 12\sqrt{\pi}\,d_0^5,
  \qquad
  \int_0^\infty e^{-x^2/(4d_0^2)}\,x^2\,dx \;=\; 2\sqrt{\pi}\,d_0^3.
  \label{eq:xi-moments}
\end{equation}
Hence the numerator carries dimension $d_0^5$ (length$^5$) and the denominator carries
$d_0^2 \times d_0^3 = d_0^5$ (length$^5$), so the ratio is dimensionless:
$12\sqrt{\pi}\,d_0^5 / (d_0^2 \times 2\sqrt{\pi}\,d_0^3) = 6$, and $\xi = 15\times 6/\pi^2 = 90/\pi^2$.
The $d_0^2$ in the denominator of equation~\eqref{eq:xi-exact} is not ``dangling'':
it is the factor that cancels the $d_0^5$ from the $x^4$ moment against the
$d_0^3$ from the $x^2$ moment, leaving a pure number.  This cancellation was verified
symbolically (Computations §verify\_track\_b).

The coefficient $\xi = 90/\pi^2 \approx 9.12$ is therefore fixed exactly; the
amplitude $\xi\,d_0^2$ is parameter-free in its coefficient, with the scale $d_0$
the Landau--Ginzburg coherence length.

What does \emph{not} depend on $\xi$ or $d_0$ is the power-law \emph{exponent}.  The
exponent $-6$ in $\delta P_{\mathrm{PST}} \propto d^{-6}$ follows from dimensional
analysis and the parity constraint alone: the correction must be even in $d_0/d$ (odd
powers would require a preferred orientation of the substrate, inconsistent with the
isotropy established in Section~3), and the leading even power is $(d_0/d)^2$, which,
multiplied by the standard $d^{-4}$ Casimir scaling, gives $d^{-6}$.  This argument uses
only the existence of a length scale $d_0$ and the symmetry of the substrate; it does not
use the values of $\xi$ or $d_0$.  The exponent $-6$ is therefore a \emph{categorical}
prediction: it cannot be changed by adjusting $\xi$ upward or downward, by moving $d_0$
within its allowed range, or by any other parametric variation within the theory.

The experimental consequence is that the test of PST's Casimir prediction has two
logically distinct layers.  The \emph{first layer} is qualitative and parameter-free:
does the measured Casimir pressure, after subtracting the Lifshitz-theory prediction
(which already accounts for finite conductivity, surface roughness, and thermal
fluctuations to high precision), show a residual that scales as $d^{-6}$?  If the residual
is consistent with zero within experimental uncertainty, PST is constrained.  If it shows
a different power law that cannot be attributed to known corrections, PST is falsified.
A $d^{-6}$ residual is a \emph{necessary} condition for PST at this order, but not
sufficient: other single-length-scale modifications of the vacuum also predict $d^{-6}$
\cite{Bordag2001}.  At the first layer PST therefore makes a categorical prediction
about \emph{the form of the correction} (the exponent), not about its amplitude.

The \emph{second layer}, conditioned on observing $d^{-6}$ scaling, is quantitative.
$\xi = 90/\pi^2$ is fixed exactly by the projection-kernel moments; the amplitude
$\xi d_0^2$ contains a single free parameter $d_0$, the Landau--Ginzburg coherence
length, which is \emph{not derived from P1--P3} and which existing
$160$--$750$~nm precision Casimir data constrains only by an upper bound,
$d_0 < 16$~nm (\S\ref{eq:d0bound}).  A measured residual at $d^{-6}$ with amplitude
$A$ therefore determines $d_0 = \sqrt{A/\xi}$ from the data rather than confirming a
parameter-free prediction.  Inconsistency between the inferred $d_0$ from different
experimental separations \emph{would} falsify PST at the second layer; a single
amplitude measurement at the precision required to extract a $d^{-6}$ residual
constitutes a determination of $d_0$, not a sharp test.

Honest comparison with parameter-rigid tests: the gravitational-wave tests of
General Relativity constrain a specific waveform morphology in which both the
exponent (post-Newtonian structure) and the amplitude (Bayesian fit to source
parameters) are predicted up to a finite list of source-parameter inputs (masses,
spins).  PST's Casimir test is exponent-rigid but not amplitude-rigid until $d_0$
is independently fixed; in this sense PST's current falsifiability is intermediate
between a parameter-free theory and a fit, closer to the parameter-free end at the
exponent layer and closer to the fit end at the amplitude layer.  The asymmetry is
real and is acknowledged in the open-research register
(\S\ref{sec:open-research-work}): deriving $d_0$ from the postulates, or anchoring
it to another PST observable, is required to upgrade the second layer to a sharp
parameter-free test.

The $d^{-6}$ scaling is separable from the standard $d^{-4}$ Casimir dependence by
fitting measurements at multiple separations.  The correction budget at relevant
separations is discussed in detail in Section~\ref{sec:casimir-budget}.  In brief: finite-conductivity
corrections enter via the Lifshitz formula as an effective $d^{-5}$ correction at leading
order in $\lambda_{\mathrm{plasma}}/d$, and this Lifshitz-corrected Casimir force
is well-known and can be subtracted with high precision.  The PST $d^{-6}$ signal would
appear as a residual after this subtraction.  Surface roughness introduces corrections
at the spatial-frequency scale of the roughness spectrum, with a known power-law
dependence; thermal corrections have a distinct temperature dependence isolable by
varying plate temperature.  The subtraction strategy is the standard one used in current
precision Casimir experiments \cite{Bordag2001}; the challenge is the accuracy required
to resolve a ${\sim}2\%$ residual at $d = 50$~nm after subtracting Lifshitz corrections
that are themselves of order 100\% at this separation.

\subsection{Observational constraint}
Precision Casimir measurements at separations $160$--$750\;\mathrm{nm}$ report
approximately $1\%$ agreement with standard QFT \cite{Decca2003,Bordag2001}.  Requiring
$|\xi|(d_0/d)^2 < 0.01$ at $d = 160\;\mathrm{nm}$ gives:
\begin{equation}
  d_0 \;<\; \frac{16\;\mathrm{nm}}{\sqrt{|\xi|}}
  \label{eq:d0bound}
\end{equation}
For $|\xi| \sim 1$ this constrains $d_0 < 16\;\mathrm{nm}$.  The PST correction becomes
detectable at the $1\%$ level for separations $d \lesssim 10\,d_0$, placing the
accessible signal at $d \lesssim 160\;\mathrm{nm}$ at the precision of existing measurements.  A
ratio test between measurements at two separations $d_1$ and $d_2$ provides a clean,
apparatus-independent signature:
\begin{equation}
  \frac{P(d_1)/P_{\mathrm{standard}}(d_1)}{P(d_2)/P_{\mathrm{standard}}(d_2)} - 1
  \;\propto\;
  \left(\frac{d_0}{d_1}\right)^{\!2} - \left(\frac{d_0}{d_2}\right)^{\!2}
  \label{eq:ratiotest}
\end{equation}
This represents a falsifiable prediction distinguishable from both standard QFT and from
Lifshitz-theory corrections.

What the theory fixes without free parameters is the \emph{form} of the
correction: the $d^{-6}$ scaling and the coefficient $\xi = 90/\pi^2 \approx 9.12$
(equation~\eqref{eq:xi-exact}).  The overall amplitude scales as $(d_0/d)^2$, with
$d_0$ the Landau--Ginzburg coherence length of the modal condensate (Section~10);
its absolute magnitude is a condensate property, naturally in the nanometre range
and many orders of magnitude above $\ell_P$, rather than a number the three
postulates pin down.  For a coherence length of order a few nanometres the
correction is at the edge of current precision near $d = 160\;\mathrm{nm}$, reaches
several percent near $d = 50\;\mathrm{nm}$, within reach of next-generation
experiments, and becomes order-unity below $d = 20\;\mathrm{nm}$.  The Casimir
measurement is accordingly a genuine predicted detectable signal, not merely a
formal bound, with its precise size set by $d_0$.


\subsection{Layman's Summary}

Having established the theoretical structure, this section identifies what PST predicts that can actually be measured.  General Relativity and quantum field theory already predict gravitational curvature and quantum fluctuations correctly; PST recovers both.  Where PST departs from standard physics is at very short distances: because the substrate has a finite grain size, the vacuum is not perfectly smooth, and this deviation appears in the Casimir effect, the tiny attractive force between two uncharged metal plates held very close together.  PST predicts that at separations below about 100 nanometres the force scales differently from standard predictions, by an amount large enough to detect with near-future precision instruments.

% 9. Grounding d_0
\section{Grounding $d_0$: Modal Coherence and the Casimir Prediction}
\label{sec:casimir}

\textit{Conditionality.}  The parameter-free content of this section --- the
$d^{-6}$ Casimir scaling and its coefficient $\xi = 90/\pi^2$ --- depends only on
the moments of the projection kernel and is independent of the magnitude of
$d_0$.  The absolute size of the Casimir signal at a given separation scales as
$(d_0/d)^2$ and is therefore set by the coherence length $d_0$, whose value is a
property of the modal condensate rather than an output of $(D, \delta, \mu)$
alone.  The mathematical content is sound within that scope.

\subsection{The default assumption that must be dropped}

The Casimir correction derived in Section~9 has a distinctive $d^{-6}$ scaling whose
amplitude is proportional to $d_0^2$.  Section~7 related $G$ and $d_0$ via
$G = \alpha_G d_0^2$ in natural units, but did not constrain $\alpha_G$.  A reader might
assume $\alpha_G \sim 1$, which would place $d_0 \approx \ell_P \approx
1.6 \times 10^{-35}$~m.  At $d = 160$~nm this gives $(d_0/d)^2 \sim 10^{-56}$:
the correction would be fifty-six orders of magnitude below the leading term and
permanently undetectable.

The assumption $\alpha_G \sim 1$ is not a result of PST.  It was never derived; it was
imported as a prior that the only relevant length scale in the theory is $\ell_P$.  This
section shows the prior is wrong: $d_0$ is not the microscopic grain of the substrate but
the coherence length of the modal condensate, which near a second-order transition sits
many orders of magnitude above the grain.  This places $d_0$ in the nanometre range, far
above $\ell_P$, so that the Casimir correction is a genuine predicted detectable signal,
not a formal statement about an unobservably small parameter.

\subsection{Two distinct scales: microscopic grain versus coherence length}

The conflation of $d_0$ with $\ell_P$ rests on identifying two physically distinct
lengths: the \emph{microscopic grain} of the substrate and the \emph{coherence length}
of the modal condensate.  The closest condensed-matter analogue makes the distinction
precise.

In BCS superconductivity the ionic crystal has lattice constant $a \approx 0.3$~nm.
This is the microscopic scale of the problem.  The characteristic scale of the
superconducting order, however, is the Cooper pair coherence length:
\begin{equation}
  \xi_{\mathrm{BCS}} \;=\; \frac{\hbar v_F}{\pi\,\Delta}
  \label{eq:bcs_xi}
\end{equation}
where $v_F$ is the Fermi velocity and $\Delta$ the superconducting gap.  Because
$\Delta/E_F$ is exponentially small for conventional superconductors
($\sim 10^{-3}$--$10^{-5}$), the coherence length satisfies
$\xi_{\mathrm{BCS}} \sim 10^3 a$ to $10^6 a$.  The scale at which the condensate is
probed exceeds the lattice constant by three to six orders of magnitude.  This is not
an accident: it is the universal consequence of near-criticality.  Near a
second-order phase transition the correlation length diverges; a system barely below
its critical point has a coherence length that can be macroscopically large even when
the underlying lattice is atomic.

PST's modal condensate has the same architecture.  The substrate carries a microscopic
scale $\ell_*$, the intrinsic grain of the distinction space $D$ at which primitive
differences are registered.  Nothing in the PST axioms forces $\ell_* = \ell_P$; and
even if it did, $d_0$ would not equal $\ell_*$.  The scale $d_0$ is defined by the
projection operator $\Pi$ as the finest resolution at which the realisation map
$\rho: D \to M$ can distinguish spacetime points.  This is the spatial scale over
which the modal order parameter $\psi$ varies: its coherence length.  In the
Landau-Ginzburg functional of Section~3:
\begin{equation}
  F[\psi] \;=\; \int \mathrm{d}^4x \left[
    \tfrac{1}{2}\,c_{\scriptscriptstyle LG}\,|\nabla\psi|^2
    \;+\; a_{\scriptscriptstyle LG}\,|\psi|^2
    \;+\; b_{\scriptscriptstyle LG}\,|\psi|^4
  \right]
  \label{eq:lgfull}
\end{equation}
the coherence length of $\psi$ is \cite{GinzburgLandau1950}:
\begin{equation}
  d_0 \;=\; \xi_\psi \;=\; \sqrt{\frac{c_{\scriptscriptstyle LG}}{2\,|a_{\scriptscriptstyle LG}|}}
  \label{eq:xi_psi}
\end{equation}
Near the modal threshold, $a_{\scriptscriptstyle LG} \propto (T/\tau - 1)$, so
$\xi_\psi \to \infty$ as $T \to \tau^+$: the correlation length diverges at the
transition, exactly as in any second-order phase transition.  The universe exists
because $T > \tau$, but nothing requires $T$ to lie far above $\tau$.  A substrate
that barely exceeds the modal threshold has $d_0 = \xi_\psi \gg \ell_*$.  How much
larger $d_0$ is depends on how close to threshold the substrate sits, a property
of the modal condensate rather than a number the postulates fix.

\subsection{Modal sublimation and the substrate scales}

The functor $\Phi: \mathcal{S} \to \mathcal{G}$ maps the precausal substrate to an
instantiated geometry with four large dimensions (three spatial, one temporal),
established by the coinstantiation argument of Section~5 and the orbital stability
requirement of Section~6.  The remaining substrate structure is carried by the
finite noncommutative internal factor (Section~\ref{sec:open-research-work}), which
has no spatial extent: it is the algebraic home of the gauge and matter structure,
not a set of compact spatial dimensions.

Two length scales characterise the instantiated theory.  The first is the
Landau--Ginzburg coherence length of the order parameter,
\begin{equation}
  d_0 \;=\; \sqrt{\frac{c_{\scriptscriptstyle LG}}{2\,|a_{\scriptscriptstyle LG}|}},
  \label{eq:coh_len}
\end{equation}
which sets the substrate discreteness scale that enters the Casimir correction.  The
second is the modal scale $\ell_* = \hbar/M_*$, the length associated with the energy
$M_*$ at which the modal condensate forms (identified below with $M_* = 4\pi m_h$).
With the Higgs identification $r_0 = v$ and
$|a_{\scriptscriptstyle LG}| = m_h^2/4$, the coherence length $d_0$ is fixed by the
Landau--Ginzburg vacuum structure rather than by any compactification, and its
numerical value is not determined by the three postulates alone.

\subsection{The fundamental modal scale and the electroweak identification}

In PST, $\ell_*$ is the length associated with the energy at which the modal condensate
forms, the scale at which the order parameter $\psi$ first acquires a non-zero
expectation value.  The current section uses a structural identification to pin $\ell_*$
to a known energy scale.

The PST order parameter $\psi$ and the Standard Model Higgs field $\phi$ occupy the
same mathematical role.  Both are complex scalars that:
\begin{enumerate}
  \item inhabit a degenerate minimum of a $\mathrm{U}(1)$-symmetric quartic potential;
  \item break that symmetry by acquiring a non-zero vacuum expectation value;
  \item produce a massive longitudinal (Higgs/radial) mode and a massless Goldstone
    direction (photon/$S^1$ vacuum manifold).
\end{enumerate}
In PST's ontological framework, the Higgs condensate is the four-dimensional projection
of the modal condensate: $\phi = \Pi(\psi)$, the order parameter as seen from inside
the instantiated geometry.  If this identification holds, then the modal condensate and
the Higgs condensate share the same vacuum structure, and $M_* = \hbar c/\ell_*$ is the
energy scale at which the full precausal modal structure becomes manifest, above the
electroweak vev $v = 246$~GeV, where the condensate itself forms, but within the regime
constrained by LHC electroweak searches.

A reader will note that the identification now places three distinct energy scales near
the condensate, and their consistency should be made explicit.  The radial-mode mass
derived in Section~7, $m_h^2 = 4\varepsilon$, and the vacuum radius $r_0^2 = \varepsilon/(2b)$
of equation~\eqref{eq:vacmanifold} together give $m_h^2 = 8b\,r_0^2$, so the three
quantities are not independent: fixing any two fixes the third through the quartic
coefficient $b$.  Identifying $r_0$ with the Higgs vacuum expectation value $v = 246$~GeV
and $b$ with one quarter of the Standard-Model Higgs self-coupling,
$b = \lambda/4$ with $\lambda \approx 0.13$, returns
$m_h = \sqrt{2\lambda}\,v \approx 125$~GeV, the measured Higgs mass
\cite{ATLASHiggs2012,CMSHiggs2012}, with no additional input.  The modal scale $M_* = 4\pi m_h \approx 1573\;\mathrm{GeV}$ is a fourth and separate
quantity (Section~\ref{sec:renorm}): it is not the vev and not the radial-mode mass
but the threshold energy $\hbar c/\ell_*$ at which the full modal structure becomes
manifest.  The Higgs mass and the vev are the low-energy observables of the modal
condensate.  All three are mutually consistent through $b = \lambda/4$ and $v < M_*$.

The Higgs field acquires its vacuum expectation value at $v = 246$~GeV.  Precision
electroweak data and LHC direct searches constrain new physics associated with the
electroweak condensate to begin above approximately $1$~TeV.

The choice of $M_*$ is settled.  $M_* = 4\pi m_h \approx 1573\;\mathrm{GeV}$
is derived as a structural prediction of PST (Section~\ref{sec:renorm}) from the
requirement that the one-loop LG quartic matches the SM Higgs self-coupling at threshold.
This derivation uses $m_h = 125.20\;\mathrm{GeV}$ as the sole measured input and requires
no free parameters beyond the measured input $m_h$.  The structural prediction
throughout the rest of the paper is $M_* = 4\pi m_h \approx 1573\;\mathrm{GeV}$.

The substrate grain length is then $\ell_* = \hbar c / M_*$.  The Casimir
correction's power-law exponent and its coefficient $\xi = 90/\pi^2$ are structural
predictions; the onset scale $d_0$ is the Landau--Ginzburg coherence length, set by
the LG dynamics.  Taking $M_* = 4\pi m_h \approx 1573\;\mathrm{GeV}$:
\begin{equation}
  \ell_* \;=\; \frac{\hbar c}{M_*}
         \;=\; \frac{197.3\;\mathrm{MeV\cdot fm}}{1573\;\mathrm{GeV}}
         \;\approx\; 1.253 \times 10^{-19}\;\mathrm{m}
  \label{eq:lstar_value}
\end{equation}

\subsection{The substrate coherence scale}

The Casimir correction below is governed by the substrate coherence length $d_0$
of equation~\eqref{eq:coh_len}.  Here $d_0$ is set by the
Landau--Ginzburg vacuum structure and is not fixed in value by the three
foundational postulates alone; accordingly the Casimir correction is a
parameter-free prediction in its power-law \emph{exponent}, with the onset scale
$d_0$ left to be determined by the Landau--Ginzburg dynamics or by experiment.

\subsection{Casimir detectability at each separation regime}

Evaluating at a representative coherence length $d_0 \approx 4.92$~nm (the theory
fixes the $d^{-6}$ form and the coefficient $\xi = 90/\pi^2 \approx 9.12$ exactly,
but not the magnitude of $d_0$), the Casimir correction of
equation~\eqref{eq:casimir} gives a concrete signal at each accessible
plate separation.

At $d = 160$~nm, the baseline of existing precision measurements \cite{Decca2003}:
\begin{equation}
  \left.\left(\frac{d_0}{d}\right)^{\!2}\right|_{d = 160\,\mathrm{nm}}
  \;=\; \left(\frac{4.92}{160}\right)^2
  \;\approx\; 9.45 \times 10^{-4}
  \label{eq:casimir_160}
\end{equation}
The PST correction with $\xi = 90/\pi^2$ is $9.12 \times 0.094\% \approx 0.86\%$.  This
lies at the edge of the current $1\%$ precision floor; the theory is consistent with all
existing data, and the correction may become detectable with current-generation instruments.

At $d = 50$~nm, accessible to torsion-pendulum and AFM-based Casimir experiments
currently under development:
\begin{equation}
  \left.\left(\frac{d_0}{d}\right)^{\!2}\right|_{d = 50\,\mathrm{nm}}
  \;=\; \left(\frac{4.92}{50}\right)^2
  \;\approx\; 9.68 \times 10^{-3}
  \label{eq:casimir_50}
\end{equation}
The correction with $\xi = 90/\pi^2$ is $9.12 \times 0.97\% \approx 8.8\%$, well above the
$1\%$ current floor and unambiguously detectable at next-generation sensitivity.

At $d = 20$~nm, near the practical lower limit for parallel-plate measurements:
\begin{equation}
  \left.\left(\frac{d_0}{d}\right)^{\!2}\right|_{d = 20\,\mathrm{nm}}
  \;=\; \left(\frac{4.92}{20}\right)^2
  \;\approx\; 0.0606
  \label{eq:casimir_20}
\end{equation}
A correction of $9.12 \times 6.1\% \approx 55\%$ over standard QFT is unambiguous: it exceeds finite
conductivity corrections (typically $2$--$5\%$ at these separations), surface roughness
corrections ($\lesssim 1\%$ for well-prepared surfaces), and thermal corrections
(subdominant below $100$~nm at room temperature \cite{Bordag2001}).

The distinguishing experimental signature throughout is the power law.  Standard QFT
and all known material corrections modify the coefficient of $d^{-4}$ or introduce
$d^{-3}$ thermal terms; none contribute at order $d^{-6}$.  The $d^{-6}$ PST term is
therefore separable by fitting measurements at multiple separations, and the
$d^{-6}$/$d^{-4}$ ratio grows as plate separation decreases, providing an unambiguous
directional signature.  Substituting the same representative $d_0 \approx 4.92$~nm
explicitly into equation~\eqref{eq:casimir}:
\begin{equation}
  \delta P_{\mathrm{PST}}(d)
  \;=\; -\frac{\xi\,\pi^2\hbar c \times (4.92\;\mathrm{nm})^2}{240\,d^6}
  \label{eq:casimir_explicit}
\end{equation}
The coefficient $\xi = 90/\pi^2 \approx 9.12$ is derived exactly from the
Gaussian moments of the substrate weight function; see equation~\eqref{eq:xi-exact}.
Measuring the $d^{-6}$ component at two separations via
the ratio test~\eqref{eq:ratiotest} determines $d_0$ without requiring knowledge of
$\xi$, and a subsequent fit to the absolute magnitude determines $\xi$.  The two
measurements together overconstrain the two parameters, providing an internal
consistency check.

\subsection{Falsifiability and the experimental programme}

The PST Casimir prediction satisfies strict experimental falsifiability.  The prediction
is refuted if:
\begin{enumerate}
  \item Precision measurements at $d = 50$~nm agree with standard QFT to better than
    $1\%$.  This constrains $d_0 < 5$~nm, pushing the implied $M_*$ above
    approximately $2$~TeV, inconsistent with the electroweak condensate identification.
  \item Multi-separation ratio tests (equation~\eqref{eq:ratiotest}) in the $20$--$50$~nm
    range show no $d^{-6}$ component at the $0.5\%$ level.
  \item A non-$d^{-6}$ power law is detected, inconsistent with PST's even-in-$d_0/d$
    expansion (any power law intermediate between $d^{-4}$ and $d^{-6}$ would falsify
    the theoretical structure, not merely constrain $d_0$).
\end{enumerate}
The prediction is confirmed if:
\begin{enumerate}
  \item A $d^{-6}$ component is detected at separations below $50$~nm, with amplitude
    consistent with $d_0 \in [6, 10]$~nm.
  \item The ratio test between separations $d_1 \approx 20$~nm and
    $d_2 \approx 160$~nm yields $(d_0/d_1)^2 - (d_0/d_2)^2$ consistent with
    $d_0 \approx 7$~nm.
  \item The fitted $d_0$ is consistent with $\ell_* = \hbar c/M_*$ for an $M_*$
    compatible with LHC measurements of the electroweak new-physics threshold.
\end{enumerate}
Current experiments \cite{Decca2003,Bordag2001} at $160$--$750$~nm agree with standard
QFT to $\sim 1\%$, consistent with $d_0 < 16$~nm from equation~\eqref{eq:d0bound}.
An onset scale $d_0$ below this bound is consistent with current Casimir data, and
the next generation of sub-$50$~nm Casimir experiments will test the prediction.

The $d^{-6}$ correction structure is independent of the numerical value of $d_0$:
its power-law exponent and its coefficient $\xi = 90/\pi^2$ are structural, while the
onset scale $d_0$ is the Landau--Ginzburg coherence length set by the LG dynamics.  A
measured $d^{-6}$ residual at sub-$50$~nm separations would support the
substrate-discreteness prediction; its absence at any separation would falsify it.
The modal scale $M_* = 4\pi m_h \approx 1.573$~TeV is a separate, collider-testable
prediction: PST expects new physics associated with the modal condensate near this
scale.

\subsection{Experimental status of $M_*$ and the ADD comparison}
\label{sec:add-constraints}

PST is superficially analogous to the ADD model \cite{ADD1998} only in involving
structure beyond the four macroscopic dimensions.  A critical difference makes
ADD collider and astrophysical bounds inapplicable to PST: the additional
structure is a finite noncommutative internal factor, not large extra spatial
dimensions, and its only associated length is the sub-fermi modal scale
$\ell_* = \hbar/M_* \approx 1.25 \times 10^{-19}$~m, far from the
millimetre-to-micron range ADD requires for its gravity-dilution mechanism.  The
ADD bounds on the fundamental scale --- LHC mono-jet/mono-photon
($M_* \gtrsim 5$--$8$~TeV), SN1987A and neutron-star KK-graviton emission --- are
all derived for macroscopic extra dimensions, with KK gravitons coupling to the
full stress-energy tensor at tree level with the four-dimensional Newton
constant.  None of those premises hold for a finite noncommutative internal
geometry, so the bounds do not transfer in their standard form.

What remains is a genuine empirical-consistency question, not a derived
constraint: whether $M_* = 4\pi m_h \approx 1.573$~TeV is consistent with the
existing data once the PST-specific coupling structure (set by the projection
operator $\Pi$) is computed.  That computation has not been performed, and the
question is listed among the open items (Section~\ref{sec:open-research-work},
item~(E)).  The coherence length $d_0$ that sets the absolute size of the Casimir
signal is a property of the modal condensate; its magnitude is not pinned by the
postulates, so the signal's amplitude is predicted in form (the $d^{-6}$ scaling
and the coefficient $\xi = 90/\pi^2$) but not in absolute size.

\subsection{Wilson coefficients and electroweak precision: $\Delta S = \Delta T = 0$}
\label{sec:wilson-ewpo}

PST's structural commitments fix the dim-6 SMEFT Wilson coefficients that
control the electroweak $S$ and $T$ parameters to zero exactly, so PST
contributes no new-physics correction to the global electroweak fit
(Computation~10, \verb|computation_10.py|, \S10).

The argument is structural.  PST's Landau--Ginzburg modal potential
$V(H^\dagger H) = -\varepsilon\,(H^\dagger H) + \tfrac{1}{4}(H^\dagger H)^2$
is a function of the $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R$ singlet
$H^\dagger H$ only; it is manifestly custodial-symmetric.  The dim-6
operators $\mathcal{O}_{HWB} = (H^\dagger \sigma^a H) W^{a\mu\nu} B_{\mu\nu}$
and $\mathcal{O}_{HD} = (H^\dagger D_\mu H)^\dagger (H^\dagger D^\mu H)$
are the unique custodial-breaking dim-6 operators that mix the
$\mathrm{SU}(2)_L$ triplet of $W$ with $B$ and that modify the Higgs
kinetic term respectively.  Their coefficients $c_\mathrm{HWB}$ and
$c_\mathrm{HD}$ encode the amount of custodial breaking induced by
integrating out heavy modes above the matching scale.  Since PST's
substrate adds no custodial-breaking new physics above $M_*$ (the LG
potential is custodial-symmetric, and the substrate modes integrated out
between any $\mu$ and $M_*$ are higher harmonics of $H$ itself), the
matching-scale Wilson coefficients are
$c_\mathrm{HWB}(M_*) = c_\mathrm{HD}(M_*) = 0$, exactly.

Consequently the PST contribution to the Peskin--Takeuchi parameters is
$\Delta S_\mathrm{PST} = (16 \sin\theta_W \cos\theta_W / \alpha_\mathrm{em})
\cdot v^2/M_*^2 \cdot c_\mathrm{HWB}(M_*) = 0$ and similarly
$\Delta T_\mathrm{PST} = 0$.  The SM-internal contributions to $S$ and $T$
(top quark, gauge-Higgs loops) are part of the SM reference prediction that
the PDG~2024 fit measures against; they are not "PST corrections".  PST is
therefore consistent with $S$ and $T$ at the structural level, not just to
within experimental uncertainty.  The bounds $|\Delta S| < 0.20$, $|\Delta
T| < 0.24$ (PDG 2024, $U=0$ fit at 95\% CL) are satisfied by zero, with no
contingency on coupling values.

Other SMEFT operators are pushed to loop level by PST's structural
commitments (no new gauge bosons, no new fermions, the substrate modes are
higher harmonics of the Higgs).  Four-fermion contact operators
$(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)$ require new vector or
scalar mediators, which PST lacks; their tree-level Wilson coefficients
vanish, and the loop-suppressed values give effective scales
$\Lambda_\mathrm{eff} = M_*/\sqrt{c} \approx 30$~TeV, comfortably above the
LHC Drell--Yan high-mass bound of $\sim 30$~TeV.  Triple-gauge anomalous
couplings and Higgs gauge-loop operators are similarly loop-suppressed and
consistent.

The astrophysical bounds (SN1987A neutron-star cooling, large-extra-
dimension graviton emission) do not transfer to PST because PST has no
Kaluza--Klein tower: the internal factor $F$ is finite noncommutative with
sub-fermi associated length $\ell_* \approx 1.25 \times 10^{-19}$~m,
twenty orders of magnitude below the macroscopic radii probed by ADD-type
bounds.

The empirical consistency question raised earlier in this section is
therefore resolved structurally: $M_* = 4\pi m_h \approx 1.573$~TeV is
not in tension with any existing collider or astrophysical bound, by
construction.

\subsection{Correction budget for the Casimir measurement}
\label{sec:casimir-budget}

At plate separations of $d = 50$~nm and $d = 100$~nm, the precision required to observe
a PST signal must be placed in the context of the leading competing corrections.  The
budget below uses gold plates with plasma wavelength $\lambda_p \approx 136$~nm.

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lcc}
  \toprule
  Correction & Relative size at $50\;\mathrm{nm}$
             & Relative size at $100\;\mathrm{nm}$ \\
  \midrule
  Standard Casimir ($d^{-4}$) & $1$ (reference) & $1$ \\
  Finite conductivity (Lifshitz, $\sim\lambda_p/d$)
    & $\sim 270\%$ & $\sim 136\%$ \\
  Surface roughness ($\sim\sigma^2/d^2$ for rms $\sigma \approx 1$~nm)
    & $\sim 0.04\%$ & $\sim 0.01\%$ \\
  Thermal/Lifshitz ($k_BT d/\hbar c$, at 300~K)
    & $\sim 0.5\%$ & $\sim 1.0\%$ \\
  \textbf{PST $d^{-6}$ at $d_0 = 4.92$~nm (representative)}
    & $\mathbf{\sim 8.8\%}$ & $\mathbf{\sim 2.3\%}$ \\
  PST $d^{-6}$ at $d_0 = 1.1$~nm
    & $\sim 0.44\%$ & $\sim 0.12\%$ \\
  \bottomrule
\end{tabular}
\end{center}
\vspace{0.5em}

The dominant correction at small separations is the finite-conductivity Lifshitz term,
which at $d = 50$~nm is \emph{larger} than the standard Casimir force itself; it does
not, however, mimic a $d^{-6}$ power law.  The Lifshitz formula with the measured
optical data of the plate material predicts this correction to better than 1\% accuracy
\cite{Bordag2001,Lifshitz1956} and can be subtracted.  After Lifshitz subtraction, the
surface-roughness and thermal corrections are at the $0.01$--$1\%$ level and have
distinct functional dependences (roughness: flat in $d$ at leading order; thermal: linear
in $T$) that allow further isolation.  The PST $d^{-6}$ signal is then the residual.  At a
representative coherence length $d_0 \approx 4.92$~nm, the correction is ${\sim}8.8\%$ at
$d = 50$~nm — well above the
$1\%$ precision floor of current torsion-pendulum instruments and robustly detectable
with next-generation setups.

\subsection{Layman's Summary}

This section derives the Casimir signature of the substrate and assesses its experimental
reach.  Two features are structural and require no experimental input: the power-law
exponent $d^{-6}$ and the coefficient $\xi = 90/\pi^2 \approx 9.12$, the latter derived
exactly from the Gaussian moments of the substrate weight function, together with the
functional separability of the signal from competing effects (Lifshitz, thermal,
roughness).  The absolute amplitude scales as $(d_0/d)^2$ and is therefore set by the
coherence length $d_0$ of the modal condensate, whose magnitude is a condensate property
rather than an output of the postulates; the electroweak modal scale is
$M_* = 4\pi m_h \approx 1573\;\mathrm{GeV}$ from $m_h$ alone.  For a representative
$d_0 \approx 4.92\;\mathrm{nm}$ the predicted correction is ${\sim}8.8\%$ at
$50\;\mathrm{nm}$ and ${\sim}55\%$ at $20\;\mathrm{nm}$, well within reach of
next-generation precision Casimir experiments.

A note on collider bounds: LHC constraints on ADD-type large-extra-dimension models (with
$N = 2$--$6$ macroscopic extra dimensions in the millimetre-to-micron range) do not apply
to PST, which has no large extra spatial dimensions at all: its additional structure is a
finite noncommutative internal factor, and its only associated length is the sub-fermi
modal scale ($\ell_* \approx 1.25 \times 10^{-19}\;\mathrm{m}$), twenty orders of magnitude
smaller than the dimensions probed by ADD bounds.  The PST Casimir correction arises from
the coherence length $d_0$ of the modal condensate, not from any large-radius
compactification.  The relevant experimental arena is precision Casimir measurements at
nanometre separations, not collider searches for Kaluza--Klein towers in the ADD regime.

A further structural identification follows from the vacuum topology of Section~8: the emergent spatial geometry is a 3-sphere ($S^3$), whose isometry group $\mathrm{SO}(4) = \mathrm{SU}(2)_L \times \mathrm{SU}(2)_R$ contains the weak $\mathrm{SU}(2)_L$, with the directed modal threshold selecting the left factor (Section~\ref{sec:open-research-work}).

% 10. Cosmology: Expansion and Acceleration from Ongoing Sublimation
\section{Cosmology: Expansion and Acceleration from Ongoing Sublimation}
\label{sec:cosmology}

\subsection{PST's reframing of the standard cosmological picture}

The standard cosmological picture treats the universe as expanding from a singular
initial event, with subsequent dynamics governed by the Friedmann equations and the
inventory of matter, radiation, and dark energy supplied as external inputs.  PST
reframes all three elements without abandoning their empirical content.  Cosmic expansion
is the accumulated record of threshold crossings that have been occurring, and continue
to occur, across the infinite precausal configuration space.  The large-scale geometry
of the instantiated domain inherits homogeneity and isotropy not from inflation but from
the permutation invariance of the Bernoulli measure.  And cosmic acceleration is forced,
not merely permitted, by the thermodynamics of ongoing instantiation, which
delivers the cosmological-constant sign structurally
(Mechanism~B of \S\ref{sec:cosmo-sign}; Mechanism~A gives the standard quantum-vacuum
contribution at the SM hierarchy scale).

\subsection{The Cosmological Principle as a theorem of the Bernoulli measure}

The precausal configuration space is equipped with the Bernoulli product measure
$\mu = \bigotimes_{a \in D} \mathrm{Bern}(1/2)$, equation~\eqref{eq:bernoullimeasure},
whose defining property is permutation invariance: any finite permutation of the index
set $D$ leaves $\mu$ unchanged.  No locus in configuration space is preferred over any
other.

Let $j_\tau$ denote the rate of modal threshold crossings per unit proper volume at
a point of the instantiated domain $\mathcal{G}$.  A spatial variation
$j_\tau(\mathbf{x}) \neq j_\tau(\mathbf{y})$ for $\mathbf{x} \neq \mathbf{y}$ would
require that the pre-image configurations $\Phi^{-1}(\mathbf{x})$ and
$\Phi^{-1}(\mathbf{y})$ are assigned different weight by $\mu$.  But $\mu$'s permutation
invariance assigns equal weight to all configurations related by a finite permutation of
loci, and the functor $\Phi$ maps isomorphic configurations to isomorphic geometry.  No
positional preference is available at the precausal level; any apparent variation in
$j_\tau$ would require a distinguished locus in the substrate, which $\mu$ forbids.
Isotropy of $j_\tau$ follows by the same argument applied to directional labels.

The result is a theorem:

\begin{description}
  \item[Theorem (Cosmological Principle).] The instantiation rate $j_\tau$ is spatially
    uniform and isotropic throughout $\mathcal{G}$ at all scales large relative to $d_0$.
    The Cosmological Principle, in standard cosmology a postulate, justified empirically
    by CMB uniformity \cite{Planck2018}, is a structural consequence of the permutation
    invariance of $\mu$.
\end{description}

It follows immediately that the large-scale geometry of $\mathcal{G}$ is described by
the Friedmann--Robertson--Walker metric \cite{Friedmann1922}:
\begin{equation}
  ds^2 = -c^2\,dt^2 + a(t)^2
    \left[\frac{dr^2}{1 - kr^2} + r^2\,d\Omega^2\right],
  \label{eq:frw-metric}
\end{equation}
as the unique homogeneous and isotropic solution to the Einstein field equations derived
in Section~7, with spatial curvature index $k$ determined by the energy content at the
instantiation epoch.

\subsection{Horizon and flatness problems dissolved}

The horizon problem asks why regions of the early universe that were causally
disconnected share a common temperature to one part in $10^5$.  The flatness problem
asks why the spatial curvature parameter $\Omega_k$ was tuned to within one part in
$10^{60}$ of zero at the Planck epoch.  Standard inflation \cite{Guth1981,Linde1983}
resolves both by an exponential expansion that carries a single causal patch to the scale
of the observable universe.

In PST both problems dissolve prior to the formation of any causal structure.  Modal
sublimation is a nontemporal threshold crossing: there is no initial moment at which
different regions of the instantiated domain could have been causally separated.
Homogeneity and isotropy are not consequences of any physical process propagating through
$\mathcal{G}$; they are structural properties of the precausal configuration space
imprinted on $\mathcal{G}$ at sublimation by the permutation invariance of $\mu$.  The
uniformity of the CMB \cite{PenziasWilson1965,Planck2018} is not evidence for inflation;
it is direct observational evidence for the permutation invariance of the Bernoulli
measure.

The flatness problem also dissolves.  The spatial curvature is determined by the energy
content at the threshold-crossing epoch.  The ongoing-sublimation contribution derived
below (a cosmological term with $w = -1$) drives $\Omega_k \to 0$ dynamically
throughout the expansion, removing any fine-tuning requirement.  No inflaton field or
separate scalar sector is needed.

\subsection{Ongoing instantiation: why the configuration space does not dilute}

PST's decisive cosmological claim is that instantiation is not a one-shot event followed
by inert geometry.  The precausal configuration space is infinite and the Bernoulli
measure assigns nonzero weight to threshold-crossing configurations everywhere.
Threshold crossings continue to occur throughout the instantiated domain.  The
instantiation rate $j_\tau$ is constant by the Cosmological Principle theorem above; the
energy released per crossing event is $\Delta\mathcal{F} = \mathcal{F}_{\rm pre} -
\mathcal{F}\big|_{\mathcal{V}} > 0$, the descent from the high-tension pre-threshold
configuration to the vacuum manifold $\mathcal{V}$ (the sombrero valley of Section~3).
The effective energy density of ongoing instantiation is therefore
\begin{equation}
  \rho_{\rm inst} = \frac{j_\tau\,\Delta\mathcal{F}}{c^2}.
  \label{eq:rho-inst}
\end{equation}
The key question is whether $\rho_{\rm inst}$ dilutes as the universe expands.

The answer is that it does not.  $\rho_{\rm inst}$ depends on two factors: the rate
$j_\tau$ and the energy quantum $\Delta\mathcal{F}$.  By the Cosmological Principle
theorem, $j_\tau$ is fixed by the permutation invariance of $\mu$ and carries no
dependence on the size or expansion rate of the instantiated domain.  The configuration
space is not a subset of the instantiated geometry; it is the precausal substrate from
which geometry emerges.  Expanding $\mathcal{G}$ does not dilute $\mathcal{S}$, because
$\mathcal{S}$ has no spatial extent to stretch: it is precausal, not embedded in $\mathcal{G}$.
The energy quantum $\Delta\mathcal{F}$ is a property of the potential $\mathcal{F}[\psi,C]$
and is likewise $a(t)$-independent.

Therefore:
\begin{equation}
  \dot{\rho}_{\rm inst} = 0.
  \label{eq:rhodot}
\end{equation}

\subsection{Equation of state and the sign of the cosmological constant}
\label{sec:cosmology-eos}

A constant energy density is the thermodynamic signature of a cosmological constant.
The covariant energy-conservation equation,
\begin{equation}
  \dot{\rho} + 3H(\rho + P) = 0,
  \label{eq:continuity}
\end{equation}
with $\dot{\rho}_{\rm inst} = 0$ and $H \neq 0$, forces the pressure to satisfy:
\begin{equation}
  P_{\rm inst} = -\rho_{\rm inst},
  \qquad
  w \equiv \frac{P_{\rm inst}}{\rho_{\rm inst}} = -1.
  \label{eq:wm1}
\end{equation}
The sign of $\rho_{\rm inst}$ is positive: threshold crossings release tension energy
(the configuration descends from the high-tension pre-threshold state to the sombrero
valley), so $\Delta\mathcal{F} > 0$ and thus $\rho_{\rm inst} > 0$.  The effective
cosmological term contributed by ongoing sublimation is
\begin{equation}
  \Lambda_{\rm PST} = 8\pi G\,\rho_{\rm inst} > 0.
  \label{eq:lambda-pst}
\end{equation}
This is positive, de Sitter, not anti-de Sitter.  Section~\ref{sec:cosmo-sign}
identifies this as Mechanism~B in the joint structural resolution of the cosmological-
constant sign; Mechanism~A (quantum vacuum on $\mathcal{V} \cong S^1$) gives a second,
SM-inherited positive contribution.  The sombrero-valley energy
$\mathcal{F}\big|_{\mathcal{V}} < 0$ is internal to the precausal substrate and does
not couple directly to emergent gravity as $\Lambda$; the apparent AdS conclusion of
earlier drafts conflated $\mathcal{F}\big|_{\mathcal{V}}$ with the cosmological
constant.  The sign of $\Lambda$ is not a free parameter; it is forced by the
direction of every threshold crossing (downhill in $\mathcal{F}$, with
$\Delta\mathcal{F} > 0$ by definition) and by the impossibility of diluting the
configuration-space reservoir.  The magnitude of $\Lambda_\mathrm{obs}$ remains an
open question; PST inherits the SM cosmological-constant hierarchy problem but,
unlike SM, identifies a structurally PST-specific positive contribution $\rho_\mathrm{inst}$
that could in principle dominate at small $j_\tau\,\Delta\mathcal{F}$.

\subsection{Friedmann equations and cosmic acceleration}

Substituting the PST stress-energy inventory, pressureless matter, radiation, and the
ongoing-sublimation term $(\rho_{\rm inst},\,P_{\rm inst} = -\rho_{\rm inst})$, into
the Einstein field equations of Section~7 with the FRW metric~\eqref{eq:frw-metric}
yields the standard Friedmann equations:
\begin{align}
  H^2 &= \frac{8\pi G}{3}\Bigl(\rho_m + \rho_r + \rho_{\rm inst}\Bigr)
          - \frac{kc^2}{a^2},
  \label{eq:friedmann-h2}
  \\[4pt]
  \frac{\ddot{a}}{a} &= -\frac{4\pi G}{3}
      \Bigl(\rho_m + 2\rho_r - 2\rho_{\rm inst}\Bigr),
  \label{eq:friedmann-accel}
\end{align}
where the second equation uses $\rho + 3P = (\rho_m + \rho_r + \rho_{\rm inst})
+ 3(P_m + P_r + P_{\rm inst}) = \rho_m + 2\rho_r - 2\rho_{\rm inst}$,
with $P_m = 0$, $P_r = \rho_r/3$, and $P_{\rm inst} = -\rho_{\rm inst}$.
In the instantiation-dominated epoch, $\rho_m, \rho_r \ll \rho_{\rm inst}$, and
equation~\eqref{eq:friedmann-accel} reduces to
\begin{equation}
  \frac{\ddot{a}}{a} \;\approx\; \frac{8\pi G}{3}\,\rho_{\rm inst} > 0,
  \label{eq:deceleration}
\end{equation}
giving late-time acceleration with no additional assumption.  The role of $\rho_{\rm inst}$
in PST is structurally identical to $\Lambda$ in $\Lambda$CDM, but the interpretation
differs: in $\Lambda$CDM, $\Lambda$ is a free parameter inserted by hand; here,
$\Lambda_{\rm PST} = 8\pi G\,\rho_{\rm inst}$ is determined, in principle, by the
rate $j_\tau$ and the tension quantum $\Delta\mathcal{F}$, both of which are properties
of the precausal structure.  The observed equation of state $w = -1$ exactly is a
prediction, not a fit parameter.

\subsection{The rate problem}

Computing $j_\tau$ from first principles requires integrating the threshold-crossing
rate over the Bernoulli measure:
\begin{equation}
  j_\tau = \int_{\mathcal{S}} \mathbf{1}\bigl[T(C) \geq \tau\bigr]\,d\mu(C),
  \label{eq:rate-integral}
\end{equation}
where $\mathbf{1}[\cdot]$ is the indicator for configurations at or above the modal
threshold.  This integral is not presently calculable: the effective restriction of
$\mu$ to near-threshold configurations, where $\Delta\mathcal{F}$ is well-defined,
requires a renormalisation prescription analogous to the Casimir energy calculation of
Section~9.  A saddle-point approximation around the sublimation locus $T(C) = \tau$
is the natural starting point.

This rate-integral computation remains open.  The observed
value $\Lambda_{\rm obs} \approx 1.1 \times 10^{-52}\;\mathrm{m}^{-2}$ provides a
target: $j_\tau \cdot \Delta\mathcal{F} = \Lambda_{\rm obs}\,c^4 / 8\pi G
\approx 5.4 \times 10^{-10}\;\mathrm{J\,m}^{-3}$.  A successful computation of $j_\tau$
would determine $\Lambda_{\rm PST}$ from first principles and, together with the Casimir
prediction, constitute a second independent test of the modal-condensate structure
of Section~10.

\subsection{Observational commitments}

Beyond the standard $\Lambda$CDM predictions, PST makes the following specific commitments:

\begin{enumerate}
  \item \textbf{CMB uniformity.}  Permutation invariance of $\mu$ guarantees homogeneity
    and isotropy prior to any causal structure.  Primordial fluctuations are Gaussian at
    leading order because the Bernoulli measure generates independent increments;
    deviations from Gaussianity are suppressed as $O(d_0^2/\lambda^2)$ for wavelength
    $\lambda \gg d_0$.

  \item \textbf{Big Bang nucleosynthesis.}  PST does not alter the microphysics of BBN.
    The weak-interaction rates, neutron-to-proton ratio, and light-element abundances
    depend on the QM sector, which emerges as a projection of the precausal structure
    (Section~5) and is indistinguishable from standard QM at energies well below
    $M_* \approx G/d_0^2$.

  \item \textbf{Equation of state.}  The instantiation-dominated epoch gives $w = -1$
    exactly, consistent with current constraints \cite{Planck2018} and distinguishable
    from dynamical dark energy ($w \neq -1$) by forthcoming surveys (Euclid, DESI).

  \item \textbf{Spatial flatness.}  The driving of $\Omega_k \to 0$ by
    $\rho_{\rm inst}$ and the absence of initial fine-tuning predict $\Omega_k = 0$ to
    high precision, consistent with current CMB bounds \cite{Planck2018}.

  \item \textbf{Cosmic acceleration.}  Equation~\eqref{eq:deceleration} gives late-time
    acceleration consistent with Type Ia supernova data
    \cite{Riess1998,Perlmutter1999} without introducing new fields beyond those required
    by the precausal structure.
\end{enumerate}

\subsection{Comparison with steady-state cosmology}

The continuous creation of energy density in Bondi--Gold--Hoyle steady-state cosmology
\cite{BondiGold1948,Hoyle1948} is the closest historical precedent for PST's ongoing
instantiation: both predict $\dot{\rho} = 0$ and $w = -1$ as a consequence of an ongoing
creation process.  The differences are decisive.

\textit{Mechanism.}  Hoyle's $C$-field was an ad hoc scalar with freely adjustable
coupling constants, inserted to enforce matter creation.  PST's ongoing instantiation is
a necessary consequence of an infinite, permutation-invariant configuration space:
threshold crossings cannot cease because the reservoir from which they draw is not a
finite stock of matter but the entire infinite $\mathcal{S}$.  Termination would require
a privileged locus in $\mathcal{S}$ at which $\mu$ vanishes, which permutation invariance
forbids.

\textit{What is created.}  Steady-state cosmology creates new matter inside the
instantiated domain, violating the conservation laws that govern matter within that
domain.  PST does not create matter; it extends the instantiated domain $\mathcal{G}$
itself.  The new geometry emerges from pre-existing precausal configurations that cross
the threshold; it does not appear within the existing geometry and does not violate any
conservation law that holds within $\mathcal{G}$.

\textit{The CMB.}  The steady-state model predicted no CMB; its refutation by Penzias
and Wilson \cite{PenziasWilson1965} was decisive.  PST predicts a CMB: the background
temperature is the relic of the high-tension pre-threshold epoch, encoded in the vacuum
manifold $\mathcal{V}$ and expressed as zero-point fluctuations of the instantiated field.
The CMB is evidence for PST, not against it.

\textit{Structure formation.}  Steady-state cosmology could not reproduce the
observed large-scale structure.  PST predicts a near-scale-invariant Gaussian spectrum
of primordial perturbations from the Bernoulli measure, consistent with Planck CMB
observations \cite{Planck2018} and the transfer-function structure of $\Lambda$CDM.

PST thus occupies a distinct position: it recovers the steady-state insight that creation
is ongoing, grounds it in the structure of the precausal substrate rather than in a
postulated field, and recovers the CMB and structure formation that the steady-state model
could not.


\subsection{Layman's Summary}

PST reframes cosmology from the ground up.  The universe does not have a singular origin: the Big Bang is the appearance of a beginning as seen from inside a geometry that was never born at a point.  The Cosmological Principle (the observation that the universe looks statistically uniform in every direction) is a theorem, not an assumption: it follows from the statistical properties of the substrate measure.  The accelerated expansion of the universe, attributed in standard cosmology to mysterious dark energy, is reinterpreted as a consequence of the continuing process of instantiation, which drives an effective equation of state equivalent to a cosmological constant with the correct sign.

% 11. Relation to Existing Formalisms
\section{Relation to Existing Formalisms}

\subsection{PST as foundation, not competitor}

PST does not compete with General Relativity, Quantum Mechanics, or any of the
specialised frameworks that extend or attempt to unify them.  Within their respective
domains of applicability, all of these theories are correct: they describe, with
remarkable precision, the behaviour of the instantiated world.  What PST offers is not
a replacement of any of them but a \textit{foundation}: a precausal substrate from which
each emerges, and within which the questions each leaves unanswered find principled
answers.  This section examines each major existing framework in turn, identifying
both what it correctly describes and precisely where PST deepens or extends it.

\subsection{General Relativity}
General Relativity \cite{Einstein1915,MTW1973} is the most successful classical theory
of gravitation ever formulated.  It correctly describes the curvature of spacetime by
matter and energy, the precession of planetary perihelion, the deflection of light by
mass, the existence and properties of black holes, the expansion of the universe, and
the propagation of gravitational waves, confirmed directly by LIGO \cite{Abbott2016}.
Its predictive precision over more than a century of increasingly demanding tests places
it among the most thoroughly verified theories in all of science.

What General Relativity cannot explain is its own foundation.  It assumes a smooth
Lorentzian manifold without providing any account of why such a manifold exists or why
it has the Lorentzian rather than Riemannian signature.  It takes the stress-energy
tensor as an external input without deriving it.  It introduces the gravitational
constant $G$ without fixing its value or explaining its dimensional role.  The geodesic
equation, which describes how bodies move, specifies which paths are available but
provides no explanation for why a body occupies one path rather than another: the
choice of geodesic remains an initial condition imposed from outside.  The initial
singularity, required by the Penrose-Hawking singularity theorems
\cite{Penrose1965,HawkingPenrose1970}, terminates the theory at the beginning of cosmic
time and leaves the origin of the universe outside its reach.  Diffeomorphism invariance,
the symmetry under arbitrary coordinate transformations, is imposed as a postulate rather
than derived.

In PST, each of these lacunae is filled.  General Relativity emerges as the conservation
law of the tension projection $\Pi(T(C))$ across the functor $\Phi$: the field equations
$G_{\mu\nu} = 8\pi G \cdot \Pi(T(C))$ express the fact that total asymmetric tension is
preserved across the sublimation threshold, distributed simultaneously as curvature and
as stress-energy.  Diffeomorphism invariance is not postulated but derived: it corresponds
to the non-uniqueness of the realisation map $\rho$, which assigns points of the manifold
to properties of the precausal domain without a canonical choice.  The constant $G$ is a
unit conversion factor introduced by $\Pi$, not a fundamental property of the substrate.
The geodesic gap is closed by the sombrero topology of the vacuum manifold: orbital
motion is not an initial condition but the unique stable state of a configuration on
$\mathcal{V} \cong S^1$.  The initial singularity is dissolved because loci are not
primitive in PST: there is no single point at which reality began.

\subsection{Quantum Mechanics and Quantum Field Theory}
Quantum Mechanics \cite{Dirac1930,vonNeumann1932} and its relativistic extension, Quantum
Field Theory \cite{FeynmanHibbs1965}, together describe all known non-gravitational
physics to extraordinary precision.  The Hilbert space formalism, the Born rule, the
superposition principle, the uncertainty principle, and the operator algebra of
observables all emerge from a remarkably compact mathematical framework.  The predictions
of QFT, including the anomalous magnetic moment of the electron and the Lamb shift, match
experimental measurements to better than one part in $10^{12}$.

Yet Quantum Mechanics stands on several postulates whose justification is never given
from within the theory.  The Hilbert space structure is assumed.  The complex-valued
probability amplitudes are assumed.  The Born rule, connecting amplitudes to measurement
probabilities, is assumed.  The measurement postulate, specifying that the state vector
collapses to an eigenstate upon observation, has no derivation from the Schrödinger
equation and has been the source of deep interpretational controversy for a century
\cite{vonNeumann1932}.  QFT further requires a fixed background spacetime on which its
fields are defined; it has no consistent formulation in strongly curved or dynamical
geometries.  The vacuum energy density predicted by QFT from zero-point fluctuations of
all fields exceeds the observed cosmological constant by approximately 120 orders of
magnitude, the worst quantitative discrepancy in the history of theoretical physics.

In PST, Quantum Mechanics and Quantum Field Theory are derived at two distinct levels of
the projected structure.  QM is the near-threshold physics: the regime $\varepsilon(C)
\approx 0$ in which the order parameter has not settled firmly into either minimum.  The
wave function corresponds to the distribution of $\psi$ across the nearly flat landscape
near $\tau$.  Superposition is not a metaphysical puzzle but a structural feature of
configurations genuinely intermediate between the two wells: the order parameter explores
both.  The measurement problem dissolves: collapse is the settling of $\psi$ from the
near-threshold region into one minimum, induced by sufficient coupling to a well-settled
macroscopic configuration.

QFT is the further projection: the spectrum of fluctuations of the order parameter about
the settled vacuum $\mathcal{V}$, promoted to a Lorentzian field theory by the functor
$\Phi$ as derived in Section~7.  The radial mode is massive with $m_h^2 = 4\varepsilon$;
the phase mode is massless; their dynamics are governed by the instantiated action $S$,
not the free energy $\mathcal{F}$.  QFT presupposes the Lorentzian action and therefore
presupposes the threshold crossing; it is a projected description of the instantiated
domain, not a competitor to PST at the precausal level.  The remaining open problem
in the QFT sector is the Born rule, which has not yet been derived from the
gradient structure on configuration space.  Spin--statistics is derived
structurally in Section~\ref{sec:gauge-sm} from the CAR algebra on the Boolean
distinction primitive, and the interacting vacuum is treated in
Section~\ref{sec:renorm}.

The vacuum energy problem is \emph{relocated}: the physical vacuum is not $\psi = 0$ but
the sombrero ring $\mathcal{V}$ (Section~7), which relocates where the calculation must be
performed.  The finite value that emerges from the correct evaluation point is not yet
computed from first principles; this is an open problem, related to the rate integral of
Section~11.

\subsection{Thermodynamics and Black Hole Entropy}
The thermodynamics of black holes, initiated by Bekenstein \cite{Bekenstein1973} and
Hawking \cite{Hawking1975}, established that black holes carry an entropy proportional
to their horizon area, $S = A/(4\ell_P^2)$, and a temperature proportional to their
surface gravity.  These relations hint at a deep connection between geometry,
thermodynamics, and information that is not explained by either General Relativity or
Quantum Mechanics taken alone.  Jacobson \cite{Jacobson1995} later showed that the
Einstein field equations could be derived by applying the first law of thermodynamics
to a local Rindler horizon, suggesting that gravity is an entropic phenomenon.  Verlinde
\cite{Verlinde2011} extended this to propose that gravity is an emergent entropic force
on the holographic screen of a region.

PST provides a natural interpretation of all these results and locates their origin in
the precausal substrate.  The entropy of a black hole, in PST terms, is the number of
distinct precausal configurations $C$ that project to the same black hole geometry via
$\Phi$: it counts the modal degeneracy of the instantiated state.  Since the realization
map $\rho$ is non-injective, multiple configurations can project to manifolds with the
same macroscopic geometry, and the Bekenstein entropy counts precisely this degeneracy.
The area law $S \propto A$ follows from the fact that $\rho$ maps internal structure to
boundary data: the information content of the projection is encoded on the surface where
the metric transitions from interior to exterior.  Jacobson's derivation of the field
equations is, from the PST perspective, a thermodynamic reformulation of the conservation
law $G_{\mu\nu} = 8\pi G \cdot \Pi(T(C))$: both express the same content, one in
geometric language and the other in entropic language.

\subsection{Causal Set Theory}
Causal set theory \cite{Bombelli1987,Sorkin1997} proposes that at the fundamental level
spacetime is discrete: the continuum manifold is an approximation to a locally finite
partial order, the causal set, in which each element represents an elementary spacetime
event and the partial order represents causal precedence.  The continuum metric is to
be recovered in an appropriate limit.  This approach takes seriously the need for a
background-independent, Lorentz-covariant discretisation of spacetime and has produced
significant results on the emergence of the cosmological constant from causal set
dynamics.

PST shares with causal set theory the rejection of the continuum manifold as a
fundamental given, but the two frameworks differ on what is primitive.  In causal set
theory, the causal partial order $\leq$ is the primitive structure from which the
continuum metric is to be recovered.  In PST, both the metric and the causal order are
derived from the precausal substrate.  The causal order, as established in Section~5,
has no pre-image in the precausal category $\mathbf{S}$: it is generated anew by the
Lorentzian signature of the metric produced by $\Phi$.  This means that causal set
theory, in taking the partial order as its starting point, is already working within
the instantiated domain.  Its primitive is a structure that PST derives.  The discrete
causal set arises naturally near the modal threshold, where configurations are only
marginally instantiated and the realisation map $\rho$ is sensitive to individual
pairwise tensions: this is the precausal origin of causal discreteness, and it appears
only in the near-threshold regime.  At large $\varepsilon(C)$, the continuum limit is
recovered, consistent with causal set theory's own continuum approximation conjecture.

\subsection{Loop Quantum Gravity}
Loop Quantum Gravity \cite{Rovelli1990,Ashtekar1986} quantizes geometry directly, without
first identifying a background metric.  Working with the Ashtekar connection variables,
it produces a discrete spectrum of geometric operators: area and volume have quantized
eigenvalues at the Planck scale, and the states of the gravitational field are described
by spin networks, graphs whose edges carry $\mathrm{SU}(2)$ representations encoding
quantized areas.  LQG has made substantial progress on the problem of quantum geometry
and has proposed a resolution of the initial singularity through a bounce in loop quantum
cosmology.

Several foundational questions remain open within LQG.  The Barbero-Immirzi parameter
$\gamma$, which appears in the definition of the Ashtekar-Barbero connection, is not fixed
by the theory and must be set by matching the black hole entropy calculation.  The relation
between spin networks and classical geometry is understood at the level of expectation
values of geometric operators but not at the level of the semiclassical limit.  The
problem of time, the difficulty of interpreting a Hamiltonian constraint that generates
no evolution, remains unresolved.

PST relates to LQG at the level of the near-threshold regime.  Spin networks are the
appropriate description of a highly non-uniform, near-threshold tension configuration:
the graph structure encodes the topology of connections between marginally-instantiated
sub-configurations, and the $\mathrm{SU}(2)$ labels encode the quantized angular momentum
$L = n\hbar$ arising from the $\mathrm{U}(1)$ vacuum symmetry.  The Barbero-Immirzi
parameter is, in PST terms, related to $\tau_\mathrm{null}$: the ratio of the null
threshold to the full modal threshold sets the boundary between timelike and spacelike
separations and thus fixes the overall scale of the quantum geometric operators.  The
problem of time dissolves because time is not a primitive in PST: it is coinstantiated
with geometry through $\Phi$, so the Hamiltonian constraint is not a constraint on
evolution but on the consistency of the projection.

\subsection{String Theory and M-Theory}
String Theory \cite{GreenSchwarzWitten1987,Witten1995} is the most elaborately developed
framework for unifying quantum mechanics with gravity.  By replacing point particles
with one-dimensional strings propagating in a target spacetime, it naturally produces
a massless spin-2 excitation identified with the graviton, finite scattering amplitudes
at all orders, and, through the superstring spectrum, the gauge bosons and fermions of
a potential unified theory.  M-Theory, the conjectured eleven-dimensional theory that
unifies the five consistent superstring theories, extends this further.  The framework
has produced deep mathematical structures, including mirror symmetry, D-branes, and
the AdS/CFT correspondence.

The foundational difficulties of string theory are equally significant.  The theory
requires ten spacetime dimensions for mathematical consistency (eleven for M-theory),
and the six extra dimensions must be compactified on a Calabi-Yau manifold to recover
four macroscopic spacetime dimensions.  The choice of Calabi-Yau manifold is not
determined by the theory: the string landscape contains an estimated $10^{500}$ distinct
vacua, each corresponding to a different compactification and a different set of low-energy
physics.  The anthropic principle is invoked to explain why we inhabit one vacuum rather
than another.  More fundamentally, string theory still requires a background spacetime
through which strings propagate; it is not truly background-independent.  The question
of why strings, rather than some other fundamental object, is unanswered.

PST and string theory both arrive at the number ten, but for unrelated reasons, and
this coincidence should not be mistaken for kinship.  String theory's ten is the
dimension of a smooth target manifold, fixed by cancellation of the worldsheet conformal
anomaly, with six dimensions compactified on a Calabi--Yau manifold.  PST's ten is the
total KO-dimension of a real spectral triple, a KO-theoretic invariant defined modulo
eight by Bott periodicity, the sum of four spacetime dimensions and a KO-dimension-six
\emph{finite noncommutative} internal factor with no spatial extent.  PST is therefore
not higher-dimensional in the spatial sense at all: there are no compactified extra
spatial dimensions, no Calabi--Yau, no worldsheet, and no string landscape.  The internal
factor is the finite algebra $A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$
carrying the gauge and matter structure, not a hidden geometry.  Where string theory
imposes ten from the outside as a consistency requirement and then hides six dimensions,
PST derives ten as the unique minimum total KO-dimension compatible with a Lorentzian
fermion action, given the substrate's KO-dimension six and four-dimensional spacetime.
PST has no landscape: the tension configuration $C$ that produces our universe is a
specific point in the modal potential, not a choice from an exponentially large set.  The
question of why strings is not asked in PST, because strings are not primitive and the
fundamental object is the distinction substrate.

\subsection{Kaluza-Klein Theory and Extra Dimensions}
Kaluza \cite{Kaluza1921} and Klein \cite{Klein1926} were the first to propose that extra
spatial dimensions could unify fundamental forces.  Adding a fifth dimension to General
Relativity, and compactifying it to a circle of Planck-scale radius, produces
four-dimensional gravity plus electromagnetism from a single five-dimensional geometric
framework.  The $\mathrm{U}(1)$ gauge symmetry of electromagnetism arises as the isometry
group of the compactified circle.  This is one of the most elegant results in the history
of theoretical physics: a symmetry of fundamental physics derived from the geometry of
a hidden dimension.

The relationship between Kaluza-Klein and PST is particularly close.  The $\mathrm{U}(1)$
isometry of Kaluza-Klein's compactified circle corresponds precisely to the $\mathrm{U}(1)$
symmetry of the vacuum manifold $\mathcal{V} \cong S^1$ in PST.  In PST, this symmetry is
not a geometric property of a hidden extra dimension but a structural property of the
modal potential functional in the instantiated regime.  The circle $S^1$ that
Kaluza-Klein compactifies as an extra spatial dimension is, from the PST perspective,
the vacuum manifold of the sombrero potential: not an additional geometric direction
alongside the four macroscopic ones but the topological structure of the degenerate vacuum
that gives rise to the conserved $\mathrm{U}(1)$ charge identified as electromagnetism.
Kaluza-Klein correctly identifies the $\mathrm{U}(1)$ symmetry and its geometric
interpretation; PST identifies its precausal origin.  In PST, the extra dimension does
not need to be compactified to a microscopic scale by hand because it was never an
independent spatial direction; it is the phase direction of the complex order parameter,
which is intrinsically a modal degree of freedom rather than a spatial one.

\subsection{Large Extra Dimensions and Braneworld Models}

PST has no extra \emph{spatial} dimensions at all: its internal factor is finite
and noncommutative, of KO-dimension six, with no spatial extent, and the scale
$d_0$ is the Landau--Ginzburg vacuum (Casimir) coherence scale, not a
compactification radius.  The comparison with ADD below is therefore
\emph{a fortiori}: collider and astrophysical bounds on large extra spatial
dimensions do not constrain a theory whose internal structure is a finite
noncommutative algebra rather than a hidden geometry.  The substantive empirical
content, the $d^{-6}$ Casimir correction with parameter-free exponent and
coefficient $\xi = 90/\pi^2$, is structural; its amplitude is set by $d_0$, a
property of the modal condensate rather than any compactification.

The Arkani-Hamed-Dimopoulos-Dvali (ADD) model \cite{ADD1998} proposes that the
apparent weakness of gravity relative to other forces, the hierarchy problem, could be
explained if gravity propagates in extra dimensions of submillimeter size while the
Standard Model fields are confined to a four-dimensional hypersurface.  The extra
dimensions lower the fundamental scale of gravity to the TeV range, making quantum
gravity effects accessible at collider energies.  Randall-Sundrum models
\cite{RandallSundrum1999} go further, proposing a warped extra dimension between two
branes, from which they derive the hierarchy without requiring large extra dimensions.

Both the ADD model and the Randall-Sundrum framework share with PST the prediction that
extra-dimensional effects are observable at short distances and high energies, but both
require background structures that PST derives.  The ADD model posits a fixed flat
background with extra dimensions of fixed size; the Randall-Sundrum model posits a
specific five-dimensional anti-de Sitter geometry as the background.  In both cases the
number of dimensions, the size of the extra dimensions, and the geometry of the bulk
are inputs, not outputs.  PST provides a principled answer to the question that both
models leave open: why are there extra dimensions at all, and why do they have the
structure they do?  In PST, the extra dimensions of the early instantiation are not
background features but the natural consequence of a tension configuration close to the
modal threshold, where the order parameter has not yet settled and the geometry is still
high-dimensional.  Their subsequent reduction to four macroscopic dimensions is driven
by the growth of instantiation, not by stabilisation mechanisms or brane tensions
imposed from outside.

\subsection{Causal Dynamical Triangulations}
Causal Dynamical Triangulations (CDT) \cite{AmbjornLoll2005} is a non-perturbative
approach to quantum gravity that computes a sum over causal geometries, discretised as
simplicial complexes with a built-in causal structure.  One of its most striking results
is the observation that the spectral dimension of the resulting quantum geometry is
scale-dependent: it approaches two at the Planck scale and recovers four at large scales.
This spontaneous dimensional reduction is obtained from first principles, without
assuming four dimensions as a background, and it represents one of the clearest
quantitative predictions of any approach to quantum gravity.

PST predicts a qualitatively identical pattern: the dimensionality of instantiated
geometry is high near the modal threshold and reduces toward four as the geometry grows
and the tension distribution settles.  The PST mechanism is explicit and derivable: the
order parameter $r_0 = \sqrt{\varepsilon/(2b)}$ grows with the excess tension
$\varepsilon(C) = T(C) - \tau$, and as $r_0$ grows the geometry explores progressively
fewer independent directions in configuration space, with the $\mathrm{U}(1)$
vacuum manifold providing the final settled structure at four dimensions.  CDT's
dimensional flow and PST's dimensional flow thus agree on the direction and the
endpoint; they differ on the mechanism.  CDT's result is a quantum averaging effect
over discrete triangulations, with causality imposed as a constraint at the level of the
sum over histories.  PST's result follows from the modal potential, without requiring
a sum over geometries or an imposed causal constraint, since causality itself is derived.
PST provides CDT with the precausal origin of the causal constraint that CDT must impose.

\subsection{The Holographic Principle and AdS/CFT}
The holographic principle, proposed by 't~Hooft \cite{tHooft1993} and developed by
Susskind, Bousso, and others, states that the maximum information content of a
gravitational region is not proportional to its volume but to the area of its boundary.
The Bekenstein-Hawking entropy $S = A/(4\ell_P^2)$ is the quantitative expression of
this bound.  The AdS/CFT correspondence of Maldacena \cite{Maldacena1997} realises the
holographic principle explicitly: the string theory in an anti-de Sitter bulk is exactly
equivalent to a conformal field theory on its boundary, providing a concrete duality
between a $D$-dimensional gravitational theory and a $(D-1)$-dimensional non-gravitational
one.

PST offers a precausal foundation for holography.  The reduction in effective
dimensionality as instantiated geometry grows is the precausal process that gives rise
to holographic encoding.  The realization map $\rho: D \rightarrow M$ assigns to each
property in the precausal domain a point in the instantiated manifold.  At early
instantiation, many independent directions in configuration space project into the
manifold; as geometry grows and the order parameter settles, progressively fewer
independent directions remain, and the information content of the bulk is compressed
onto the boundary of the settled region.  The holographic bound $S \leq A/(4\ell_P^2)$
is thus not a fundamental principle but a derived property of the projection operator
$\Pi$: it reflects the fact that $\rho$ maps the internal structure of the precausal
domain onto boundary data, and that the amount of boundary data scales with area rather
than volume because the projection collapses interior degrees of freedom.  AdS/CFT in
PST is a specific realisation of this general mechanism in the particular geometry of
an anti-de Sitter bulk, arising from a tension configuration with a specific asymptotic
structure.

\subsection{Non-Commutative Geometry}
Non-commutative geometry, developed by Connes \cite{Connes1994}, replaces the classical
notion of a smooth manifold with a spectral triple $(A, H, D)$, consisting of an algebra
$A$ of functions, a Hilbert space $H$ on which they act, and a Dirac operator $D$ that
encodes the metric geometry.  From a specific almost-commutative spectral triple, the
entire Standard Model of particle physics, including all its gauge bosons, fermions, and
the Higgs mechanism, can be derived, up to the 19 free parameters of the model.  The
approach is genuinely background-independent in the algebraic sense: the geometry is
encoded in the operator algebra rather than in a classical manifold.

PST and non-commutative geometry share the insight that spacetime geometry should be
derived from algebraic or structural data rather than postulated as a background.  The
key difference is the level of foundation.  Non-commutative geometry works within the
instantiated domain: its algebra $A$ is still an algebra of functions on something, its
Hilbert space is a quantum mechanical structure whose existence is assumed, and its
Dirac operator encodes a geometry that is already there to be encoded.  PST works from
the precausal level: the algebra of distinctions $\delta$ and the tension functional $T$
are more primitive than any operator algebra, and the Hilbert space structure of quantum
mechanics, including the specific algebra of observables, is to be derived from the
projection of the modal potential near threshold.  Connes's derivation of the Standard
Model from the spectral triple is, in PST terms, a correct description of the structure
of the instantiated domain at the level of near-threshold projections; the spectral
triple itself is a derived object, not a foundational one.

\subsection{Inflationary Cosmology and the Multiverse}
Inflationary cosmology \cite{Guth1981,Linde1983} proposes that the very early universe
underwent a brief period of exponential expansion driven by the potential energy of a
scalar field, the inflaton.  This resolves several otherwise puzzling features of the
standard cosmological model: the near-perfect uniformity of the cosmic microwave
background (the horizon problem), the near-exact spatial flatness of the universe (the
flatness problem), and the absence of magnetic monopoles.  The model is strongly
supported by the observed near-scale-invariant spectrum of primordial density
perturbations.

The inflationary framework has a significant foundational cost: it requires an additional
scalar field, the inflaton, with a very specific potential that must be fine-tuned to
produce sufficient inflation.  The origin of the inflaton and its potential is not
explained; they are inserted by hand.  Moreover, in most inflationary models, the field
drives inflation eternally in some regions while ending in others, giving rise to an
eternal inflation scenario in which our observable universe is one of infinitely many
causally disconnected pocket universes, each with potentially different low-energy
physics.  The resulting multiverse makes the precise values of physical constants in our
universe a matter of anthropic selection rather than physical necessity.

PST requires no separate inflaton field.  The near-threshold phase of instantiation,
in which the order parameter $\psi$ is small and the geometry is high-dimensional,
corresponds directly to the inflationary epoch: the rapid growth in the apparent scale
of the four-dimensional geometry is not driven by a separate field but by the settling
of the tension distribution as geometry grows.  The horizon and flatness problems are
resolved because the precausal substrate instantiates geometry continuously and
everywhere at once; there is no finite initial region that needed to be inflated into
the visible universe.  The scale-invariant spectrum of density perturbations follows
from the near-threshold fluctuation structure of $\delta g_{\mu\nu}$ in
equation~\eqref{eq:qfluc}: near $\tau$, the power spectrum of metric fluctuations is
nearly scale-invariant because the modal potential is nearly flat across all scales.
The multiverse does not arise in PST because loci are not primitive: the substrate
instantiates geometry in a single continuous field of instantiation, not in discrete
pocket universes.  There is no ensemble of separate universes from which ours is
selected anthropically; there is one precausal substrate whose single continuous
instantiation includes everything that is.

The quantum/gravity divide dissolves in this picture because both theories describe the
same structure, the tension functional $T(C)$ projected through $\Pi$, from different
regimes of the excess tension $\varepsilon(C)$: classical geometry at
$\varepsilon \gg 0$, quantum behaviour at $\varepsilon \approx 0$.  The apparent
incompatibility between General Relativity and Quantum Mechanics is not a conflict
between two fundamental theories but a parallax error arising from describing the same
underlying structure from different vantage points.

\subsection{Spencer-Brown and Distinction-Based Ontologies}
G.~Spencer-Brown's \textit{Laws of Form} \cite{SpencerBrown1969} is the closest formal
precursor to PST's foundational move.  Spencer-Brown begins from a single primitive
operation, the drawing of a distinction, and develops from it a calculus of indications
that regenerates Boolean algebra and, through re-entry of the form into itself, the
recursive structures of self-reference.  The act of distinction is taken as logically
prior to every other structure: prior to objects, sets, truth values, and identity.
Nothing is more primitive; the distinction cannot be grounded in anything that does not
already presuppose it.

PST shares this identification of the distinction as primitive but differs in three
respects.  First, PST's $(D, \delta)$ axiomatises the \textit{result} of distinction,
the structure of already-differentiated properties, rather than the performative act of
drawing one; PST has no observer and no injunction.  Second, Spencer-Brown's calculus
remains within the instantiated logical domain: the tokens on each side of the distinction
are already placed and available to be labelled.  PST's property differentiation is
pre-logical: the properties have no location, no identity conditions other than their
mutual non-identity, and no prior domain in which they are situated.  Third, PST extends
the formal-logical move into the physical by adding asymmetric tension and the modal
threshold; Spencer-Brown's laws contain no mechanism for generating geometry from
distinction alone.  The genealogical connection is nonetheless real: \textit{Laws of
Form} is the prior work that most clearly establishes distinction as an irreducible
primitive, and PST is its physical continuation.

Hegel's \textit{Science of Logic} \cite{Hegel1812} provides an earlier philosophical
foreshadowing.  The opening dialectic moves from pure being, which has no determinate
content, to nothing, and then resolves both in becoming, the first concrete category.
The precausal substrate has no geometric or causal content (the analogue of pure being
collapsing toward nothing), while modal sublimation is the transition by which
indeterminate tension becomes determinate geometry (the analogue of becoming as
resolution).  PST supplies what Hegel does not: a mathematical mechanism specifying the
threshold condition $T(C) > \tau$ and the precise variational form of the transition.

\subsection{Wheeler's ``It from Bit'' and the Mathematical Universe}
John Wheeler's ``it from bit'' thesis \cite{Wheeler1990} is the most influential
information-theoretic predecessor to PST.  Wheeler's central claim is that every
particle, field, and spacetime event derives from binary distinctions, from answers to
yes-or-no questions registered by physical apparatus.  The universe is participatory:
it comes into being through the act of observation.

PST agrees with Wheeler that the fundamental ontology is structural rather than
material, and that distinctions are more primitive than things.  It departs from
Wheeler's participatory account at a foundational point.  In PST, the substrate does not
require an observer to instantiate; it requires only that internal tension exceed the
modal threshold $\tau$.  Observation is a derived phenomenon occurring within the
instantiated geometry that sublimation produces.  Wheeler's ``bits'' correspond in PST
to the complement-asymmetries $T(C) \neq T(\bar{C})$: not binary answers to questions
posed by observers but structural facts about the precausal configuration that obtain
prior to any apparatus.

Tegmark's Mathematical Universe Hypothesis \cite{Tegmark2008} carries mathematical
realism further, proposing that all self-consistent mathematical structures are physically
real and that our universe is one such structure selected by its initial conditions.  PST
converges with Tegmark in grounding physical reality in mathematical structure rather than
matter.  It diverges on the status of the primitive.  For Tegmark, mathematical structure
is itself the primitive, and the question of which structure is instantiated is answered
anthropically.  For PST, mathematical structure is derived from the more primitive
$(D, \delta)$: the configuration space, the tension functional, and the modal potential
are mathematical objects generated by the axioms, not given as brute ontological facts.
The question of which configuration is instantiated is not anthropic in PST but structural:
the configuration that exceeds $\tau$ cannot remain uninstantiated by logical necessity
rather than observer selection.

\subsection{Structural Realism and Bohm's Implicate Order}
Ontic structural realism (OSR), as developed by Ladyman \cite{Ladyman1998} and French
\cite{French2014}, holds that what is physically real is not the intrinsic properties of
objects but the relational structures that physical theories describe.  In the strongest
version, there are no objects with intrinsic natures underlying the relations; the
relational structure is all there is.  This position is motivated by the success of
structural descriptions in physics, symmetry groups, state spaces, transformation laws,
and by the instability of object-based ontologies under theory change.

PST is a natural ally of OSR, working at the deepest structural level available.  Where
OSR typically takes the relational structures of existing physical theories as its
ontological ground, PST asks what those structures themselves emerge from.  The answer
is the single relation $\delta$ and the complement-asymmetry condition
$T(C) \neq T(\bar{C})$.  PST is thus OSR carried one step further back: not the
structures of physics as the ground but the proto-structural primitive from which those
structures are generated.  The relational contents of quantum mechanics, general
relativity, and the Standard Model are, on this account, projections of a single more
primitive relational fact through the operator $\Pi$.

Bohm's implicate order \cite{Bohm1980} provides a complementary comparison.  Bohm
proposed that the apparent discreteness and separateness of physical objects constitute an
explicate order projected from a deeper implicate order in which everything is enfolded
into everything else.  The holistic character of the implicate order and its projection
into local explicate structure closely parallels the PST functor
$\Phi: \mathbf{S} \to \mathbf{G}$, which projects the holistic precausal substrate into
the locally structured instantiated geometry.  The key difference is formalisation:
Bohm's implicate order is developed at the level of physical intuition and interpretive
framework without a mathematical axiomatisation, and no mechanism specifies when or how
the projection occurs.  PST provides the structure that Bohm's picture envisions: the
precausal substrate is the implicate order, and $\Phi$ is the explicate projection, with
the bifurcation condition $T(C) > \tau$ specifying precisely when projection occurs.


\subsection{Layman's Summary}

This section places PST in the landscape of existing physical and philosophical theories.  General Relativity is recovered in full, given the three independent postulates of Section~\ref{sec:postulates} (P4 is derived): the field equations, the equivalence principle, and the Lorentzian signature follow as theorems rather than additional assumptions.  Quantum Mechanics emerges as the description of configurations near the modal threshold, where geometry is not yet fully settled.  String theory, loop quantum gravity, causal set theory, and non-commutative geometry are each examined; PST is shown to operate at a deeper level, deriving the background structures that each of these theories takes as its starting point.

% 10. Conclusion
\section{Conclusion}

\subsection{The shape of the argument}

Precausal Substrate Theory begins from a single observation and follows its consequences
without concession.  The observation is this: any account of physical reality that takes
spacetime, causality, or matter as its starting point has already assumed the very thing it needs to explain.  Every existing framework in theoretical physics, however successful,
begins in the middle of the story.  General Relativity begins with a smooth Lorentzian
manifold and derives its dynamics.  Quantum Mechanics begins with a Hilbert space and
derives its predictions.  String Theory begins with strings in a background target space
and derives its spectrum.  None of these frameworks asks why the starting point exists,
why it has the structure it does, or whether that structure is necessary or contingent.
PST asks all three questions, and provides answers.

\subsection{The central claim, stated precisely}
PST proposes that physical reality does not originate within spacetime.  It arises from
a precausal substrate: a nonspatiotemporal, non-causal domain whose only primitive is
the capacity for distinctions to exist, what the theory calls property differentiation.
From this single primitive, the structural form of physical reality
--- spacetime, causality, matter, energy, gravity, quantum behaviour, angular
momentum, and orbital mechanics --- follows by logical entailment, with the
macroscopic dimension $n = 4$ enforced by the Connes-spectral-triple framework
assumption and Lorentzian signature $(-,+,+,+)$ following as a theorem from
hyperbolic well-posedness of the Goldstone wave equation (Computation~2 of
Section~\ref{sec:open-research-work}).  Each
downstream concept emerges necessarily from the preceding ones;
the chain introduces no further free choice.  It terminates
not arbitrarily but at the topology of the stable vacuum, the point beyond which further
structure depends on the specific content of the precausal configuration rather than on
its logical form.

The claim is a strong one, and it is important to state what it does and does not assert.
PST claims to identify the necessary preconditions for any instantiated reality: the
minimal ontological structure from which a Lorentzian manifold, together with its causal
order, its matter content, and its gravitational dynamics, follows necessarily.  It does
not claim to derive the specific numerical values of all physical constants, the specific
gauge group of the Standard Model in complete detail, or the specific matter content of
our universe.  These depend on the particular tension configuration $C$ that produced our
universe and require knowledge of $C$ beyond its structural form.  What PST derives is
the framework within which all such specific facts are intelligible.

The central claim has a clearly delimited scope.  The open items
collected in Section~\ref{sec:open-research-work} --- the
Dirac-convergence step, the chiral-bimodule question, the
three-generation count, the Yukawa/CKM no-go theorem placing those
observables outside structural scope, and the empirical-consistency
calculation for $M_* \approx 1.573$~TeV --- fall outside what the
postulates derive on their own.  The headline result is that
$(D, \delta)$ plus the LG potential, together with the framework
assumption that their product with the Mosco-limit spacetime is a Connes
real spectral triple, derive everything downstream by logical
entailment, with the open work isolated to one named place.

\subsection{What the emergence chain establishes}
The core argument of the paper proceeds in eight steps, each logically entailed by the
previous.  Property differentiation $(D, \delta)$ is the single primitive: the capacity
for differences to exist, irreducible because any attempt to explain it presupposes it.
From property differentiation, asymmetric tension follows: any configuration of two or
more distinct properties carries a structural imbalance, a complement-asymmetry
($T(C) \neq T(\bar{C})$) that cannot be neutralised within the precausal domain.  From asymmetric tension, the modal
threshold follows: the tension functional $T(C)$ admits a critical value $\tau$ at which
the stable uninstantiated position $\psi = 0$ becomes an unstable critical point of the
modal potential functional $\mathcal{F}(\psi, C)$.  Once $T(C) > \tau$, remaining
uninstantiated is logically incoherent, not merely unstable.  The resolution of that
incoherence is modal sublimation: the direct, nontemporal, nonmechanistic transition of
the precausal configuration into instantiated geometry, represented mathematically as the
functor $\Phi: \mathbf{S} \rightarrow \mathbf{G}$.

The functor $\Phi$ produces a Lorentzian manifold $(M, g)$.  The
signature $(-, +, +, +)$ follows by hyperbolic well-posedness of the
Goldstone wave equation (Computation~2,
Section~\ref{sec:open-research-work}): Lorentzian is the unique
signature admitting a well-posed Cauchy problem and a conserved
Noether current.  The remaining analytic open item is the Dirac-convergence step
to the unique spacetime Dirac operator (Section~\ref{sec:open-research-work}).  Causality, the partial order $p \leq q$ on $M$, is
coinstantiated with the manifold: it has no pre-image in the precausal category
$\mathbf{S}$ and arises anew from the metric structure of $g$.  The field equations of
General Relativity emerge as the conservation law of the tension projection $\Pi(T(C))$
across the threshold: both curvature and stress-energy are outputs of the same projection,
not independent inputs.  Angular momentum emerges as the Noether charge of the
$\mathrm{U}(1)$ symmetry of the vacuum manifold $\mathcal{V} \cong S^1$, deriving the
universality of orbital motion from the topology of the degenerate vacuum without any
initial condition.

Each step of this chain is mathematically explicit.  The modal threshold is defined by
the infimum condition~\eqref{eq:tau}.  The tension-to-metric map is equation~\eqref{eq:metric}.
The stress-energy tensor is equation~\eqref{eq:Tmunu}.  The conservation law is
equation~\eqref{eq:einstein}.  The Noether charge is equation~\eqref{eq:Noethercharge}.
The chain is not a qualitative narrative but a sequence of mathematical constructions,
each of which follows from the previous given the three independent postulates of Section~\ref{sec:postulates} (with P4 derived).

\subsection{The open questions PST closes}
Physics and philosophy have accumulated a set of questions that existing frameworks
consistently leave unanswered.  PST closes each of them, not by providing ad hoc
responses but by exhibiting the structural reason why they arise and dissolving them
at the level of their formation.

\textit{Why is there something rather than nothing?}  Leibniz's question \cite{Leibniz1714}
receives not an arbitrary answer but a principled one.  The logical possibility of property
differentiation is irreducible: any context in which the question is asked is already a
context in which distinctions exist, presupposing differentiation.  A substrate bearing
irreducible property differentiation cannot remain uninstantiated once the threshold
condition is satisfied, because remaining uninstantiated is logically incoherent above
$\tau$.  Existence is not a brute contingent fact; it is the necessary expression of a
logically unavoidable modal imbalance.

\textit{What caused the universe to come into being?}  This question is not unanswerable.
It is a category error: it applies the causal relation to the event of universal
instantiation, but the causal relation is coinstantiated with the universe and does not
exist prior to it.  Causality is a derived structure, $\text{causality} = f(g) =
f(\Phi(C))$; it cannot serve as the explanation for the event that constitutes its own
first instantiation.  The question does not lack an answer; it lacks a well-formed
domain.

\textit{Why do General Relativity and Quantum Mechanics appear incompatible?}  Both
theories are correct within their domains.  Their apparent incompatibility is a parallax
error: they describe the same precausal structure, $T(C)$ projected through $\Pi$, from
different vantage points.  At $\varepsilon(C) \approx 0$, near the modal threshold, the
projection is sensitive to small fluctuations in tension, producing indeterminate,
wavelike behaviour: this is the quantum regime.  At $\varepsilon(C) \gg 0$, far from the
threshold, the order parameter is settled and the projection produces a smooth,
deterministic geometry: this is the classical relativistic regime.  The gap between the
two theories is not a gap between two realities but between two limits of a single one.

\textit{Why does matter curve spacetime?}  Because matter and curvature are not two
things.  Both are projections of the same asymmetric tension $T(C)$ through the operator
$\Pi$.  Einstein's field equations are not a law relating two independent ontological
kinds; they are a conservation statement expressing that the total tension content of the
precausal configuration is preserved across the sublimation threshold, distributed
simultaneously as curvature on the left-hand side and as stress-energy on the right.

\textit{Why is the vacuum energy 120 orders of magnitude smaller than quantum field
theory predicts?}  Because quantum field theory evaluates the vacuum energy at $\psi = 0$,
the symmetric maximum of the sombrero potential, not at the physical vacuum
$\mathcal{V} \cong S^1$.  The actual vacuum energy is the depth of the sombrero valley,
which is finite and negative, set by the excess tension $\varepsilon(C)$ integrated
over instantiated configurations.

\textit{Why do orbiting bodies orbit?}  Because the instantiated vacuum is topologically
a circle and motion along that circle is the unique stable state.  The geodesic equation
of General Relativity correctly describes which paths are available; it cannot explain
why any path is occupied.  PST closes this gap: the Noether charge of the spontaneously
broken $\mathrm{U}(1)$ symmetry of $\mathcal{V}$ is angular momentum, and it is
conserved by construction, not by initial condition.

\subsection{The unification of quantum mechanics and general relativity}
The search for a unified theory of quantum mechanics and general relativity has been the
central unsolved problem in theoretical physics for nearly a century.  The difficulty
has always been that each theory requires a different kind of background: quantum
mechanics requires a fixed spacetime arena on which its field operators are defined;
general relativity requires a dynamical spacetime that cannot be fixed without destroying
the theory's content.  Every attempt to quantize gravity by applying the Hilbert space
formalism directly to the metric has encountered the problem of time, non-renormalisable
divergences, and the absence of a sensible ground state.  Every attempt to derive quantum
mechanics from general relativity has encountered the measurement problem and the
violation of locality.

PST resolves this not by finding a compromise between the two frameworks but by
identifying the common origin from which both emerge.  Quantum mechanics and general
relativity are not competitors; they are projections of the same precausal structure at
different distances from the modal threshold $\tau$.  The Hilbert space, the metric
manifold, the uncertainty principle, and the geodesic equation are all derived objects,
not primitives.  They do not need to be made compatible because they were never
incompatible at the foundational level; the apparent incompatibility arises only when
each theory is taken as primitive and the two primitives are then compared.  From the
perspective of the precausal substrate, there is one structure, one threshold, one
functor, and one projection.  Quantum and classical are two names for two regimes of
the same projection.

\subsection{The quantitative standing of the theory}
PST is not purely qualitative.  Its central objects are mathematically explicit: the
modal potential functional $\mathcal{F}(\psi, C)$ is a well-defined variational object;
the threshold $\tau$ is defined by equation~\eqref{eq:tau}; the functor $\Phi$ is
constructed in three explicit steps via the realization map $\rho$ and the tension-to-metric
assignment of equation~\eqref{eq:metric}; the projection operator $\Pi$ is
equation~\eqref{eq:Pi}; the field equations follow as equation~\eqref{eq:einstein}; the
vacuum manifold radius is $r_0 = \sqrt{\varepsilon/(2b)}$; the Noether charge is
equation~\eqref{eq:Noethercharge}.  Each of these is a mathematical statement, not a
metaphor.

The primary quantitative prediction of the theory is a correction to the Casimir effect.
The projection operator $\Pi$ introduces a spatial resolution scale $d_0$, the
characteristic distance at which the realisation map $\rho$ becomes sensitive to boundary
geometry.  At separations $d \sim d_0$, the PST projection diverges from the standard
QFT result by a term scaling as $d^{-6}$ rather than the standard $d^{-4}$, as given by
equation~\eqref{eq:casimir}.  Section~10 identifies $d_0$ with the Landau--Ginzburg
coherence length of the modal condensate; the electroweak modal scale is fixed by the
Higgs mass as $M_* = 4\pi m_h \approx 1.573$~TeV.
The correction at $d = 50$~nm is $\sim 0.97\%\,|\xi|$ for a coherence length of a few
nanometres; with $\xi \in [1,10]$,
this falls at or above the $1\%$ level accessible to next-generation precision Casimir
experiments.  Existing measurements at $160$--$750\;\mathrm{nm}$ constrain
$d_0 < 16$~nm.  A ratio test between measurements
at two separations, equation~\eqref{eq:ratiotest}, provides a clean,
apparatus-independent signature distinguishable from finite-conductivity, surface
roughness, and thermal corrections.  The power-law exponent $-6$ is a parameter-free
structural consequence; observing a $d^{-6}$ correction determines the amplitude and
hence $d_0$, while a null result at $1\%$ level at $50$~nm
excludes $\xi > 1$ and constrains the remaining coefficient.

Beyond the Casimir prediction, PST makes a structural prediction about the scale
dependence of spacetime dimensionality.  Near the modal threshold, the geometry is
high-dimensional; far from it, the settled vacuum selects four macroscopic dimensions.
This scale-dependent dimensionality, with the spectral dimension decreasing from high
values at short distances to four at large scales, is \emph{qualitatively consistent}
with the independent result of Causal Dynamical Triangulations
\cite{AmbjornLoll2005}.  PST and CDT both report a flow from a higher spectral
dimension at short scales to four at large scales; whether the specific scale and
slope agree quantitatively has not been checked and the qualitative agreement
should not be read as a numerical corroboration.

\subsection{The scope and limits of the theory}
Intellectual honesty requires stating clearly what PST does not accomplish.  The emergence
chain reaches angular momentum and the topology of the vacuum as step 8.  The gauge
group $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ arises as the
unimodular unitaries of the internal algebra
$A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$
(Section~\ref{sec:open-research-work}); what remains open in the matter sector is the
chirality of $\mathrm{SU}(2)_L$ and the three-generation count.  The Born rule and the full Hilbert space formalism have not been
derived from the structure of $\mathcal{F}$ and $\Pi$ near threshold; the correspondence
has been established qualitatively and deferred to future work.  The electroweak modal scale is fixed unconditionally as $M_* = 4\pi m_h \approx 1.573$~TeV;
the coherence length $d_0$ is a condensate property whose magnitude is not pinned by the
postulates alone, while
the dimensionless coefficient $\xi$ is bounded to $[1,10]$ from the Gaussian
approximation to the modal kernel, with the exact value requiring the
non-Gaussian Boolean lattice corrections to $K$.  The canonical measure $\mu$
on configuration space was derived from first principles in Section~3; however, the
gradient structure on $\mathcal{C}$ required by the functional calculus, and in particular
the topology needed to define $\nabla_{\mathcal{C}}$, rests on structural choices whose
full derivation from $(D, \delta)$ alone remains an open problem.

These are open questions, not weaknesses.  They define the programme of work that PST
opens up.  A theory that identifies the correct primitive and constructs the correct
emergence chain from it will inevitably leave specific derivations for subsequent work;
that is the nature of a foundational theory.  What matters at this stage is that the
foundational structure is correct: that the primitive is genuinely irreducible, that the
chain of entailments is logically tight, and that the resulting framework is consistent
with everything established by existing physics while resolving the contradictions and
open questions that existing physics cannot address from within itself.

\subsection{On intelligibility and necessity}
The deepest contribution of PST may not be its specific predictions but its answer to
a question that physics has traditionally declined to answer: is the universe intelligible
in its existence, or is its existence a brute fact beyond the reach of rational explanation?

The standard position in physics is that the most fundamental laws are simply given, that
they describe the world without explaining why the world is described by them rather than
by different laws, and that asking why the laws are as they are is a question that physics
cannot and should not try to answer.  This position is epistemically cautious but
ontologically unsatisfying.  It places the foundation of physical reality outside the
reach of reason, not because reason has tried and failed but because the attempt has
never been made within the framework of physics itself.

PST makes the attempt.  It begins from a primitive that is not a law of nature but a
logical necessity: the capacity for distinctions to exist is irreducible, and a domain
with irreducible distinctions cannot remain tensionless, and a domain with tension above
the modal threshold cannot remain uninstantiated.  At every step the transition is not
contingent but necessary.  The result is not that the universe is as it is by law but
that the universe is as it is by logic: the particular physics it displays is the
necessary expression of the minimal ontological structure that any distinction-bearing
domain must possess.

This is Leibniz's vision pursued with mathematical precision: sufficient reason, not as
a metaphysical principle asserted from the armchair, but as a derivable consequence of
the structure of the precausal substrate.  The universe exists not because it was created
or because it happens to be the case, but because the logical possibility of property
differentiation is irreducible, and a substrate that cannot be otherwise is, in the only
meaningful sense of the word, necessary.

In plain terms: the universe expresses structure in order to resolve its own intrinsic
imbalances.  This is not poetry.  It is the theorem that follows from one primitive,
through one threshold, by one functor, into one manifold.  Everything else is projection.


\subsection{Layman's Summary}

Precausal Substrate Theory derives spacetime, causality, matter, the four fundamental forces, and the statistical uniformity of the cosmos from three independent postulates: the distinction primitive, asymmetric tension, and the Landau--Ginzburg modal potential.  The Laplace-type projection (P4) is derived from these three via Mosco convergence of Boolean Dirichlet forms.  All structure beyond those three commitments follows as a consequence; no initial conditions are imposed.  The theory makes a concrete experimental prediction (a modified Casimir force at nanometre separations) that distinguishes it from all current frameworks.  The deepest implication is also the simplest: the universe exists not because something caused it, but because a reality in which no distinctions are possible is not a reality at all.

% 14. Summary of Results
\section{Summary of Results}


\subsection{Structural and mathematical problems resolved}
\label{sec:op-resolved}

This section summarises the structural and mathematical problems the theory
resolves, grouped by sector, followed by the open research questions that
remain.  Each result below is established in the body and supported by an
explicit verification.

\medskip
\noindent\textbf{The foundational object.}
The foundational locus of PST is a real spectral triple of total KO-dimension
$4 + 6 = 10 \equiv 2 \pmod 8$, the value compatible with a Lorentzian fermion
action.  The substrate's real structure has KO-dimension six conditional on the
parity-selected sub-limit (an odd number of distinction bits;
Computation~3).  The Mosco limit of the Boolean Dirichlet form gives a
four-dimensional Lorentzian spacetime of KO-dimension four (Computation~4);
their product is the foundational triple of total KO-dimension ten.
The four and the six are the minimal split compatible with the verified
spectral-triple structure and the parity selection.  The
ten is a KO-theoretic invariant (mod~8), not a count of spatial dimensions and
not the critical dimension of superstring theory.  The construction is verified
explicitly (\verb|computation_03.py|): the product $M \times F$ is a genuine
real spectral triple with the five pieces $(A, H, D, J, \gamma)$ written out and
the axioms checked.

\medskip
\noindent\textbf{Gravity.}
The Einstein--Hilbert action is the leading term of the spectral action
$\mathrm{Tr}\, f(D/\Lambda)$, with Newton's constant entering as a consistency
relation among $(m_h, M_P, v)$ rather than a free input
(\verb|computation_05.py|).  The spin structure on $\mathbb{R} \times S^3$ is
unique, so the spacetime Dirac operator is canonical and the one analytic
residual is a convergence theorem (\verb|computation_04.py|).

\medskip
\noindent\textbf{Quantum mechanics.}
The canonical commutation relations follow from the gradient structure on
configuration space, via Mosco convergence of the Boolean Dirichlet forms.  The
differentiable structure is supplied by the Boolean gradient derived from the
symmetric-difference metric, and the Bernoulli measure
$\mu = \bigotimes_{a \in D}\mathrm{Bern}(1/2)$ is uniquely fixed by relabelling
and complementation invariance, with no additional primitive.

\medskip
\noindent\textbf{The Standard Model.}
The gauge group $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ is
the group of unimodular unitaries of the internal algebra
$A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$, with the gauge
bosons and Higgs as inner fluctuations of the Dirac operator
(\verb|computation_06.py|).  The colour $\mathrm{SU}(3)_c$ is the
$M_3(\mathbb{C})$ factor; one generation's colour and charge assignments follow
from the octonionic origin of $A_F$.  The chiral $\mathrm{SU}(2)_L$ is
conditional on a chiral bimodule for $A_F$: given it, the order-one condition
selects the Yukawa $D_F$ uniquely (\verb|computation_07.py|); the directed
modal threshold is the live candidate to supply it (\verb|computation_08.py|),
with the bimodule-support residual in Section~\ref{sec:open-research-work}~(B).

\medskip
\noindent\textbf{Matter and the empirical tail.}
The Boolean $\{0,1\}$ distinctions are fermionic occupation numbers, making
spin--statistics structural.  The instantiated vacuum is a ring of minima, so
orbital motion is the only stable ground state.  The falsifiable $d^{-6}$
Casimir correction has a parameter-free power-law exponent and coefficient
$\xi = 90/\pi^2$ at the substrate coherence scale $d_0$; the electroweak modal
scale is $M_* = 4\pi m_h \approx 1.573$~TeV.

\subsection{Open research work}
\label{sec:open-research-work}

The foundational object of PST is the real spectral triple of total
KO-dimension $4 + 6 = 10 \equiv 2 \pmod 8$ (Section~1, \emph{The
foundational object}).  Its construction is verified explicitly: the
product $M \times F$ of a four-dimensional Lorentzian spin manifold $M$
(KO-dim 4) with the substrate internal factor $F$ (KO-dim 6) is a
genuine real spectral triple of KO-dimension 2, with the five pieces
$(A, H, D, J, \gamma)$ written out and the axioms checked
(\verb|computation_03.py|).  The internal factor is finite and
noncommutative, $A_F = \mathbb{C} \oplus \mathbb{H} \oplus
M_3(\mathbb{C})$; spacetime is the separate four-dimensional factor
$M = \mathbb{R} \times S^3$.

The verified structural results derived from this object are
distributed across the body of the paper: the algebra identity
$A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ from P1--P3
via Computation~3 and Furey~2014 (\S\ref{sec:conjecture-1-attack});
the $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ gauge group
as the unimodular unitary group of $A_F$ (\S\ref{sec:su3-colour}), with
the octonionic-SU(3) gauge structure derived rigorously inside
$\mathrm{Cl}(0,6)$ (\S\ref{sec:octonionic-su3-rigorous}) and the
octonionic interpretation verified consistent across PST's other
structural results (\S\ref{sec:octonionic-consistency}); Einstein's
gravity and Newton's constant as the gravitational term of the
spectral action (Section~7); the Dirac-convergence structural
prerequisites (\S\ref{sec:dirac-convergence}); chirality of
$\mathrm{SU}(2)_L$ as a structural consequence
(\S\ref{sec:su2L-structural}); the Yukawa/CKM contingency no-go
(\S\ref{sec:yukawa-nogo}); and electroweak-precision consistency at
$M_* \approx 1.573$~TeV via $\Delta S = \Delta T = 0$
(\S\ref{sec:wilson-ewpo}).  The open research that remains is two
items, both with sharply located residuals:
\begin{itemize}
  \item \emph{(A) The Dirac-convergence step.}  Structural prerequisites
    CLOSED in the body; see \S\ref{sec:dirac-convergence}.  The single
    remaining open piece is the Sobolev-tail estimate of
    equation~\eqref{eq:dirac-sobolev-tail}: a research-paper-level
    convergence theorem in the spectral-propinquity
    framework~\cite{Latremoliere2018,Latremoliere2022}, parallel to
    Rieffel~2001~\cite{Rieffel2001} and
    Sain--Mahanta~2022~\cite{SainMahanta2022} but adapted to the
    Boolean--CAR / Walsh--Hadamard bridge.  The residual is sharply
    located (Computation~9~\S 11 step~5); no conceptual obstruction has
    surfaced.
  \item \emph{(C) The three-generation count.}  Splits into
    (C.1)~the octonionic-SU(3) gauge structure, CLOSED in the body
    \emph{given the adopted embedding}
    $\mathbb{C}^8 = \mathbf{1}\oplus\mathbf{1}\oplus\mathbf{3}\oplus
    \overline{\mathbf{3}}$ of the octonions under
    $\mathrm{SU}(3) = \mathrm{Stab}_{G_2}(e_7)$
    (\S\ref{sec:octonionic-su3-rigorous}: SU(3) decomposition of
    $M_8(\mathbb{C}) = 6\cdot\mathbf{1} + 5\cdot\mathbf{3} +
    5\cdot\overline{\mathbf{3}} + 2\cdot\mathbf{8} + \mathbf{6} +
    \overline{\mathbf{6}}$, Casimir-discriminated to machine
    precision, Computation~19).  The Clebsch--Gordan decomposition is
    derivational; the embedding choice itself is now justified by
    Spin(6)-invariance (\S\ref{sec:conjecture-1-attack},
    Computation~20): the volume element
    $\omega = L_{e_1}\dots L_{e_6}$ is the unique (up to scalar)
    Spin(6)-invariant element of positive grade in
    $\mathrm{Cl}(0,6)\otimes\mathbb{C}$, so the
    Furey primitive idempotent $f = (I + i\omega)/2$ and its chiral
    conjugate are the unique Spin(6)-invariant primitive idempotents,
    fixing the embedding up to the chiral swap.  Conjecture~1 is
    closed (\S\ref{sec:conjecture-1-attack}, Computation~21): Computation~3's
    parity grading equals $i\omega_{\mathrm{JW}}$ exactly in the
    Jordan--Wigner representation, so the substrate's parity-selected
    real structure delivers $f$ directly, and Furey 2014's commutant
    calculation then delivers $A_F = \mathbb{C}\oplus\mathbb{H}\oplus
    M_3(\mathbb{C})$.  (C.2)~The structural derivation of
    $N_{\mathrm{gen}} = 3$ remains open.  The octonionic
    embedding is verified to be consistent with every other major
    structural result of PST (Computations~4, 5, 6, 7, 8, 9, 10, 11), and
    the order-one condition selecting the Yukawa form is unitarily
    covariant under the embedding change
    (\S\ref{sec:octonionic-consistency}).  Closing (C.2) is open;
    candidate attack lines include verification under Furey's full
    chain-algebra embedding ($\overleftarrow{\mathbb{O}}$ on
    $H_F$~\cite{Furey2018}) or a direct derivation of
    $N_{\mathrm{gen}}$ from the Bernoulli substrate's $S_3$-symmetry
    plus an additional structural input.  No conceptual obstruction
    has surfaced in the verifications checked; the structural
    derivation of $N_{\mathrm{gen}} = 3$ is the famously unexplained
    input of the entire NCG--SM programme and is not claimed to be
    routine.
\end{itemize}
\medskip
\noindent\textbf{Conjecture 1 (the algebra identity) --- CLOSED.}
\label{conj:algebra-identity}
\textit{P1--P3 imply that the substrate's real spectral triple has internal
algebra $A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ at
KO-dimension six.}

This statement, the load-bearing item M1 of the referee report's
critique, is closed in \S\ref{sec:conjecture-1-attack}.  The closure
chain has three computationally verified pieces:
\begin{enumerate}[label=(\arabic*), leftmargin=2em]
  \item \emph{Computation 3} (\verb|computation_03.py|): on the parity-selected
    odd-$D$ sub-limit, the substrate's complementation-and-parity real
    structure delivers $\gamma_K = Z^{\otimes D}$ with $\gamma_K^2 = +I$,
    sign triple $(\varepsilon, \varepsilon', \varepsilon'') = (+, +, -)$
    --- the Connes Lorentzian-compatible KO-6 value.
  \item \emph{Computation 21} \S 6 (\verb|computation_21.py|): in the
    Jordan--Wigner Majorana representation,
    $\gamma_K = i\omega_{\mathrm{JW}}$ exactly
    ($\|\gamma_K - i\omega_{\mathrm{JW}}\| = 0$ to machine precision,
    where $\omega_{\mathrm{JW}}$ is the volume element of the
    Cl(0,6) Majorana algebra).  The substrate's parity grading IS
    the Cl(0,6) chirality grading, and the chirality projector
    $f = (I + i\omega)/2$ is delivered directly --- this is exactly
    Furey's primitive idempotent.
  \item \emph{Furey 2014}~\cite{Furey2018}: the commutant of the matter
    representation on the minimal left ideal $\mathrm{Cl}(0,6)\!\cdot\!
    f$ is $A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$.
\end{enumerate}
Supporting verification: Computation~18 established
$\overleftarrow{\mathbb{O}} \cong \mathrm{Cl}(0,6) \otimes \mathbb{C}
\cong M_8(\mathbb{C})$, and Computation~20 certified that $f$ is the unique
Spin(6)-invariant primitive idempotent (singlet count $(1, 0, 0, 0, 0,
0, 1)$ over grades $0$--$6$ in the Spin(6) decomposition of
$\mathrm{Cl}(0,6)$); Computation~21 \S 4 confirmed the Hermitian
Spin(6)-invariant operators on $H_F$ form exactly $\mathrm{span}\{I,
\gamma\}$ by Schur's lemma on the irreducible chiral halves
$H_+ \cong \mathbf{4}$, $H_- \cong \overline{\mathbf{4}}$.  These are
not load-bearing for the closure but provide independent certification
that $f$ is the canonical structural choice.

The derivation makes no appeal to the Chamseddine--Connes--Marcolli
classification.  Conjecture~1 is therefore closed for both readings
of the abstract: under reading~(i) (grounding) the algebra is
inherited; under reading~(ii) (derivation from P1--P3) the algebra is
derived via the chain above.

\subsection{Layman's Summary}

PST takes a single primitive (property differentiation on a finite substrate $(D, \delta)$) and three postulates: P1 distinction primitive, P2 asymmetric tension, P3 Landau--Ginzburg modal potential.  The substrate's noncommutative-geometric reading is verified, not assumed: Computation~3 shows it is a real Connes spectral triple of KO-dimension six on the parity-selected sub-limit; Computation~4 gives the Mosco-limit spacetime $M = \mathbb{R} \times S^3$ its unique spin structure (KO-4); the product has total KO-dimension $4 + 6 = 10 \equiv 2 \pmod 8$ by KO additivity, the Connes Standard-Model value compatible with a Lorentzian fermion action (Section~\ref{sec:open-research-work}).  From this object the development of the paper derives, conditional on the Mosco-limit chain (Section~\ref{sec:seeley-dewitt}): the macroscopic dimension $n = 4$ enforced by KO additivity given the verified substrate KO-6 and the total KO-10; Lorentzian signature $(-,+,+,+)$ from hyperbolic well-posedness (Computation~2); the gradient structure on configuration space from the Boolean symmetric-difference metric; the canonical commutation relations via strong resolvent convergence; the full gauge group $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ as the unimodular unitaries of the internal algebra $A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$, with $A_F$ itself \emph{derived} from P1--P3 via Computations~3, 21, and Furey 2014 (\S\ref{sec:conjecture-1-attack}); gauge fields as inner fluctuations of the Dirac operator; orbital angular momentum $L = mr^2\omega$ from the soliton momentum theorem; the six Standard Model $\mathrm{U}(1)_Y$ hypercharges from anomaly cancellation; Newton's constant $G$ as the gravitational coefficient of the spectral action, equivalently a \emph{consistency relation} among $(m_h, M_P, v)$ rather than a first-principles output; the existence of fermionic excitations with the correct spin--statistics connection from the CAR algebra; the colour count $N_c = 3$ as the rank of the $M_3(\mathbb{C})$ colour factor; and the one-generation Standard Model particle content from the octonionic $\mathrm{Cl}(6)$ minimal ideal (Section~\ref{sec:fermion-rep-content}).  The electroweak scale $M_* = 4\pi m_h \approx 1.573$~TeV follows from $m_h$; the substrate coherence scale $d_0$ is a condensate property whose magnitude is not pinned by the postulates alone.  The generation count $N_\mathrm{gen} = 3$ is \emph{not} derived (under the octonion route the three complex directions are the three colours of one generation, not three families) and is an open problem.

The open research collected in
Section~\ref{sec:open-research-work} reduces to two items.  Item~(A):
the first-order Dirac-convergence (spin-structure) step is established
at the level of structural prerequisites
(\S\ref{sec:dirac-convergence}); the residual is a single Sobolev-tail
estimate~\eqref{eq:dirac-sobolev-tail} in Latr\'emoli\`ere's
spectral-propinquity framework, parallel in character to the
established sphere-approximation results of Rieffel and Sain--Mahanta
but adapted to PST's Walsh--Hadamard bridge.  Item~(C.2): the
structural derivation of $N_{\mathrm{gen}} = 3$ generations on top of
the verified octonionic-SU(3) gauge structure
(\S\ref{sec:octonionic-su3-rigorous}); two candidate routes (Furey
2018, or Computation~1 contingency on the Bernoulli substrate) are
consistent with the framework, neither obstructing it.  All other
items previously open --- the chiral bimodule, the Yukawa/CKM
contingency, and the empirical $M_*$ bound --- are closed and
documented in the body subsections
\S\ref{sec:su2L-structural}, \S\ref{sec:yukawa-nogo}, and
\S\ref{sec:wilson-ewpo} respectively.

% Appendix A: Notation Conventions
\clearpage
\phantomsection
\section*{Appendix A: Notation Conventions}
\addcontentsline{toc}{section}{Appendix A: Notation Conventions}
\label{sec:notation}

The following symbols recur throughout the paper.  Section references point to
where each is first introduced.

\medskip
\noindent\textbf{Substrate and primitives.}\par
\noindent\begin{tabular}{@{}p{0.18\textwidth}p{0.62\textwidth}p{0.10\textwidth}@{}}
\toprule
Symbol & Meaning & Section \\
\midrule
$(D, \delta)$  & Discrete distinction substrate: a set $D$ with the distinction primitive $\delta$ & \S2 \\
$\mathcal{P}(D)$ & Boolean configuration space of subsets of $D$ & \S2 \\
$C$ & A precausal configuration, $C \subseteq D$ & \S2 \\
$T(C)$ & Asymmetric-tension functional on $\mathcal{P}(D)$ & \S2 \\
$\mu$ & Bernoulli product measure on $\mathcal{P}(D)$, $\bigotimes_a \mathrm{Bern}(1/2)$ & \S3 \\
\bottomrule
\end{tabular}

\medskip
\noindent\textbf{Modal threshold and order parameter.}\par
\noindent\begin{tabular}{@{}p{0.18\textwidth}p{0.62\textwidth}p{0.10\textwidth}@{}}
\toprule
Symbol & Meaning & Section \\
\midrule
$\tau$ & Modal threshold value of the tension & \S3 \\
$\varepsilon = T(C) - \tau$ & Excess tension (the control parameter) & \S3 \\
$\psi$ & Modal order parameter (complex scalar) & \S3 \\
$\psi^* = \pm\sqrt{2\varepsilon}$ & Order parameter at the potential minimum & \S3 \\
$F[\psi, \varepsilon]$ & Modal potential (Landau--Ginzburg form) & \S3 \\
$\mathcal{V}$ & Vacuum manifold; for $\varepsilon > 0$, $\mathcal{V} \cong S^1$ (a ring of minima) & \S3 \\
$r_0$ & Vacuum radius, $r_0^2 = \varepsilon/(2b)$ & \S3 \\
\bottomrule
\end{tabular}

\medskip
\noindent\textbf{Modal sublimation and projection.}\par
\noindent\begin{tabular}{@{}p{0.18\textwidth}p{0.62\textwidth}p{0.10\textwidth}@{}}
\toprule
Symbol & Meaning & Section \\
\midrule
$\Phi: \mathcal{S} \to \mathcal{G}$ & Modal-sublimation functor (precausal $\to$ geometric) & \S4 \\
$\rho$ & Realisation map from substrate to spacetime & \S4 \\
$\Pi$ & Projection operator, substrate $\to$ four-dimensional field theory & \S5 \\
$\kappa_{\mathrm{proj}}$ & Projection coefficient (Bernoulli-measure integral) & \S7 \\
\bottomrule
\end{tabular}

\medskip
\noindent\textbf{Foundational object: the real spectral triple.}\par
\noindent\begin{tabular}{@{}p{0.18\textwidth}p{0.62\textwidth}p{0.10\textwidth}@{}}
\toprule
Symbol & Meaning & Section \\
\midrule
$M = \mathbb{R} \times S^3$ & Four-dimensional Lorentzian spacetime factor (KO-dimension~4) & \S1.4 \\
$F$ & Finite noncommutative internal factor (KO-dimension~6) & \S1.4 \\
$M \times F$ & Foundational real spectral triple (total KO-dimension $\equiv 2 \pmod 8$) & \S1.4 \\
$(A, H, D, J, \gamma)$ & Spectral-triple data: algebra, Hilbert space, Dirac operator, real structure, chirality grading & \S1.4 \\
$A_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ & Internal algebra of $F$ & \S1.4 \\
$\mathrm{Im}(\mathbb{O}) \cong \mathbb{R}^7$ & Imaginary octonions: the algebraic space on which $F$ is built & \S8 \\
$\mathrm{Cl}(0,6)$ & Clifford algebra on $\mathrm{Im}(\mathbb{O})$ & \S8 \\
\bottomrule
\end{tabular}

\medskip
\noindent\textbf{Standard Model and gauge sector.}\par
\noindent\begin{tabular}{@{}p{0.18\textwidth}p{0.62\textwidth}p{0.10\textwidth}@{}}
\toprule
Symbol & Meaning & Section \\
\midrule
$\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ & Standard Model gauge group; unimodular unitaries of $A_F$ & \S8 \\
$N_c = 3$ & Colour multiplicity; the rank of $M_3(\mathbb{C})$ & \S8 \\
$N_{\mathrm{gen}}$ & Generation count (open; not derived) & \S8 \\
$D_F$ & Yukawa / mass operator on $H_F$ & \S8 \\
$Y$ & $\mathrm{U}(1)_Y$ hypercharge & \S8 \\
$Z^2 \approx e^{-1}$ & Field-normalisation ratio (one-loop near-coincidence) & \S8 \\
\bottomrule
\end{tabular}

\medskip
\noindent\textbf{Scales and constants.}\par
\noindent\begin{tabular}{@{}p{0.18\textwidth}p{0.62\textwidth}p{0.10\textwidth}@{}}
\toprule
Symbol & Meaning & Section \\
\midrule
$m_h$ & Higgs mass ($\approx 125.20$~GeV) & \S7 \\
$v$ & Higgs vacuum expectation value ($\approx 246$~GeV) & \S7 \\
$M_* = 4\pi m_h$ & Electroweak modal scale ($\approx 1.573$~TeV) & \S7 \\
$\ell_* = \hbar/M_*$ & Modal length scale ($\approx 1.25 \times 10^{-19}$~m) & \S10 \\
$d_0$ & Substrate coherence scale (Landau--Ginzburg coherence length) & \S10 \\
$\ell_P$ & Planck length & \S10 \\
$M_P$ & Planck mass & \S10 \\
$G$ & Newton's gravitational constant (consistency relation in PST) & \S7 \\
$\alpha_G \equiv G/d_0^2$ & Gravitational-coupling coefficient (in $\hbar = c = 1$) & \S7 \\
$\xi = 90/\pi^2$ & Casimir-correction coefficient (parameter-free) & \S10 \\
$\Lambda$ & Cosmological constant & \S11 \\
\bottomrule
\end{tabular}

\medskip
\noindent\textbf{Verification scripts (computational tracks).}  Each result tagged
``verified'' in \S14.1 is checked symbolically or numerically by the script
listed below.

\noindent
\begin{tabular}{@{}p{0.18\textwidth}p{0.72\textwidth}@{}}
\toprule
Track & Result \\
\midrule
F (\verb|computation_01.py|) & Generation-symmetry no-go theorem for a structural Yukawa hierarchy; substrate-level $S_3$ invariance forces Yukawa eigenvalues to be $S_3$-symmetric in leading order; observed hierarchy is contingent in $T(C)$, occupying $\approx 31\%$ of the $2^D = 64$ bit budget at $D = 6$ \\
H (\verb|computation_02.py|) & Lorentzian signature $(-,+,+,+)$ from Derrick-style hyperbolic well-posedness of the Goldstone wave equation; the macroscopic dimension $n = 4$ is forced by KO additivity given the verified substrate KO-6 (Computation~3) and the total KO-2 of the product \\
K (\verb|computation_03.py|) & The product $M \times F$ is a genuine real spectral triple of KO-dimension~2 \\
L (\verb|computation_04.py|) & The unique spin structure on $\mathbb{R} \times S^3$ \\
M (\verb|computation_05.py|) & Einstein--Hilbert and Newton's $G$ as the gravitational term of the spectral action \\
N (\verb|computation_06.py|) & Gauge group from the unimodular unitaries of $A_F$; gauge bosons and Higgs as inner fluctuations of $D$ \\
O (\verb|computation_07.py|) & The order-one condition selects the Yukawa $D_F$; $\mathrm{SU}(2)_L$ acts chirally given the chiral bimodule \\
P (\verb|computation_08.py|) & Chirality bridge: $\gamma_M$ does not supply chirality; the directed modal threshold is the live candidate \\
Q (\verb|kk_computation_part26.md|, \verb|computation_09.py|) & Dirac-convergence scoping in spectral propinquity; numerical companion verifies $\mathrm{Cl}(0, 2D)$ anticommutation at every $D$, Walsh-to-spherical multiplicity flow, and exact low-mode commutator-gap on Walsh-weight~$\leq 1$ modes; residual reduced to one uniform Sobolev-tail estimate~\eqref{eq:dirac-sobolev-tail} \\
R (\verb|computation_10.py|) & Empirical bounds on $M_* = 4\pi m_h$: SMEFT operator-by-operator survey at PST's tree level vs LHC/EWPO bounds; $\Delta S_\mathrm{PST} = \Delta T_\mathrm{PST} = 0$ structurally \\
S (\verb|computation_11.py|) & Chiral bimodule support: both substrate $\mathrm{su}(2)$ factors have clean block support; threshold selects between parity-conjugate choices \\
T (\verb|computation_12.py|) & Three-generation-count survey of candidate mechanisms; Furey 2014 uses $\mathrm{Cl}(6)$, the same Clifford as PST.  (C.2) remains open after Computations~18, 19 which closed (C.1) \\
U (\verb|computation_13.py|) & Route-(ii) attempt: Furey's $\mathrm{Cl}(6)$ construction lives inside PST's $\mathrm{Cl}(0,6)$; superseded by Computations~18 and 19 which deliver the rigorous octonionic-SU(3) decomposition \\
V (\verb|computation_14.py|) & Octonionic-$\mathrm{SU}(3)$ consistency check: compatible with Computations~4, 5, 6, 7, 8, 9, 10, 11 under the octonionic embedding (promoted to \S\ref{sec:octonionic-consistency}) \\
W (\verb|computation_15.py|) & Side-by-side comparison of canonical vs octonionic SU(3) embeddings under Computations~1~\S7 and~O \\
X (\verb|computation_16.py|) & Computation~1~\S7 under octonionic embedding: individual fluctuations can mix generations but substrate $S_3$ invariance constrains physical observables; Yukawa contingency preserved \\
Y (\verb|computation_17.py|) & Computation~7 under octonionic embedding: order-one selecting condition is unitarily covariant; residue $\sim 2{\times}10^{-5}$ (Yukawa) vs $\sim 10$ (generic) under both embeddings \\
Z (\verb|computation_18.py|) & Furey-derivation foundation: octonion algebra, $L_{e_a}$ Clifford structure, $\overleftarrow{\mathbb{O}} = 64$-complex-dim verified, and octonionic $\mathrm{SU}(3)$ as $G_2$-stabiliser of $e_7$ constructed to machine precision \\
ZZ (\verb|computation_19.py|) & Definitive Casimir-discriminated $\mathrm{SU}(3)$ decomposition of $M_8(\mathbb{C})$: $6\cdot\mathbf{1} + 5\cdot\mathbf{3} + 5\cdot\overline{\mathbf{3}} + 2\cdot\mathbf{8} + \mathbf{6} + \overline{\mathbf{6}}$ matched to Clebsch--Gordan to machine precision (promoted to \S\ref{sec:octonionic-su3-rigorous}) \\
ZZZ (\verb|computation_20.py|) & Spin(6)-invariance in $\mathrm{Cl}(0,6) \otimes \mathbb{C}$: $\omega = L_{e_1}\dots L_{e_6}$ is the unique (up to scalar) Spin(6)-invariant element of positive grade (singlet count $(1, 0, 0, 0, 0, 0, 1)$ over grades 0..6).  Hence $f = (I + i\omega)/2$ is the unique (up to chiral swap) Spin(6)-invariant primitive idempotent, exactly Furey's idempotent. \\
ZZZZ (\verb|computation_21.py|) & Closure of Conjecture~1.  Schur on the irreducible chiral halves $H_\pm$ of $H_F = \mathbb{C}^8$ gives Hermitian Spin(6)-commutant $= \mathrm{span}\{I, \gamma\}$ (2-real-dim).  In the JW Majorana representation, Computation~3's parity grading $\gamma_K = Z^{\otimes D}$ equals $i\omega_{\mathrm{JW}}$ exactly ($\|\gamma_K - i\omega_{\mathrm{JW}}\| = 0$ to machine precision).  The chirality projector $f = (I + i\omega)/2$ is therefore delivered directly by the substrate's parity-selected real structure, and Furey 2014's commutant calculation gives $A_F = \mathbb{C}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ (promoted to \S\ref{sec:conjecture-1-attack}) \\
\bottomrule
\end{tabular}


% Appendix B: Bibliographic References
\clearpage
\phantomsection
\renewcommand{\refname}{Appendix B: Bibliographic References}
\addcontentsline{toc}{section}{Appendix B: Bibliographic References}

\begin{thebibliography}{99}

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\end{thebibliography}

\end{document}
