Appendix D Comparison to Other Theories

This appendix tabulates the substrate-style and emergent-physics theories that share posture with PST: each posits a layer beneath spacetime, causality, or the Standard Model from which the macroscopic structure is supposed to follow. The narrative comparison sits in Chapter 17; the table below condenses each theory along five axes for side-by-side reading.

Inclusion criterion. A theory is included if it either (i) posits a pre-geometric primitive from which spacetime or causality is derived, or (ii) derives a Standard-Model-relevant structure from an underlying programme. Theories PST recovers as effective descriptions (General Relativity, Quantum Mechanics, Quantum Field Theory) are not in the comparison set; for those see Chapter 17.

Axes. The columns are: the theory with its attribution; the substrate primitive (what is taken as fundamental); the spacetime-emergence mechanism (how, if at all, spacetime is derived); Standard-Model coverage (whether gauge group, matter content, or families are addressed); and the relation to PST (whether PST agrees, extends, contrasts, or sits orthogonal). Entries are deliberately compressed; the cited references, where given, supply the full treatment.

Theory	Substrate primitive	Spacetime emergence	Standard-Model coverage	Relation to PST
Reference row.
PST (Meelker, 2026)	Property differentiation with $V_{7}$-valued directional content $(D,\delta,v)$	Mosco convergence of $V_{7}$-Boolean Dirichlet form to Laplace–Beltrami on $\mathbb{R}\times S^{3}$ along $\hat{\tau}$	$\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ derived; one generation from $\mathrm{Cl}(0,6)$ minimal ideal; $N_{\mathrm{gen}}=3$ from quaternionic sub-algebras of $\mathbb{O}$ containing $\hat{\tau}$
Pre-geometric / combinatorial substrate.
Wolfram Physics Project (Wolfram, Gorard, 2020+) [123]	Spatial hypergraph with local rewrite rules	Causal graph from update sequence; $(3{+}1)$D from rule-space exploration	Programmatic; gauge groups not yet derived from rules	Closest in posture (no presupposed geometry; everything from substrate updates); differs on substrate nature (computation vs. distinction-plus-direction); PST derives SM from internal algebra rather than searching rule space
’ t Hooft CA Interpretation ( ’ t Hooft, 2014) [124]	Deterministic Planck-scale cellular automaton	Spacetime lattice presupposed; QM emerges from CA bookkeeping	Programmatic; SM not derived	Shares “QM is derived” thesis; PST’s Mosco-convergence route delivers CCR from substrate without requiring determinism
Group Field Theory (Oriti, Freidel, Pereira, 2000+) [125]	Second-quantised spin-network vertices on a Lie group	Spacetime as condensate phase of GFT (Bose–Einstein-like)	Silent on matter content	Combinatorial substrate analogue of PST’s algebraic substrate; both invoke limit theorems; GFT yields LQG-style quantum geometry, PST yields a Connes spectral triple
Causal Set Theory (Bombelli et al., 1987+) [4, 5, 232]	Locally finite partially ordered set of events	Spacetime continuum as dense Lorentz-invariant “sprinkling” limit; order plus number fixes the metric [223]	Scalar fields carried on the order [226]; gauge structure and fermions not derived	Pre-geometric combinatorial substrate; PST adds $V_{7}$-valued directional content beyond bare order, supplying the gauge structure the order lacks. Three points of contact: everpresent $\Lambda$ against PST’s exact $w=-1$ (§16.9); the spectral dimension, which rises at short scales on the full causal set as PST’s does between the grain and $d_{0}$ (§20.6, Computation 207); and the sprinkling theorem [225], that no finite-valency graph is Lorentz invariant. PST’s discreteness passes the theorem exactly when the realised positions are independent, Poisson-type, which A2’s weak equidistribution does not fix (a lattice satisfies it and fails) and which P3’s maximum-entropy clause supplies along with the density: given the density, maximum entropy is independence, and the rate A3 states conditions only the modes above the Mosco scale, so the conditioned cloud keeps dispersion one under boosts at the grain (Computation 208, Computation 211); the explicit scaling placement cannot test it, having no Euclidean side (Computation 209)
Loop Quantum Gravity (Rovelli, Smolin, 1986+) [6]	Spin-network states quantising $\mathrm{SU}(2)$ holonomies and fluxes	Continuum spacetime as semiclassical limit of spin-foam histories	Not derived; matter coupled by hand	Spin-network combinatorial substrate; PST’s substrate is algebraic (Walsh modes / Cl(0,6)) rather than spin-network; both aim at background-independent quantum geometry
Penrose Twistor Theory (Penrose, 1967+) [126]	Twistor space $\mathbb{T}\cong\mathbb{C}^{4}$ encoding null geodesics + helicity	Spacetime points as projective null lines in twistor space (Klein correspondence)	Conformal kinematics; gauge structure later via twistor-string/amplituhedron	Pre-spacetime spinor programme; PST’s distinction substrate is logically prior to both spacetime and the twistor encoding
Energetic Causal Sets (Cortês–Smolin, 2014) [127]	Events with energy–momentum + causal poset	Spacetime as derived from energetic causal poset	Programmatic	Extends causal sets with dynamical input; PST has richer substrate ($V_{7}$ directionality + Bernoulli measure) and a spectral-triple emergence route
Hardy Causaloids (Hardy, 2005+) [128]	Causaloid: operational data structure encoding probabilistic causal correlations	Causal structure primitive; geometry derived	Operational, silent on specific gauge/matter	Operational pre-geometric framework; PST is an ontological pre-geometric framework with explicit derivation chain
Causal Fermion Systems (Finster, 2010+) [129]	Measure on operators of finite spatial rank in a Hilbert space (no a priori spacetime)	Spacetime + causal structure emerge from operator-support overlap; Lorentzian signature from causal action principle	Derives Dirac sea + a Standard-Model-like gauge structure (chiral fermions, weak isospin) from the causal action	Closest mathematical kin: both use Hilbert-space-with-extra-structure as substrate; CFS uses measure theory + causal action, PST uses Connes spectral triples + Mosco convergence; CFS gauge derivation runs through a different mechanism
BFSS Matrix Models (Banks–Fischler–Shenker–Susskind, 1996) [130]	Large-$N$ matrices of $D0$-brane positions and spinor partners	11-D supergravity as the $N\to\infty$, ’t Hooft limit; spacetime as eigenvalue distribution	Low-energy SUGRA spectrum	Matrix-model substrate; PST is spectral-triple substrate; both invoke large-$N$/coarse-graining limits but in mathematically distinct categories
General Substrate Theory (GST)	Causal pacing in place of geometry; physical laws as coherence-conserving structural adaptations	Spacetime replaced by causal-pacing relations; relativistic and quantum-discreteness phenomena recovered from coherence-conservation constraints	Programmatic; not specific on gauge/matter content	Shares pre-geometric posture; PST commits to a specific substrate ($V_{7}$-directional distinguishability) and a definite emergence theorem (Mosco convergence) rather than an adaptive coherence-pacing principle
“Linkon” pre-geometric substrate proposal	“Linkons”: primitive relations that self-organise	Continuous information fields and gravity emerge from coherence among linkons rather than from a pre-existing spatial manifold	Programmatic	Closest in stated spirit, “relations as primitive” parallels PST’s distinguishability; PST commits to specific algebraic structure ( $V_{7}$, Bernoulli measure, Landau–Ginzburg potential ) and a Connes spectral-triple realisation, where that proposal as described remains schematic
Substrate Theory of Everything (Reed, 2024–2026) [143]	Single scalar substrate field with Mexican-hat potential and $\mathrm{U}(1)$ gauge structure; eight ontological postulates; two-frame ontology distinguishing undisturbed substrate ($F_{1}$) from perturbed observable form ($F_{2}$)	Spacetime emerges from substrate perturbations; Newton, special relativity, general relativity recovered as theorems	Standard-Model gauge structure and particle-mass cascade derived from $\mathrm{U}(1)$ cascade; dark matter / dark energy and matter–antimatter asymmetry addressed; testable predictions at HL-LHC, Gaia DR4, CMB B-modes, LISA	Most parallel in posture: another substrate from which spacetime, gravity, and SM physics are derived, with a Mexican-hat-style potential at the centre. Differences are technical: Reed uses a scalar substrate with a single $\mathrm{U}(1)$ producing the SM gauge cascade, where PST uses a $V_{7}$-directional distinguishability substrate, $G_{2}$-invariant LG potential on $V_{7}$, and the Connes spectral-triple route; PST’s gauge group emerges as the unimodular unitaries of $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ (itself derived from P1–P3) rather than from a single $\mathrm{U}(1)$ cascade
(continued: 2026 release)			Adds a proposed resolution of the cosmological-constant problem, reducing the quantum-field-theory discrepancy from ${\sim}122$ to ${\sim}1$ orders of magnitude via a holographic bound on $F_{2}$ substrate displacement	On the cosmological constant the two part company on what is being claimed: Reed offers a numerical resolution of the vacuum-energy discrepancy, where PST fixes the location, sign and equation of state $w=-1$ of the accelerating component from ongoing sublimation and leaves the static valley depth to renormalise like any vacuum energy, relocating and disarming the 120-orders mismatch rather than resolving it numerically (§9.1)
Replication-Based Pre-Geometric Cosmology (Montanari, 2025) [142]	Finite “unstructured energy” whose sole intrinsic property is vibration, supported on an open bounded spectral interval $(\omega_{\min},\omega_{\max})$ (finiteness forbids eternally stable configurations); evolved by competing excitation vs. randomisation in an energy-conserving Langevin/Fokker–Planck regime, yielding a metastable unstructured state	Not derived in Paper I (deferred to Papers II/III). Only pre-geometric precursors arise: emergent time as maximal-correlation ordering $C(t,t^{\prime})=\int a(\omega,t)a(\omega,t^{\prime})\,d\omega$ plus a discrete replication tick $\Delta\tau$; an acyclic pre-causal partial order (light-cone analogue); and a Nyquist minimum resolution $\sim 1/\omega_{\max}$ from spectral compactness	None; purely cosmological, with no gauge group, matter content, or families addressed (axioms PG1–PG9 + qualitative dynamics only). Persistent structure arises solely via a $1\to n$ replication instability of coherent spectral units (eigenpatterns of a universal excitation drift), consuming finite substrate energy and terminating on depletion	Shares PST’s pre-geometric posture, finiteness as the load-bearing constraint, emergence of time and causality from substrate dynamics, and a competing order/disorder instability echoing PST’s tension-threshold ($G_{2}$-invariant Landau–Ginzburg) sublimation. But Montanari is cosmology-only and outline-level (axioms; no concrete spacetime derivation or predictions yet), whereas PST carries the same posture through to a derived $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$, three generations, and falsifiable numbers (a $d^{-6}$ Casimir correction at $d_{0}=2.2$ am, bounded by data at $2.3$ nm; $\lambda_{\mathrm{SM}}(M_{*})=\mathrm{e}^{-1}/4$ at $1.4\sigma$)
Emergent-gravity programme: gravity as effective description of underlying microphysics.
Sakharov Induced Gravity (Sakharov, 1967) [131]	Standard-Model quantum-field vacuum fluctuations	Einstein–Hilbert action as one-loop effective action of matter fields	Presupposes SM matter	Historical precursor of all emergent-gravity proposals; PST replaces “vacuum fluctuations” with a precausal substrate and derives the SM rather than presupposing it
Thermodynamics of Spacetime (Jacobson, 1995) [19]	Local Rindler-horizon degrees of freedom (unspecified)	Einstein equation derived as the Clausius equation of state $\delta Q=T\mathrm{d}S$ on local horizons	Not addressed	Parallel “gravity from underlying microphysics” claim; PST specifies the microphysics ($V_{7}$-substrate) and recovers the same Einstein equation via the spectral action
Atoms of Spacetime (Padmanabhan, 2002+) [132]	Microscopic “atomic” degrees of freedom of spacetime horizons	Gravity as horizon thermodynamics; equipartition over the surface degrees of freedom yields Einstein equations	Not addressed	Same emergent-gravity claim as Jacobson; PST’s route is spectral-action (matter + gravity coupled at leading order) rather than horizon-thermodynamic
Entropic / Emergent Gravity (Verlinde, 2010, 2016) [18, 133]	Holographic information / quantum entanglement between regions	Gravity as entropic force from holographic counting; modified inertia in low-acceleration regime	Dark-matter phenomenology suggested; SM not derived	Emergent-gravity programme; PST’s gravity emerges from spectral action on the substrate, not from entanglement entropy bookkeeping
Quantum-information substrate: spacetime from entanglement.
ER$=$EPR / It from Qubit (Maldacena–Susskind 2013; Van Raamsdonk 2010) [134, 135]	Entanglement structure of the boundary quantum theory	Bulk geometry from entanglement entropy (Ryu–Takayanagi); ER bridges as EPR pairs	Inherits boundary CFT’s matter content; not derived	Spacetime-from-entanglement vs. PST’s spacetime-from-Boolean-Dirichlet-Mosco; complementary programmes; PST gives a substrate finite in internal structure and an explicit limit theorem, IfQ relies on holography
Tensor-Network Emergence (Swingle 2009; Vidal MERA) [136, 137]	Hierarchy of tensor-network entanglement (MERA / HaPPY codes)	AdS geometry from MERA scale layers; bulk locality from network connectivity	Silent on matter content	Operational realisation of It-from-Qubit; geometry from QI structure rather than from a logical primitive
Algorithmic Information Substrate	Algorithmic information / computational complexity of the substrate’s update history	Quantum vs. classical regime separated by a critical algorithmic-complexity threshold ($\sim 50$ bits in some proposals): below threshold the substrate maintains quantum superposition; above it, classical irreversibility	Not derived; treated as effective consequence of complexity-threshold crossing	Complementary in mechanism: PST yields the quantum/classical transition via Mosco convergence of Boolean Dirichlet forms and the modal threshold $\tau$ rather than via an algorithmic-complexity cut-off; both share a discreteness $\to$ continuum scaling structure
Foundational / philosophical substrate proposals.
Octonion-Substrate Programme (Dixon 1994; Manogue–Dray 1999+; Furey 2014+) [138, 139, 121, 122]	Division-algebra chain (e.g. $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$); octonions as fundamental	Spacetime not derived; programme is internal-algebra-focused	One generation of fermions from minimal ideal of Cl(6); $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ from unimodular unitaries	Closest technical kin: PST extends Furey 2014 by deriving $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ from P1–P3 (rather than positing the chain algebra) and pairs the internal algebra with explicit spacetime emergence; PST supplies a different selector for the generation count, reading $N_{\mathrm{gen}}=3$ off the quaternionic sub-algebras of $\mathbb{O}$ that contain the modal-sublimation direction $\hat{\tau}$ rather than off the algebra alone; the programme reaches three generations by its own route (Furey 2018), and PST inherits its colour–generation distinguishability caveat rather than removing it (C.4)
Constructor Theory (Deutsch & Marletto, 2013+) [140]	Counterfactual possible/impossible transformations	Spacetime presupposed; not derived	Not addressed	Shares “logical primitive” stance with PST’s distinguishability postulate; orthogonal mathematical machinery (possibility/impossibility lattices vs. Connes spectral triples)
Spencer-Brown Distinction Ontologies (Spencer-Brown, 1969; Kauffman; Varela) [61]	The act of making a distinction	Not formalised; mostly philosophical	Silent	PST is the closest formalisation of “distinction as primitive” with an explicit derivation chain to physics; shares the foundational starting point but commits to additional structure ($V_{7}$ directionality, Bernoulli measure, Landau–Ginzburg potential)
Wheeler’s “It from Bit” / MUH (Wheeler 1989; Tegmark 1998+) [62, 63]	Binary informational acts (Wheeler) / all mathematical structures (Tegmark)	Programmatic; no specific derivation	Programmatic	Programmatic ancestor; PST instantiates the philosophy with concrete substrate + derivation chain rather than leaving emergence as a slogan
Information Substrate Theory (Kriger, 2026) [141]	“Information” as the name for differentiatedness in being; a single axis-free substrate of which Shannon entropy, Kolmogorov complexity, thermodynamic entropy etc. are specific projections (along time, space, energy, …)	Spacetime not derived; affirmed as one projection / coordinate reading of the same substrate	Affirms existing SM; not derived	Closest in ontology: both PST and IST take differentiatedness as the foundational primitive (PST’s “property differentiation” $\leftrightarrow$ IST’s “information as the name of differentiatedness”). PST commits to a specific algebraic realisation ($V_{7}$-directional distinctions, Bernoulli measure, $G_{2}$-invariant Landau–Ginzburg potential, Connes spectral triple) and an explicit derivation of spacetime and the Standard Model; IST keeps the substrate axis-free and reads existing physics off it as projections rather than deriving the projections from postulates

D.1.  Conclusion of comparison

The table holds twenty-seven entries, PST and twenty-six others. PST is the only entry in the table that does all five of the following at once.

1. It posits a pre-geometric substrate finite in internal structure, not a finite universe.

2. It derives spacetime as a limit theorem from it.

3. It derives the structural ingredients of the $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ gauge group from the three postulates.

4. It places $N_{\mathrm{gen}}=3$ in Furey’s six-$\mathrm{SU}(3)$-triplet decomposition of $\mathrm{Cl}(0,6)$ as the natural reading.

5. It delivers a specific Standard-Model coupling value, $\lambda_{\mathrm{SM}}(M_{*})=b\cdot\mathrm{e}^{-1}=\mathrm{e}^{-1}/4=0.0920$, where $b=1/4$ is the Landau–Ginzburg modal quartic supplied by P3, matching the observed value at $1.2\%$. The substrate-side factor $\mathrm{e}^{-1}$ is derived from the postulates; the SM-side matching is reduced within PST to a field-strength-renormalisation identity resting on the emergence premise F4 (§12.6.2.1) plus the renormalisation-group equation [RGE] test, so it is not parameter-free (§19.7).

D.2.  The two nearest entries

Two rows deserve a word, because a quick reading of them might suggest they come closer to the five axes than they do.

Substrate Theory of Everything (Reed, 2024–2026) [143] reaches further into the Standard Model than any other entry in the table, and its posture is the most parallel to PST’s. The route, however, is a different one. Reed’s substrate is a continuous scalar field that already carries a $\mathrm{U}(1)$ gauge structure and a Mexican-hat potential, laid down in eight ontological postulates; the Standard Model is then reached by a cascade from that $\mathrm{U}(1)$. PST’s substrate is finite in internal structure and carries no gauge group at all: the gauge group appears only at the end of the chain, as the unimodular unitaries of $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$, and that algebra is itself derived from P1–P3 rather than assumed. On axis (i), then, Reed’s substrate is not finite in internal structure; on axis (iii), its gauge content is posited rather than derived.

The divergence is sharper than a difference of machinery, and it runs in both directions. Reed derives a particle-mass cascade, which PST does not claim and, by the structural-scope theorem of §14.3, cannot claim: the fermion mass and mixing spectrum sits in the contingent configuration content $T(C)$, not in the layer fixed by the postulates. The 2026 release goes further still and offers a numerical resolution of the cosmological-constant problem, reducing the quantum-field-theory discrepancy from some $122$ orders of magnitude to about one via a holographic bound on substrate displacement. PST does not claim that either. What PST fixes structurally is the location of the vacuum, the sign, and the equation of state $w=-1$ of the accelerating component, all carried by ongoing sublimation. The magnitude of the static valley depth renormalises like any vacuum energy and is not derived. On PST’s own account, then, the 120-orders mismatch is relocated and disarmed rather than resolved numerically (§9.1). The two theories differ, in short, on what counts as a derivation, and on how much of the observed world is supposed to follow from the substrate at all. Where they make the same claim they can be compared; where Reed claims more, the comparison is not that PST scores lower on a shared axis, but that PST does not put those quantities in the derivable layer in the first place.

Causal Fermion Systems (Finster, 2010+) is the closest mathematical kin, and it does meet axes (i) and (ii): the substrate is a measure on operators of finite rank, and spacetime together with its causal structure emerges from operator-support overlap. On axis (iii) it derives a Standard-Model-like gauge structure, chiral fermions and weak isospin, from the causal action principle rather than the Standard Model gauge group itself, and it makes no structural commitment on the generation count or on any specific coupling value.

These five axes are, by construction, PST’s own load-bearing deliverables. The table is therefore best read not as a ranking on a neutral scale, but as a precise statement of what PST supplies that no surveyed alternative, on its own stated terms, does.