Maximal Mutual Determination: from Qubit Substrate to the Standard Model via E8 Foam

Maximal Mutual Determination: from Qubit Substrate to the Standard Model via E₈ Foam — v5.1 This work articulates a programme of structural derivation in fundamental physics: the content of spacetime, quantum mechanics, gravity, matter, and the Standard Model emerges as derived consequences of a single variational principle on a minimal substrate, rather than being separately postulated. The substrate is a continuous qubit field φ : ℳ → ℂℙ¹ on a smooth manifold, with dynamics governed by an action functional Sφ = Skin + SWZP combining a Dirichlet kinetic term with a Wess–Zumino–Pancharatnam topological term. Its stationary patterns form a definite chain C₃ → K₄ → cluster → 600-cell → 2I → E₈-roots → E₈-lattice → foam, realised dynamically through stochastic vertex-by-vertex margin growth at rate Γvertex = λ, with λ identified with the foam-graph percolation critical density pc ≈ 1/175. The framework's organising principle is maximal mutual determination (PMMD): the structural relations among substrate, chain stages, and macroscopic content mutually constrain each other to a unique self-consistent configuration. Principal derived content. Standard Model — Rigorous identification of the foam vacuum with the irreducible 112-dimensional representation V₁₁₂ of W(E₈); the three SM generations arise from its Z₃-cyclic decomposition. The Koide identity Q = 2/3 is derived as a theorem; the Koide angle θ = 2/9 follows from the algebraic invariant C₂(3)/(2h∨). Charged-lepton mass ratios at 10⁻⁵ precision; unified coupling α−1E₈ ≈ 25.4 with RG matching at 1% at MZ; Cabibbo angle via the Fritzsch relation at 1.5%. Gravity — Einstein–Hilbert effective action, graviton wave equation, spin-2 from foam homotopy, strong equivalence principle from c-universality, and the Bekenstein–Hawking area law all emerge as theorems from the foam at percolation criticality. The closing relation ℓ*/ℓP = 2φ⁵ cross-validates the foam-cellular scale with the explicit μ₉ = 1/(4φ⁶). Cosmology — Λ ≈ 4.17×10⁻¹²³ from the V₁₁₂ vacuum (factor 2.7 from observed); baryon asymmetry ηB ≈ 5.97×10⁻¹⁰ at 0.56% match via the cosmological 3-fold merger, verified numerically by direct Monte Carlo (53,611 events, z = −0.39σ); CMB photon-to-baryon ratio nγ/nB ≈ 1.7×10⁹ at ~10%. Matter–antimatter and dark sector — The substrate ℤ/2 symmetry 𝒞 : k → −k, φ → φ̄ generates the matter-antimatter asymmetry as the stochastic residue of the 3-fold merger algebra at percolation, with all three Sakharov criteria automatically satisfied. Dark matter is structurally identified with Standard Model matter propagating in the temporally opposite direction within the same percolating foam, with the gravitational cluster integrity theorem ensuring both ±t projections contribute additively to the cluster's gravitational mass. Spacetime emergence — The 4D Lorentzian macroscopic spacetime emerges from the 8D E₈ lattice via cut-and-project to the chain's D₄ base, with the Lorentzian signature reconstructed from the foam causal phase within the Bombelli–Lee–Meyer–Sorkin causal-sets framework. Falsifiable structural conjecture. The framework predicts a deep connection between Standard Model matter content (via its trace-anomaly c-coefficient) and foam percolation criticality: fcSM · pc ≈ 1, with two structurally distinct readings — the algebraic form fcalg = 183 (Spin(10)-complete) at 4.5% deviation, and the IR-effective form fcIR = 176.25 (sub-seesaw, νR integrated out) at 0.7% deviation. Consistency with established physics. General relativity, quantum mechanics, Lorentz invariance, causality, and unitarity are all articulated as derived consequences of the substrate dynamics rather than as separate postulates. Quantum mechanics emerges from the Kähler form ω on ℂℙ¹ via Pancharatnam phase accumulation; the substrate's continuous indeterminacy structure is the geometric origin of ℏ-dependent interference. Structure and content. The framework is articulated across 8 Parts, 291 pages: Part I (foundations and axioms); Parts II–IV (chain stages stage-by-stage with explicit chirality-propagation verification); Part V (Standard Model phenomenology including the sector-precision hierarchy theorem); Part VI (relational ontology and complete gravitational sector); Part VII (epistemic stratification — every quantitative claim classified by Stratum 1–5); Part VIII (chain uniqueness rigorisation programme in 7 sprints). Total formal content: 56 theorems, 10 lemmas, 17 propositions, 17 corollaries, 7 conjectures, 3 foundational axioms, 16 definitions, 97 remarks, 144 bibliographic references. The framework's six bridge claims to standard physics (Lorentz invariance, SM gauge invariance, unitarity, locality, mass spectrum, Einstein–Hilbert gravity) are articulated at Stratum 2 conditional or better, with the joint promotion path through verification of foam Poisson universality and dBS = 4 dimensionality. v5.1 release notes. v5.1 is a conceptual reorganisation of v5.0: no new theorems or quantitative predictions are added. The framework's fundamental dynamics is articulated as stochastic vertex-by-vertex margin growth at the substrate's discrete-graph level, with the chain's discrete stage-transition instantons recovered as the effective description at the level of stage patterns. The Stage 7-to-Stage 8 transition is rewritten replacing the previous Stratum-3 thermodynamic-limit framing with a Stratum 2 conditional regime articulation. A detailed v5.0 → v5.1 changelog is included as a standalone Part at the end of the document.