Precausal Substrate Theory

Abstract

Precausal Substrate Theory (PST) proposes that spacetime, causality, matter, and the fundamental forces are not themselves fundamental but arise from logic. The sole primitive is property differentiation (P1): the capacity for distinctions to exist at all. Property differentiation carries an asymmetric vector tension (P2). Modal sublimation past a Landau-Ginzburg threshold (P3) produces instantiated geometry along the substrate’s net direction; causality is born together with geometry. These three are the theory’s only postulates and every later structure is a derivation. The Big Bang itself is one such derivation: a perspectival artifact rather than a first event.

From these three primitives a concrete mathematical structure follows. The substrate is a real Connes spectral triple of KO-dimension six whose Mosco limit yields spacetime M = ℝ × S³, and the Standard-Model internal algebra A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ) follows via Furey’s Cl(0,6) chain-algebra construction and Dixon’s hyperspinor synthesis. General relativity, quantum mechanics, the Standard Model, fermionic matter, and N_gen = 3 all emerge from this single spectral-triple structure. PST predicts a d⁻⁶ Casimir correction at nanometre separations with substrate coherence scale d₀ ≲ 5.3 nm. Among twenty-five surveyed substrate theories, PST is the only one delivering a finite pre-geometric substrate, spacetime as a limit theorem, the Standard-Model gauge structural ingredients, and the three-generation count within a single framework.

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

Overview

The Core Mathematics

The substrate is a real Connes spectral triple of KO-dimension six. Its Mosco limit yields spacetime M = ℝ × S³ with KO-dimension four; the product M × F has total KO-dimension ten ≡ 2 (mod 8), the Connes Standard-Model value. The substrate’s parity grading equals the Cl(0,6) chirality, and Furey’s chain-algebra construction on ℂ ⊗ 𝕆 ≅ Cl(0,6) delivers SU(3)_c × U(1) on one generation. The emergent Lorentzian Dirac sector Cl(1,3) ≅ M₂(ℍ) supplies an internal SU(2) via right-ℍ-multiplication on the spinor module. Combined via Dixon’s hyperspinor framework T = ℂ ⊗ ℍ ⊗ 𝕆, these synthesise to A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ), whose unimodular unitaries are SU(3) × SU(2) × U(1).

The Unified Physics

The same spectral-triple structure delivers the four otherwise-disjoint foundations of contemporary physics. General relativity: the Einstein-Hilbert action emerges as the leading term of the spectral action. Quantum mechanics: canonical commutation relations arise via Mosco convergence of Boolean Dirichlet forms. The Standard Model: gauge bosons and the Higgs appear as inner fluctuations of the Dirac operator. Matter: Boolean {0,1} distinctions are fermionic occupation numbers via the CAR algebra, making spin-statistics structural. Generations: N_gen = 3 is read structurally from Furey 2014’s six-SU(3)-triplet decomposition of Cl(0,6).

Predictions

PST predicts a d⁻⁶ Casimir correction at nanometre separations. Conditional on a Gaussian convolution-kernel form (assumed), the coefficient ξ = 90/π² ≈ 9.12 is fixed, giving d₀ ≲ 5.3 nm. The electroweak modal scale M_* = 4π m_h ≈ 1.573 TeV is an NDA strong-coupling estimate, not a parameter-free derivation. The scalar sector exhibits tree-level custodial SU(2) symmetry, protecting ΔT; loop and new-states corrections to S and T are not computed here. The cosmological-constant equation of state w = −1 is structurally compatible with the substrate’s pre-spatial framing via a steady-state-style non-dilution mechanism, conditional on that framing; magnitude, d₀, the Yukawa hierarchies, and CKM mixing are contingent.

Standing among substrate theories

Compared to 25 other substrate-style & emergent-physics theories, PST is the only one that simultaneously (i) posits a finite pre-geometric substrate, (ii) derives spacetime as a limit theorem from that substrate, (iii) derives the structural ingredients of the Standard-Model gauge group from the three postulates, with the full A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ) following via Dixon’s hyperspinor synthesis, and (iv) places the three-generation count in Furey’s six-SU(3)-triplet decomposition of Cl(0,6) as the natural reading. No other entry delivers all four.