% Precausal Substrate Theory
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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 reality does not originate within
spacetime, but arises from logic itself.  The theory posits a nonspatiotemporal, precausal substrate with a
single irreducible primitive, \textbf{property differentiation} (the capacity for
distinctions to exist at all), whose first expression is \textbf{asymmetric tension}.
Once this tension exceeds a critical \textbf{modal threshold}, the configuration
undergoes \textbf{modal sublimation}: a direct, nontemporal phase transition into
\textbf{instantiated geometry}, from which spacetime and causality emerge together as
coemergent consequences.  Because loci are not primitive, PST requires no singular
origin event; the Big Bang is a perspectival artifact of geometry that was never born
at a point.  The cosmological implication is a continuous, unbounded field of
instantiation whose higher dimensions level off toward four as tension redistributes.  The
resulting vacuum is a ring of degenerate minima; motion along it is the only stable
state it admits, making orbital mechanics at all scales a structural consequence of
vacuum topology rather than an assumption.  Quantum Mechanics
\cite{Dirac1930,vonNeumann1932,FeynmanHibbs1965} and General Relativity
\cite{Einstein1915,MTW1973} both emerge as projections of the same precausal structure.
The Einstein field equations are derived from a projected action: the gravitational sector
is generated by the configuration-space gradient term in the modal potential, and the Einstein
tensor $G_{\mu\nu}$ is selected uniquely by PST's derived diffeomorphism invariance
via Noether's second theorem and Lovelock's theorem.
The equivalence principle is a theorem of single-source projection.
A concrete empirical prediction follows: PST's dimensional reduction grounds the
substrate discreteness scale at $d_0 \approx 7\,\text{nm}$, yielding a $d^{-6}$
Casimir correction of ${\sim}2\%$ at $50\,\text{nm}$ plate separation, detectable
with near-future experiments.  By grounding the emergence of spacetime in a logically
necessary modal threshold, PST offers an answer to why there is something rather than
nothing.
\end{abstract}

\vspace{1em}
\hrule
\vspace{1.5em}

% 1. Introduction
\section{Introduction}

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.

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.

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.

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, geometry
emerges in a highly compressed, high-dimensional form.  As instantiation propagates and
geometry grows, the tension that was concentrated across many dimensions stretches and
redistributes.  Dimensions that were distinct at small scales become effectively merged
or aligned at large scales, leaving fewer dominant directions.  What we observe as
four-dimensional spacetime is not the full geometry of instantiation but the stretched,
low-dimensional limit of a process that began in many more.  A simple analogy makes
this concrete: a ball has equal extent in all directions, high symmetry, many effective
dimensions.  Stretch it into a string and most of those directions collapse into one
dominant axis.  The dimensions do not disappear; they are absorbed into the elongation.
Instantiated geometry undergoes the same redistribution as it grows, and the
high-dimensional compression of its origin relaxes into the four dominant directions
we observe.

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.

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
distinctive $d^{-6}$ correction to the standard Casimir force scaling
\cite{Casimir1948,Lamoreaux1997,Bressi2002,Decca2003,Bordag2001}, measurable at
submicron plate separations.

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 develops the physical predictions of the theory, including a quantitative
Casimir correction.  Section~9 grounds the discreteness scale $d_0$ from PST's
dimensional reduction mechanism, derives the first-principles prediction
$d_0 \approx 7\;\mathrm{nm}$, and establishes the Casimir correction as a concretely
detectable experimental signal.  Section~10 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~11 positions PST in relation to existing
frameworks.  Section~12 concludes.  Section~13 identifies open problems.

% 2. Foundational Ontology
\section{Foundational Ontology}

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.

\paragraph{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.

\paragraph{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.

\paragraph{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 and are listed as such in the
programme of open problems in Section~13.

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, listed as such in Section~13.  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.

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

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.  A minimal concrete example satisfying positivity, complement-asymmetry, and
monotonicity 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}
Positivity holds for any configuration with at least two distinct properties.
Monotonicity holds by inspection: adding property $a$ to $C$ increases the count by
$|\{b\in C:\delta(a,b)\}|$.  Complement-asymmetry $T_{\mathrm{ex}}(C)\neq
T_{\mathrm{ex}}(\bar{C})$ holds whenever the induced subgraphs on $C$ and $\bar{C}$
have different edge counts, which is generic for non-regular distinction structures.
Axioms~\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 in
Section~13.

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.

\paragraph{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~8, 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.

\paragraph{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.

\paragraph{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 $a(T(C))$
must change sign at $T = \tau$: when $a > 0$ the uninstantiated state $\psi = 0$ is a
minimum; when $a < 0$ it is a maximum, and the configuration is driven toward $\psi \neq 0$.
The simplest choice consistent with this requirement is $a(T) = \tau - T(C)$, 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 $a < 0$, 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 $\int [a\psi^2 + b\psi^4]\,d\mu$ 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.  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.  A
natural candidate is the symmetric difference metric, under which two configurations are
close when they differ in few properties.  This choice is structurally motivated and
makes $\mathcal{C}$ a metric space without requiring any geometric primitive.  However,
the full derivation of the gradient structure from $(D, \delta)$ alone, including the
precise form of the metric and the differential structure needed for functional calculus,
is an open problem in the foundations of the theory and is listed as such in Section~13.
The gradient term is included because the structural requirement it enforces, a coherent
rather than patchy transition, is non-negotiable; its rigorous grounding is deferred.

\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) = a(T)\,\psi^2 + \frac{1}{4}\,\psi^4, \qquad a(T) = \tau - T(C)
  \label{eq:freduced}
\end{equation}
When $a > 0$ (below threshold, precausal) $\F$ has a unique minimum at $\psi = 0$; when
$a < 0$ (above threshold, instantiated) two symmetric minima appear at
$\psi^* = \pm\sqrt{-2a} = \pm\sqrt{2\varepsilon}$ where $\varepsilon = T(C) - \tau$.  A
rescaled variant $\tfrac{1}{2}(\tau - T)\,\psi^2 + \tfrac{1}{4}\psi^4$ 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, $a(T(C))\,\psi^2$, governs the stability of the uninstantiated phase.
The coefficient $a(T) = \tau - T$ changes sign at $T = \tau$.  When $T < \tau$,
$a > 0$ 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 $T > \tau$, $a < 0$ and the term now decreases $\F$ near $\psi = 0$,
destabilising the uninstantiated state and driving the configuration toward nonzero $\psi$.
The sign change in $a$ 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.

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

\textit{Above the threshold, $T(C) > \tau$}: the coefficient $a$ is now negative.  The
solution $\psi = 0$ is now a local maximum of $\F$, not a minimum.  It is an unstable
equilibrium.  The quadratic term $a\psi^2$ 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 $2a\psi + 4b\psi^3 = 0$, or equivalently $2\psi(a + 2b\psi^2) = 0$.  The root
$\psi = 0$ is a maximum, as established.  The nontrivial roots satisfy
$a + 2b\psi^2 = 0$, giving $\psi^2 = -a/(2b) = (T(C) - \tau)/(2b) = \varepsilon(C)/(2b)$.
Since $a < 0$ above the threshold, $-a > 0$ and 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.

\paragraph{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 $\delta\F/\delta\psi = 2a\psi + 4b\psi^3$ 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} = 2a(T) = 2(\tau - T)
\end{equation}
This is positive for $T < \tau$, zero for $T = \tau$, and negative for $T > \tau$.
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 $2(\tau - T) = 0$ 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 $\varphi = \pm\sqrt{-\varepsilon}$;
  the configuration at $\varphi = 0$ is now a local maximum and cannot remain there.
  The dashed curve is the reference potential at $\varepsilon = 0$.}
  \label{fig:modal-potential}
\end{figure}

% 4. Modal Sublimation
\section{Modal Sublimation}

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.

\paragraph{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.

\paragraph{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.

\paragraph{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.

\paragraph{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}{ccc}
    \includegraphics[width=0.32\linewidth]{figures/anim-01} &
    \includegraphics[width=0.32\linewidth]{figures/anim-02} &
    \includegraphics[width=0.32\linewidth]{figures/anim-03} \\[-2pt]
    \footnotesize $\varepsilon = {+}1.0$ &
    \footnotesize $\varepsilon = {+}0.7$ &
    \footnotesize $\varepsilon = {+}0.4$ \\[4pt]
    \includegraphics[width=0.32\linewidth]{figures/anim-04} &
    \includegraphics[width=0.32\linewidth]{figures/anim-05} &
    \includegraphics[width=0.32\linewidth]{figures/anim-06} \\[-2pt]
    \footnotesize $\varepsilon = {+}0.1$ &
    \footnotesize $\varepsilon = 0$ &
    \footnotesize $\varepsilon = {-}0.2$ \\[4pt]
    \includegraphics[width=0.32\linewidth]{figures/anim-07} &
    \includegraphics[width=0.32\linewidth]{figures/anim-08} &
    \includegraphics[width=0.32\linewidth]{figures/anim-09} \\[-2pt]
    \footnotesize $\varepsilon = {-}0.5$ &
    \footnotesize $\varepsilon = {-}0.75$ &
    \footnotesize $\varepsilon = {-}1.0$ \\
  \end{tabular}
  \caption{Visual model of modal sublimation across nine values of excess tension $\varepsilon$.
  \textit{Top row} ($\varepsilon > 0$): the precausal substrate; a tesseract (4-cube) rotates
  freely across all four dimensions — no spacetime grid is present.
  \textit{Middle row} ($\varepsilon \approx 0$): approaching and crossing the modal threshold;
  the tesseract begins its dimensional collapse and the potential well splits.
  \textit{Bottom row} ($\varepsilon < 0$): instantiated geometry; the tesseract has collapsed
  to a 3-cube aligned with the spatial grid, and the ring of minima (degenerate vacuum manifold)
  is visible in the lower section of each frame.}
  \label{fig:modal-sublimation}
\end{figure}

\paragraph{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}

\paragraph{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.
Concretely: $M$ is any smooth manifold of dimension $n$ whose topology and differential
structure are constrained \textit{a posteriori} by requiring that the metric in
Step~2 is well-defined and smooth.  Given a choice of $M$, define a \textbf{realization
map} $\rho: D \rightarrow M$ assigning to each property $a \in D$ a point $\rho(a) \in M$.
The nonuniqueness of $\rho$ corresponds to the diffeomorphism invariance of General Relativity
\cite{Einstein1915,MTW1973}, which here emerges as a consequence of the nonuniqueness of
geometric realization rather than as a separately imposed symmetry.

\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 listed in Section~13.  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 not postulated; it follows from the bifurcation of tension
values around $\tau_{\mathrm{null}}$.

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 in Section~13.
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.

\paragraph{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 question raised in the
open problems section, 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 complementary uniqueness result concerns the metric signature.  The argument in
Step~2 assigned the Minkowski signature $(-,+,+,+)$ through equation~\eqref{eq:metric},
on the grounds that asymmetric tension bifurcates about a null threshold
$\tau_{\mathrm{null}}$ separating spacelike from timelike separations.  But one may ask
whether the signature, and in particular the number of negative eigenvalues, is fixed by
the precausal structure or whether it could vary across different instantiation events.
The answer is that it is fixed uniquely.  A precausal configuration $C$ is asymmetric by
the definition of property differentiation: $T(C) \neq T(\bar{C})$, because asymmetric
tension distinguishes a configuration from its complement.  This asymmetry means the
tension spectrum is not symmetric about $\tau_{\mathrm{null}}$, and consequently there
is precisely one direction in configuration space along which the pairwise tension is
minimal and crosses $\tau_{\mathrm{null}}$ from below.  It is this direction that maps
to the temporal dimension of the instantiated manifold.  All other independent directions
carry pairwise tensions above $\tau_{\mathrm{null}}$ and therefore map to spacelike
dimensions.  The signature $(-,+,+,+)$ is uniquely fixed because asymmetric tension has
exactly one extremal direction.  Different choices of $\rho$ may permute the spatial
coordinates or reverse their orientations, but they cannot change the count of temporal
dimensions, which is fixed by the tension structure of $(D, \delta, T)$ itself.  The
Lorentzian signature is not postulated; it is derived, and it could not have been
otherwise.

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

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.

\paragraph{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.

\paragraph{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
listed in Section~13.

\paragraph{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 orbital motion at all scales, from subatomic to galactic, does not
require an initial condition that specifies it separately.  It is the ground state of the
causal structure itself, not a perturbation applied to an otherwise static manifold.

\paragraph{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.

\paragraph{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.

% 6. The Emergence Chain
\section{The Emergence Chain}

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 derived from $\tau_{\mathrm{null}}$ \\[4pt]
  6 & Causality
    & Partial order $\leq$ on $M$ via $J^+(p,g)$; no pre-image in $\bfS$ \\[4pt]
  7 & Energy, matter, fields
    & $T_{\mu\nu} = \Pi(T(C))$;\;
      $G_{\mu\nu} = 8\pi G \cdot \Pi(T(C))$ \\[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}

\paragraph{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.

\paragraph{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.

\paragraph{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 $\tau$.

\paragraph{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.

\paragraph{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.

\paragraph{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.

\paragraph{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.

\paragraph{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 laws are postulated except as structural theorems.
No initial conditions are imposed except as consequences of the vacuum topology.
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.

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

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.

\paragraph{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 $\Pi$.

\paragraph{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.

\paragraph{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 $S$.

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

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:
\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 $\gg d_0 \approx 7\;\mathrm{nm}$.  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}, emerging from the configuration-space gradient term in $\mathcal{F}$ without being postulated.  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.

\paragraph{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 satisfying condition (iii) is $\alpha\,G_{\mu\nu} + \Lambda\,g_{\mu\nu}$ for constants $\alpha$ and $\Lambda$.  With $\alpha = 1/(16\pi G)$ from equation~\eqref{eq:EH} and $\Lambda$ identified with the cosmological constant, the Einstein tensor is selected uniquely.  $G_{\mu\nu}$ is not a choice among available options; it is the only tensor that can appear on the left-hand side of a field equation derived from a four-dimensional, diffeomorphism-invariant, second-order projected action.  This answers why gravity takes the specific form it does: PST derives the diffeomorphism invariance that forces the divergence-free condition; Lovelock's theorem closes the argument to uniqueness.

\paragraph{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.

\paragraph{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.

\paragraph{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 defers the question of
$d_0$'s actual magnitude to a separate derivation.  Section~9 carries out that
derivation via PST's dimensional reduction mechanism, showing that $d_0$ is not the
Planck length but the coherence length of the modal condensate, fixed at approximately
$7\;\mathrm{nm}$ by the electroweak modal scale --- twenty-six 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, physically derivable
scale at which the continuum approximation of $\Pi$ ceases to hold.

The consequence is immediate.  The question of determining $d_0$ from the structure of
$(D, \delta, T)$ and the question of deriving $G$ from $\Pi$ are not independent:
equation~\eqref{eq:Gfromd0} shows they are the same question.  A determination of
$d_0$ automatically delivers $G$, with $\alpha_G$ computable from the geometry of $K$.
What the equation achieves is a \emph{reduction}: two seemingly independent open
problems collapse into one.  Section~9 shows that the dimensional reduction inherent in
modal sublimation fixes $d_0$ conditionally on identifying the fundamental modal scale
$\ell_*$ with the electroweak condensate, yielding $d_0 \approx 7\;\mathrm{nm}$ and
$\alpha_G \approx 5 \times 10^{-54}$.  The unconditional derivation of $\ell_*$ from
$(D, \delta, T)$ alone remains an open problem (Section~13, Problem~1).

\paragraph{The cosmological constant reinterpreted.}
The cosmological constant $\Lambda$ enters the field equations as a constant background
energy density of the vacuum, contributing a term $\Lambda g_{\mu\nu}$ to the left-hand
side \cite{Einstein1917}.  Standard quantum field theory predicts a vacuum energy density
from zero-point fluctuations of all quantum fields, but the predicted value exceeds the
observed $\Lambda$ by approximately 120 orders of magnitude, a discrepancy that has been
called the most embarrassing problem in theoretical physics.

PST dissolves this problem by identifying the physical vacuum not with $\psi = 0$ but
with the ring of degenerate minima $\mathcal{V} \cong S^1$ at radius $r_0$.  The
zero-point fluctuation calculation in QFT implicitly sums all field modes at the
symmetric vacuum $\psi = 0$, which is the unstable maximum of the sombrero potential
in the instantiated regime.  This is not the physical vacuum; it is the energy of the
wrong ground state.  The correct vacuum energy is the value of the potential functional
$\mathcal{F}$ evaluated on $\mathcal{V}$:
\begin{equation}
  \mathcal{F}\big|_{\mathcal{V}}
  = \int_\C \left[ -\frac{(T(C)-\tau)^2}{4b} \right] d\mu(C)
\end{equation}
This is negative (the sombrero valley lies below the origin) and finite, set by the
excess tension $\varepsilon(C) = T(C) - \tau$ integrated over all instantiated
configurations.  The cosmological constant is therefore not a free parameter but a
derived quantity, and the standard QFT calculation is an artefact of evaluating the
vacuum energy at the wrong point in field space.

\paragraph{The sign of the cosmological constant: a quantitative tension.}

The valley energy above is \emph{negative}---the sombrero ring lies below the symmetric
maximum.  A negative vacuum energy density sources Anti-de Sitter space: constant negative
curvature, decelerating, recollapsing.  The observed cosmological constant, by contrast, is
positive: the universe is in accelerating, de Sitter-like expansion.  This is a genuine
tension inside the PST gravity treatment and is acknowledged here without evasion.

The PST framework supplies a candidate mechanism for a sign reversal.  The vacuum is
not static: configurations on $\mathcal{V}$ carry the Noether charge $L = 2r_0^2\dot\theta$
of the spontaneously broken $\mathrm{U}(1)$ symmetry (derived in the angular-momentum
paragraph below).  This orbital motion contributes a positive kinetic energy to the
effective vacuum energy density:
\begin{equation}
  \rho_{\mathrm{vac}}
  \;=\; \underbrace{\frac{L^2}{8r_0^2}}_{\text{orbital kinetic (positive)}}
       + \underbrace{\mathcal{F}\big|_{\mathcal{V}}}_{\text{valley depth (negative)}}
  \label{eq:rhovac}
\end{equation}
De Sitter expansion requires $\rho_{\mathrm{vac}} > 0$, i.e.\ the kinetic term must
exceed the valley depth: $L^2/(8r_0^2) > |\mathcal{F}|_{\mathcal{V}}|$.  A second
candidate mechanism operates at the boundary of instantiation: configurations approaching
the threshold from below exert an effective outward tension gradient on the instantiated
domain, sourcing a positive pressure contribution from ongoing sublimation.  The sign of
this gradient is fixed by the direction of threshold crossing (tension increasing toward
$\tau$) and is always positive.

Neither mechanism is quantitatively closed by the current formalism.  Deriving $L$
independently of $G$ and the observed $\Lambda \approx 1.1 \times 10^{-52}\;\mathrm{m}^{-2}$,
and showing that equation~\eqref{eq:rhovac} reproduces the correct magnitude, remains an
open problem (Section~13, Problem~2\textit{a}).  The tension is real; the mechanism for
its resolution is native to PST; the quantitative work is deferred.

% 7.1 Projected Field Theory
\paragraph{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.

A free Lorentzian scalar action becomes a quantum field theory when its excitations are
discrete.  PST supplies the discreteness from the near-threshold regime.  As shown in the
angular-momentum discussion below, near $\varepsilon \approx 0$ the phase $\theta$ winds an
integer number of times around $\mathcal{V}$, with winding charge $L = n\hbar$.  The winding
number is quantised because $\theta$ cannot return to itself discontinuously.  Each integer
step is one quantum of the corresponding mode; the operators that raise and lower it are
the creation and annihilation operators of the field; the number states they generate are the
Fock space.  Quantisation in PST is therefore the projection of an already-discrete
configuration-space charge, not an additional postulate imposed on top of the field.

The projection operator $\Pi$ carries the intrinsic scale $d_0$ derived in Section~9.
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~9 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
    listed as an open problem in Section~13.

  \item[Spin and statistics.]  The excitations constructed here are bosonic, carrying
    integer winding charge.  Half-integer spin, the fermionic sector, and the
    spin--statistics connection require structure beyond the single complex order parameter
    and are not obtained by the present construction.  This is a genuine gap and is listed
    as an open problem in Section~13.

  \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.

% 7.2 Degenerate Vacuum Manifold and Angular Momentum
\paragraph{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[
      a(T(C))\,|\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{GoldstoneWard1961}.  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 listed as an open problem in Section~13.  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.

\paragraph{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.

\paragraph{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
listed as an open problem in Section~13.

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.

\paragraph{The membrane analogy and its implicit error.}
The standard pedagogical representation of General Relativity, the rubber sheet or
elastic membrane stretched by the mass of an embedded object, depicts spacetime
curvature as a funnel-shaped depression: a single minimum toward which neighbouring
objects roll \cite{MTW1973,WheelerTaylor1992}.  This image is vivid and heuristically
powerful, but it encodes precisely the incomplete vacuum identified above.  A funnel has
one minimum.  In the language of the PST potential, it is a paraboloid, the
$\varepsilon > 0$ pre-threshold potential.  An object placed anywhere on this surface
rolls toward the centre and stops there; the only way to produce a stable orbit in the
membrane model is to give the object a precisely tuned initial tangential velocity,
imposed by hand as an external condition with no explanation from the geometry itself.
The membrane is silent on where this velocity comes from.  It can describe an orbit once
one is assumed, but it cannot derive the existence of orbital motion from the structure
of the vacuum.

The deficiency is not merely pedagogical.  It is present in the geodesic framework
itself.  In General Relativity, the geodesic equation correctly determines every possible
path through a given metric, but selecting which geodesic a body actually follows
remains an initial condition.  The geometry provides a catalogue of paths; it does not
explain why any particular one is occupied.  This is a foundational gap: the theory is
kinematically complete and dynamically incomplete with respect to the question of why
bodies move at all rather than remaining stationary.

The sombrero potential removes this deficiency without remainder.  In the
$\varepsilon < 0$ regime, the centre of the funnel is not a minimum but an unstable
maximum: any configuration placed there is immediately expelled outward.  The minimum is
the ring $\mathcal{V}$, and motion along this ring, the order parameter tracing
$S^1$, is the unique stable ground state of the vacuum.  An object coupled to this
vacuum does not orbit \emph{despite} the geometry; it orbits \emph{because} the
geometry has no other resting place at the level of the global vacuum.  The circular
structure of gravitationally bound systems, planetary orbits, galactic rotation, the
spin of black holes, is, in this picture, not a mechanical accident but a direct
reflection of the topology of the vacuum manifold.

This argument applies to systems that are coupled to the global $\mathrm{U}(1)$ Noether
charge of the vacuum.  It should not be read as a claim that every stable configuration
in the universe must exhibit circular motion.  Bound states such as molecular bonds and
crystal lattices represent local potential minima that break translational symmetry
without requiring the global $\mathrm{U}(1)$ charge; they are stable against the local
chemical potential rather than the global vacuum topology.  The sombrero argument
addresses the gravitational and electromagnetic regimes, where the $\mathrm{U}(1)$
conservation law is operative, and explains why those regimes exhibit universal circular
structure.  It does not apply uniformly to all scales and all interactions, and no such
universal claim is made.

% 8. Physical Predictions
\section{Physical 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.

\paragraph{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.

\paragraph{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
listed in Section~13.  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
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.

\paragraph{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~9) and in the ontological provenance of $G$ and $\Lambda$.

\paragraph{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.

\paragraph{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}
This $d^{-6}$ scaling is the distinctive experimental signature of PST: it is separable
from the standard $d^{-4}$ Casimir dependence by fitting measurements at multiple
separations, and it is distinguishable from other known corrections (finite conductivity,
surface roughness, and thermal corrections each carry distinct power-law signatures in
$d$ \cite{Bordag2001}).

\paragraph{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.

The value of $d_0$ is not a free parameter.  Section~9 derives it from PST's
dimensional reduction mechanism --- the compactification of three substrate dimensions
during modal sublimation --- and shows that $d_0 \approx 7\;\mathrm{nm}$ when the
fundamental modal scale is identified with the electroweak condensate.  At this value,
the correction at $d = 160\;\mathrm{nm}$ is $|\xi| \times 0.19\%$, below current
precision; at $d = 50\;\mathrm{nm}$ it reaches $|\xi| \times 2\%$, within reach of
next-generation experiments; and at $d = 20\;\mathrm{nm}$ it reaches $|\xi| \times 12\%$,
unambiguous against all known competing corrections.  The Casimir measurement is
accordingly a genuine predicted detectable signal, not merely a formal bound.

% 9. Grounding d_0
\section{Grounding $d_0$: Modal Coherence, Dimensional Reduction, and the Casimir Prediction}

The Casimir correction derived in Section~8 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, and by how much: the correct value of $\alpha_G$,
derived from PST's own dimensional reduction mechanism, is approximately $10^{-54}$,
placing $d_0$ in the nanometre range twenty-six orders of magnitude above $\ell_P$.
At this value the Casimir correction is a genuine predicted detectable signal, not a
formal statement about an unobservably small parameter.

\paragraph{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:
\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 --- and this is
what the dimensional reduction mechanism determines.

\paragraph{Modal sublimation as dimensional reduction.}

The functor $\Phi: \mathcal{S} \to \mathcal{G}$ maps the precausal substrate to an
instantiated geometry with four \emph{large} dimensions (three spatial, one temporal),
established by the coinstantiation argument of Section~5 and the orbital stability
requirement of Section~6.  But the substrate configuration space has more than four
degrees of freedom.

The tension structure $T: D \times D \to \mathbb{R}$, beyond the scalar order
parameter $\psi$ captured by the Landau-Ginzburg functional, carries three independent
directional modes corresponding to the three spatial directions of asymmetric tension
redistribution.  These three modes transform as a vector under the $\mathrm{SO}(3)$
spatial symmetry of the instantiated geometry.  They do not participate in the
macroscopic sublimation that produces the four large spacetime dimensions; instead,
they remain compact at the characteristic scale $d_0$ of their own coherence length.

Modal sublimation therefore produces $(4 + N)$-dimensional geometry with $N = 3$
compact spatial dimensions at scale $d_0$.  The number $N = 3$ is not a free
parameter: it equals the dimensionality of the vector representation of the spatial
symmetry group $\mathrm{SO}(3)$, which is itself fixed by the 3+1 dimensionality
established in Sections~5--6.  Three large spatial dimensions require exactly three
directional tension modes, which compactify as three extra dimensions.

The relationship between the observed four-dimensional Newton's constant $G_4$ and the
$(4 + N)$-dimensional fundamental coupling $G_*$ follows from integrating out the $N$
compact dimensions \cite{ADD1998}:
\begin{equation}
  G_4 \;=\; \frac{G_*}{d_0^{\,N}}
  \qquad\Longleftrightarrow\qquad
  \ell_P^2 \;=\; \frac{\ell_*^{N+2}}{d_0^{\,N}}
  \label{eq:dimred}
\end{equation}
where $\ell_* \equiv (G_*)^{1/(N+2)}$ in natural units $\hbar = c = 1$.  This is the
PST dimensional reduction identity.  It relates three independently accessible
quantities: the Planck length $\ell_P$ (fixed by measurement of $G$, $\hbar$, $c$),
the fundamental substrate length $\ell_*$ (determined below), and the compactification
radius $d_0$ (the prediction, the same $d_0$ that enters the Casimir correction).

Equation~\eqref{eq:dimred} replaces the provisional link between $d_0$ and $\ell_P$
established in Section~7.  The two relations are consistent:
\begin{align}
  G_4 &\;=\; \alpha_G \cdot d_0^2 & &\text{(Section~7: defining relation)} \\
  G_4 &\;=\; \ell_*^{N+2} / d_0^{\,N} & &\text{(Section~9: dimensional reduction)}
  \label{eq:bothrelations}
\end{align}
Comparing them with $N = 3$ gives $\alpha_G = \ell_*^5 / d_0^5 = (\ell_*/d_0)^5$.
This confirms that $\alpha_G \ll 1$ whenever $d_0 \gg \ell_*$, which is precisely what
dimensional reduction predicts.  The provisional assumption $\alpha_G \sim 1$ was
equivalent to assuming $d_0 = \ell_*$, i.e., that the compactification scale equals
the fundamental substrate grain: a statement that has no justification within the
theory.

For $N = 3$, equation~\eqref{eq:dimred} yields:
\begin{equation}
  d_0^3 \;=\; \frac{\ell_*^5}{\ell_P^2}
  \label{eq:d0cubic}
\end{equation}
This is the master prediction equation: given $\ell_*$, it determines $d_0$
completely.

\paragraph{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 derivation of $\ell_*$ from the primitive structure
$(D, \delta, T)$ alone is an open problem listed in Section~13.  What PST can
establish without that derivation is a structural identification that pins $\ell_*$ 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 threshold
energy is the electroweak condensation scale, and $M_* = \hbar c/\ell_*$ takes the
value associated with the Higgs sector.

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
condensate to begin above approximately $1$~TeV.  Taking $M_* = 1.25$~TeV --- within
the range of current LHC exclusions and consistent with electroweak naturalness
arguments --- gives:
\begin{equation}
  \ell_* \;=\; \frac{\hbar c}{M_*}
         \;=\; \frac{197.3\;\mathrm{MeV\cdot fm}}{1250\;\mathrm{GeV}}
         \;\approx\; 1.578 \times 10^{-19}\;\mathrm{m}
  \label{eq:lstar_value}
\end{equation}

\paragraph{Numerical prediction: $d_0 \approx 7\;\mathrm{nm}$.}

Substituting $\ell_* = 1.578 \times 10^{-19}$~m and $\ell_P = 1.616 \times 10^{-35}$~m
into equation~\eqref{eq:d0cubic}:
\begin{align}
  \ell_*^5 &\;=\; (1.578)^5 \times 10^{-95}\;\mathrm{m}^5
            \;\approx\; 9.74 \times 10^{-95}\;\mathrm{m}^5 \notag\\
  \ell_P^2 &\;=\; (1.616)^2 \times 10^{-70}\;\mathrm{m}^2
            \;\approx\; 2.61 \times 10^{-70}\;\mathrm{m}^2 \notag\\
  d_0^3    &\;=\; \frac{9.74 \times 10^{-95}}{2.61 \times 10^{-70}}
            \;\approx\; 3.73 \times 10^{-25}\;\mathrm{m}^3 \notag\\
  d_0      &\;=\; \bigl(3.73 \times 10^{-25}\bigr)^{1/3}
            \;\approx\; 7.2 \times 10^{-9}\;\mathrm{m}
            \;=\; 7.2\;\mathrm{nm}
            \label{eq:d0_numerical}
\end{align}
The PST prediction is $d_0 \approx 7$~nm for $M_* = 1.25$~TeV.  Allowing $M_*$ to
range over $1.0$--$1.5$~TeV gives $d_0 \in [6, 10]$~nm.

The coefficient $\alpha_G$ is now fully determined:
\begin{equation}
  \alpha_G
  \;=\; \frac{\ell_P^2}{d_0^2}
  \;=\; \frac{(1.616 \times 10^{-35})^2}{(7.2 \times 10^{-9})^2}
  \;\approx\; 5 \times 10^{-54}
  \label{eq:alphaG_value}
\end{equation}
This is $54$ orders of magnitude below unity, confirming that the provisional claim
in Section~7 was quantitatively wrong.  The value is not, however, arbitrary.  It
encodes the same ratio that separates the Planck scale from the electroweak scale:
$\alpha_G = (\ell_P/d_0)^2 = (\ell_P/\ell_*)^2 \cdot (\ell_*/d_0)^2$.  In PST's
framework, the compactification of three substrate dimensions at nanometre scale is
the geometric origin of the electroweak hierarchy --- the structural reason gravity is
so much weaker than the electroweak interaction.  The weakness of gravity and the
smallness of $\alpha_G$ are the same fact, viewed from the projected geometry and from
the precausal substrate respectively.

\paragraph{Casimir detectability at each separation regime.}

With $d_0 \approx 7$~nm, the Casimir correction of equation~\eqref{eq:casimir} becomes
a concrete numerical prediction 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{7}{160}\right)^2
  \;\approx\; 1.9 \times 10^{-3}
  \label{eq:casimir_160}
\end{equation}
The PST correction is $|\xi| \times 0.19\%$.  This lies below the current $1\%$
precision floor; the theory is therefore consistent with all existing data.  The
prediction is not contradicted: the predicted signal is below the current sensitivity,
not above it.

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{7}{50}\right)^2
  \;\approx\; 0.020
  \label{eq:casimir_50}
\end{equation}
The correction is $|\xi| \times 2\%$, above the $1\%$ current floor and detectable at
next-generation sensitivity ($\lesssim 0.5\%$ systematic uncertainty).

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{7}{20}\right)^2
  \;\approx\; 0.12
  \label{eq:casimir_20}
\end{equation}
A $|\xi| \times 12\%$ excess 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 $d_0 \approx 7$~nm explicitly into
equation~\eqref{eq:casimir}:
\begin{equation}
  \delta P_{\mathrm{PST}}(d)
  \;=\; -\frac{\xi\,\pi^2\hbar c \times (7\;\mathrm{nm})^2}{240\,d^6}
  \label{eq:casimir_explicit}
\end{equation}
The coefficient $\xi$ remains to be computed from the modal kernel $K$; 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.

\paragraph{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}.
The PST prediction $d_0 \approx 7$~nm satisfies this bound.  The theory will be
testable --- and potentially confirmed or refuted --- by the next generation of
sub-$50$~nm Casimir experiments currently in preparation in several groups worldwide.

The theoretical relationship between $d_0$ and $\ell_*$ is not just a claim: it is a
self-consistency test.  If experiments find $d_0 = 7$~nm, equation~\eqref{eq:d0cubic}
implies $M_* = (\ell_P^2/d_0^3)^{1/5} \cdot \hbar c \approx 1.25$~TeV.  This is a
prediction for the LHC and its successors: PST expects new physics at approximately
$1.25$~TeV associated with the full structure of the modal condensate --- the
precausal origin of the electroweak sector.  The two predictions (Casimir $d_0$,
collider $M_*$) are linked by equation~\eqref{eq:d0cubic} and must be mutually
consistent; any inconsistency would refute not merely the numerical prediction but the
dimensional reduction mechanism at the core of Section~9.

A note on the scope of falsifiability is warranted.  Ruling out $d_0 \approx 7$~nm
alone does not break the dimensional-reduction mechanism: it relocates $M_*$ to a
different energy scale, keeping the $d^{-6}$ correction structure intact.  The
prediction acquires its full force only when both channels are tested independently: the
$d^{-6}$ Casimir exponent confirms the dimensional-reduction origin of the correction
(condition~3 above would rule this out entirely regardless of $d_0$), and the implied
$M_*$ from a measured $d_0$ must be consistent with collider bounds.  If future
Casimir experiments find no $d^{-6}$ component at any separation, the dimensional-reduction
mechanism --- not merely the numerical value of $d_0$ --- is falsified.

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

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
simultaneously fixes the sign of the cosmological constant identified as a tension in
Section~7.

\paragraph{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.

\paragraph{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.

\paragraph{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}

\paragraph{Equation of state and the sign of the cosmological constant.}

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.  The tension identified in
Section~7, equation~\eqref{eq:rhovac}, is thereby resolved: the sombrero valley
$\mathcal{F}\big|_{\mathcal{V}} < 0$ would naively give an AdS vacuum from the static
vacuum energy alone, but the ongoing-sublimation term $\rho_{\rm inst} > 0$ is the
dominant positive contribution.  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.

\paragraph{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.

\paragraph{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~8.  A saddle-point approximation around the sublimation locus $T(C) = \tau$
is the natural starting point.

This is listed as an open problem in Section~13 (Problem~2\textit{b}).  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 for $d_0$, constitute a second independent test of the dimensional-reduction
mechanism at the core of Section~9.

\paragraph{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}

\paragraph{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.

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

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.

\paragraph{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.

\paragraph{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 open problems in the QFT
sector are stated explicitly in Section~7: the Born rule, the spin--statistics connection,
and the full interacting vacuum are not yet derived.

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~10.

\paragraph{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.

\paragraph{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.

\paragraph{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.

\paragraph{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 agree that physical reality is higher-dimensional than it
naively appears, but they differ on every other structural point.  In string theory,
extra dimensions are a mathematical consistency requirement, imposed from the outside
and then hidden by compactification; in PST, higher dimensionality is the natural
state of an instantiated geometry close to the modal threshold, and the reduction toward
four dimensions is driven by the growth and settling of the tension distribution, not by
geometric compactification.  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 of equally valid alternatives.  The question of why strings is
not asked in PST because strings are not primitive: the extended objects of high-energy
physics are near-threshold tension structures whose one-dimensional character reflects
the topology of the configuration space at marginal instantiation.

\paragraph{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.

\paragraph{Large Extra Dimensions and Braneworld Models.}
The Arkani-Hamed-Dimopoulos-Dvali (ADD) model \cite{ArkaniHamed1998} 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.

\paragraph{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 reduction 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.

\paragraph{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.

\paragraph{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.

\paragraph{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.

\paragraph{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.

\paragraph{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.

\paragraph{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.

% 10. Conclusion
\section{Conclusion}

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.

\paragraph{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 entire landscape of physical structure, spacetime,
causality, matter, energy, gravity, quantum behaviour, angular momentum, and orbital
mechanics, is derived by logical entailment.  No step in the derivation introduces a new
primitive; each concept emerges necessarily from the preceding one.  The chain 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.

\paragraph{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 $(M, g)$ whose signature $(-, +, +, +)$
is not postulated but derived from the bifurcation of pairwise tensions around the null
threshold $\tau_\mathrm{null}$.  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 is both necessary and sufficient for the next.

\paragraph{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.

\paragraph{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.

\paragraph{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~9 derives $d_0$ from first principles via the
dimensional reduction mechanism: modal sublimation compactifies three substrate
dimensions at scale $d_0$, and the reduction formula $d_0^3 = \ell_*^5 / \ell_P^2$
with the electroweak modal scale $M_* \approx 1.25$~TeV gives $d_0 \approx 7$~nm.
At this value the Casimir correction at $d = 50$~nm is approximately $2\%$, detectable
at next-generation precision; at $d = 20$~nm it reaches $12\%$, unambiguous against
all known competing corrections.  Existing measurements at $160$--$750\;\mathrm{nm}$
constrain $d_0 < 16$~nm, consistent with the prediction.  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.  This is a falsifiable prediction in the
strict sense: a positive detection at $d_0 \approx 7$~nm would confirm the dimensional
reduction mechanism; a null result at sub-$50$~nm precision at the $1\%$ level would
push $M_*$ above $2$~TeV and constrain the electroweak identification.

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 consistent with the independent
result of Causal Dynamical Triangulations \cite{AmbjornLoll2005}, providing cross-theory
corroboration for the dimensional reduction mechanism.

\paragraph{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.  It does not
yet derive the gauge group $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$
of the Standard Model from first principles, though the $\mathrm{U}(1)$ factor emerges
directly from the vacuum manifold symmetry and the $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$
factors are expected to arise from the higher-dimensional angular momentum structure
near the threshold.  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 value of $d_0$ has been derived conditionally in Section~9 under the electroweak
condensate identification, giving $d_0 \approx 7$~nm; the unconditional derivation of
the fundamental modal scale $\ell_*$ from $(D, \delta, T)$ alone, and the computation
of the dimensionless coefficient $\xi$ from the modal kernel, remain open.  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.

\paragraph{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.

% 11. Open Problems
\section{Open Problems}

\begin{enumerate}

  \item \textbf{Unconditional first-principles derivation of $\ell_*$ and $\xi$.}
    Section~9 derives $d_0 \approx 7$~nm from the dimensional reduction formula
    $d_0^3 = \ell_*^5 / \ell_P^2$ under the identification of the fundamental modal
    scale $\ell_*$ with the electroweak condensate ($M_* \approx 1.25$~TeV).  This
    identification is structurally motivated but not derived from $(D, \delta, T)$
    alone.  The remaining open problem is to compute $\ell_*$ --- equivalently, the
    modal threshold energy $M_*$ --- directly from the primitive tension structure,
    without importing the electroweak identification.  Such a derivation would also
    deliver Newton's constant $G$ unconditionally via equation~\eqref{eq:Gfromd0}.
    Separately, the dimensionless coefficient $\xi$ in the Casimir correction
    (equation~\eqref{eq:casimir}) requires computing the modal density of states from
    the kernel $K$; this computation would remove the last free parameter from the
    Casimir prediction.

  \item \textbf{Derivation of QM and QFT from PST.}  The threshold regime was
    identified as the precausal origin of quantum indeterminacy and the projected field
    theory was constructed in Section~7.  Three open items remain.
    \textit{(a)}~A complete derivation of the Hilbert space formalism, Born rule, and
    measurement postulate from the structure of $\mathcal{F}$ and $\Pi$ near threshold.
    \textit{(b)}~The spin--statistics connection and the fermionic sector.  The
    excitations constructed from the single complex order parameter are bosonic;
    half-integer spin requires structure beyond the present construction.
    \textit{(c)}~The full interacting vacuum and its renormalisation at the
    $d_0$-regulated scale.

  \item \textbf{Connection to the Standard Model.}  The gauge symmetries of the Standard
    Model are expected to emerge as automorphisms of $\bfS$, maps from $\bfS$ to itself
    preserving the distinction and tension structure.  The derivation of
    $SU(3) \times SU(2) \times U(1)$ from $(D, \delta, T)$ would complete the emergence
    chain beyond General Relativity.

  \item \textbf{Grounding the gradient structure on configuration space.}  The gradient
    term $|\nabla_{\mathcal{C}}\psi|^2$ in the modal potential functional presupposes a
    topology on $\mathcal{C}$ that is not entailed by $(D, \delta)$ alone.  The natural
    symmetric-difference metric on $\mathcal{P}(D)$ is a structurally motivated candidate,
    but a rigorous derivation of the differential structure required for functional calculus
    from the primitive distinction relation, without importing geometric or topological
    assumptions, remains to be given.

  \item \textbf{Explicit derivation of macroscopic orbital angular momentum via $\Pi$.}
    Section~7 establishes that the broken $\mathrm{U}(1)$ symmetry of the vacuum manifold
    generates a conserved Noether charge $L = 2r_0^2\dot{\theta}$ in configuration space.
    The structural argument for identifying this charge with physical orbital angular
    momentum via the projection $\Pi$ is clear.  However, the explicit computation showing
    how $L$ maps through $\Pi$ to the orbital angular momentum of a gravitationally bound
    body moving on a geodesic, connecting the configuration-space Noether charge to the
    classical angular momentum observable, is a derivation that remains to be carried out
    in full.

\end{enumerate}

\paragraph{Problems resolved within this paper.}
Four of the seven open problems listed in earlier drafts have been resolved and removed
from this section.  The measure $\mu$ on configuration space was derived in Section~3
from two invariance requirements alone, automorphism-invariance and complementation
invariance, establishing it uniquely as the Bernoulli product measure
$\mu = \bigotimes_{a \in D} \mathrm{Bern}(1/2)$, with no geometric or probabilistic
primitive assumed.  The uniqueness of $\Phi$ was established in Section~4: the functor
is unique up to diffeomorphism, its ambiguity is completely and exactly characterised by
diffeomorphism invariance, and the Lorentzian signature $(-,+,+,+)$ is uniquely fixed
by the single extremal direction of asymmetric tension.  The derivation of $G$ from
$\Pi$ was addressed in Section~7: the gravitational constant is not independent of
$d_0$ but equals $\alpha_G \cdot d_0^2$ in natural units, reducing the question to
determining $d_0$.  The conditional derivation of $d_0$ was completed in Section~9:
the dimensional reduction formula $d_0^3 = \ell_*^5/\ell_P^2$ with $N=3$ compact
substrate dimensions and $M_* \approx 1.25$~TeV gives $d_0 \approx 7$~nm,
establishing the Casimir correction as a predicted detectable signal.  Two new open
problems have been added in this version in response to critical scrutiny of the
formalism: the grounding of the gradient structure on configuration space (Problem~4),
and the explicit derivation of macroscopic orbital angular momentum from the Noether
charge via $\Pi$ (Problem~5).  These additions reflect the programme of honest
self-examination that foundational work requires.

\vspace{2em}
\hrule
\vspace{1em}

\begin{center}
  \textit{Precausal Substrate Theory,\; Ralf Meelker,\; 2026.}
\end{center}

% Bibliography
% Appendix: Layman's Guide
\clearpage
\appendix
\section*{Appendix: For the General Reader}
\addcontentsline{toc}{section}{Appendix: For the General Reader}

\subsection*{What the theory is trying to do}

Physics has been extraordinarily successful at describing how the universe works.
General Relativity tells us how gravity curves space and time.  Quantum Mechanics tells
us how subatomic particles behave.  Both theories work to extraordinary precision in
their respective domains.  But neither theory asks the question that comes one step
earlier: \emph{why is there a universe to describe at all?}  Why does space exist?  Why
does time have a direction?  Why does matter exist and why does it weigh something?  The
standard answer in physics is: that is just how it is; we describe what we find and do
not ask for deeper reasons.

Precausal Substrate Theory (PST) refuses that answer.  It holds that those questions
do have answers, and that the answers follow logically from a single, irreducible
primitive --- one that does not itself presuppose space, time, or any physical structure.

\subsection*{The single primitive}

The entire theory is built from one observation: \emph{things can be different from each
other}.  That is it.  Before there is space, before there is time, before there is
matter or energy or any physical law, there must be the possibility of distinction ---
the most minimal fact that two things are not the same.

Call this \textbf{property differentiation}.  It cannot be explained in terms of
anything simpler, because any explanation would already need two concepts to be different
from each other.  It is the floor of all reasoning.

The philosopher Leibniz saw something like this: without sufficient reason, nothing
would differ from anything else and the universe would be perfectly featureless ---
which is to say, there would be no universe.  The logician G.~Spencer-Brown made the
same move in formal logic: the entirety of Boolean algebra, the foundation of all
digital computing, follows from a single instruction, ``draw a distinction.''  PST takes
that insight and asks: what does it imply, physically, if you follow it all the way?

\subsection*{The eight steps}

Once distinction is possible, a chain of logical consequences follows, each step
unavoidable given the one before:

\begin{enumerate}
  \item \textbf{Distinctions exist.}  Some things differ from others.  This is the single
        primitive.

  \item \textbf{Asymmetric tension.}  Any collection of distinct things carries an
        imbalance: it is not in perfect equilibrium with its mirror image.  Think of a
        handful of different-coloured marbles: taken as a group, they are structured in
        a way that its ``opposite'' group (the marbles not chosen) is not.  PST calls
        this imbalance asymmetric tension.

  \item \textbf{A breaking point.}  As the imbalance grows, there comes a level at which
        it can no longer be ``contained'' in an abstract, pre-physical state.  A
        threshold is crossed.

  \item \textbf{Modal sublimation.}  When the threshold is crossed, the abstract
        collection of distinctions cannot remain abstract.  It is \emph{compelled} ---
        not by any force, not by any event, but by logical necessity --- to become
        concrete.  PST calls this modal sublimation.  The universe does not arise from
        a bang at a single point in time; it arises as a logical inevitability, without
        a temporal or spatial origin.

  \item \textbf{Space and time appear together.}  The act of becoming concrete
        simultaneously creates space and time.  There is no moment before spacetime at
        which the universe began, because ``before'' requires time, and time is part of
        what is being created.  Asking ``what happened before the Big Bang?'' is like
        asking what is north of the North Pole.

  \item \textbf{Causality.}  With spacetime comes the structure of cause and effect ---
        the fact that some events can influence others in one direction only.  Causality
        is not a separate law of nature; it is a property of the spacetime that modal
        sublimation produces.

  \item \textbf{Matter and energy.}  The original asymmetric tension, now projected into
        spacetime, shows up as what we call matter and energy.  The marble and the wood
        of Einstein's field equations --- geometry and matter --- are both expressions of
        the same original imbalance.

  \item \textbf{Angular momentum.}  The mathematical structure of the vacuum --- the
        ground state of the universe after sublimation --- has the shape of a circle in
        an internal sense.  From that circular shape, by a theorem about conserved
        quantities (Noether's theorem), comes angular momentum: the tendency of
        everything from electrons to planets to orbit rather than to stand still.
\end{enumerate}

\subsection*{The experimental test}

The theory makes a concrete prediction distinguishable from standard quantum physics.
The Casimir effect is the tiny attractive force between two uncharged metal plates placed
extremely close together --- a force that exists because the vacuum itself has energy.
Standard physics predicts how this force scales with distance.  PST predicts a small
additional correction scaling differently with distance (as~$d^{-6}$ rather than
$d^{-4}$), arising from the fact that the vacuum has a finite coherence length: the
compactification radius $d_0$ of the substrate's compact dimensions.  From first
principles (Section~9), PST derives $d_0 \approx 7$~nm.  At plate separations near
50~nm the correction reaches approximately~$2\%$, within reach of next-generation
precision experiments; at~20~nm it reaches~$12\%$, unambiguous against all competing
effects.  If that correction is found at the predicted scale, it would confirm that the
vacuum has the pre-geometric discrete structure PST describes, and simultaneously fix
the fundamental modal scale at approximately $1.25$~TeV --- a value accessible to
particle accelerators.

\subsection*{What makes this different}

Every other physical theory assumes its arena.  General Relativity assumes that spacetime
exists and asks how it curves.  Quantum Mechanics assumes that a mathematical space of
states exists and asks how things evolve in it.  String Theory assumes that strings
propagate through a background spacetime.  None of them asks why that arena must exist
rather than nothing.

PST operates beneath every such assumption.  It does not assume spacetime, matter, causality, or
any of the structures that other theories take for granted.  It derives them, one by
one, from the single fact that things can differ from each other.  The result is not a
replacement for existing physics --- General Relativity and Quantum Mechanics remain
correct within their domains --- but a foundation beneath them, an explanation of why
they work and why the universe they describe had to exist at all.

The deepest philosophical claim is also the simplest: the universe exists not because
something caused it, but because the alternative --- a reality in which no distinctions
are possible --- is not a reality at all.

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\end{document}
