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

Version 5.2 of a framework that articulates a programme of structural derivation: the content of physics—spacetime, quantum mechanics, gravity, matter, and the Standard Model—is to emerge as derived consequences of a single variational principle on a minimal substrate, rather than being separately postulated. This version refines and improves on earlier ones (v2, v2.1, v3.0, v4.0, v5.0, and v5.1; all on Zenodo). Framework summary The framework articulates physics as the structural unfolding of a single variational principle on a minimal substrate (a continuous qubit field φ: M → CP1) through a chain of nine progressive stationary stages, terminating at percolation criticality on the E8 foam. The chain articulates C3 → K4 → cluster → 600-cell → 2I → E8-roots → E8-lattice → foam, with derivation rather than postulation at each step. Principal structural results (status at v5.2) Three generations of matter from the W(E8) irreducible representation V112's ℤ3-cyclic decomposition, with the ℤ3 subgroup structurally identified at v5.2 as the cyclic group generated by the Coxeter element of an A2 ⊂ E8 (Proposition prop:Z3-A2-Coxeter-identification, Stratum 1, GAP-verified). Koide identity Q = 2/3 as a theorem of the ℤ3-cyclic structure; charged-lepton mass ratios at 10−5 precision. Foam at percolation criticality: direct Monte-Carlo through L = 12 (N ≈ 4.3 × 108 vertices, 128 trials, ~30h wall-time). FSS extrapolation from L ∈ {8, 10, 12} rejects the algebraic conjecture fcalg⋅pc = 1 at > 8σ and is consistent with the IR-effective form at ≤ 2.2σ (Proposition prop:fc-FSS-discrimination, v5.2 Stratum 1). Cosmological constant three structural readings: Λ ≈ 4.17×10−123 (V112-only, factor 2.64, Stratum 5); Λ ≈ 7.08×10−123 (V112+58, cW = 17/24, factor 1.55, Stratum 2 at v5.2 via Corollary cor:V112-W6-decomposition); Λ ≈ 1.0×10−122 (full-E8 sampling, 9% match, Stratum 3). Equation of state wDE = −1 exact (no phantom crossing). Baryon asymmetry ηB ≈ 5.97×10−10 (0.56% match to observed) from the 3-fold cosmological merger algebra; three Sakharov criteria automatically satisfied. Matter-antimatter, dark matter, and primordial black holes (v5.2 structural unification): dark matter is structurally identified with ordinary SM matter propagating in the temporally opposite direction within the same percolating foam (Theorem thm:dm-time-direction). At v5.2 the per-vertex heterogeneity of the temporal allocation τ(v) is shown to force, by elementary topology, the existence of τ-demarcation surfaces within every cosmologically extended cluster; these are event horizons (Theorem thm:tau-demarcation-PBH, Stratum 2). Three Stratum-2 corollaries follow: foam-scale mass spectrum dominantly intermediate-mass (102–105 M⊙, consistent with GW190521); ΩDM/ΩB identified as the τ-volume ratio Vτ=−1/Vτ=+1; and Bekenstein–Hawking microstates as τ-allocation configurations on horizon foam-cells. Dark matter, primordial black holes, and the Bekenstein–Hawking A/4 entropy coefficient are thereby unified into a single structural mechanism. Gravitational sector: Einstein–Hilbert effective action, graviton wave equation, graviton spin-2 from foam homotopy, strong equivalence principle from c-universality, and Bekenstein–Hawking 1/4 coefficient—all foam-emergent. The closing relation ℓ*/ℓP = 2φ5 cross-validates the foam-cellular scale. Three emergent facets: space, time, and quantum mechanics emerge as three structurally distinct facets of the substrate φ: M → (CP1, ω): space from the source manifold via chain progression, time from the substrate ℤ/2 via cosmological selection, and quantum mechanics from the Kähler form ω via Pancharatnam phase accumulation. v5.2 release content (v5.1 → v5.2) Three principal structural additions and one substantive reformulation: New Proposition (Stratum 1, GAP-verified): the ℤ3 subgroup of W(E8) producing the Koide decomposition V112 = 58⊕27⊕27 is the cyclic group generated by the Coxeter element of an A2 ⊂ E8. Centraliser order |W(A2)×W(E6)| = 311 040; 2240 such elements form exactly the smallest of the four order-3 conjugacy classes in W(E8). Independent verification via Python construction of W(E8) (scripts e8_group_theory.py, option_B_V112_koide.py). New Proposition (Stratum 1): FSS extrapolation from L ∈ {8, 10, 12} Monte-Carlo data rejects the algebraic trace-anomaly conjecture fcalg⋅pc = 1 at > 8σ and confirms the IR-effective form at ≤ 2.2σ (linear FSS giving 0.4σ match). First direct experimental discrimination between an algebraic substrate-level prediction and its IR-projected form. Promotion of the IR-effective trace-anomaly–percolation duality from Stratum 2–3 numerical near-miss to Stratum 1. New Theorem (Stratum 2) and three corollaries: structural unification of dark matter, primordial black holes, and Bekenstein–Hawking microstate counting through the geometry of τ-demarcation surfaces in percolating foam clusters. The derivation rests only on existing structural elements (foam-level decoupling, foam causal phase, Bombelli–Lee–Meyer–Sorkin construction) and elementary graph topology; no additional structural assumption is introduced. Reformulation of Proposition cosmological-phase-selection: the macroscopic Lorentzian signature is articulated as a statistical regularity of the partial order recovered from the BLS combinatorial dimensionality, rather than as pointwise uniformity of Φ(v) across the cluster. Resolves an internal inconsistency with the per-vertex τ-bit structure (which guarantees visible and dark matter coexist in the same percolating cluster). Promotion to Stratum 2: the intermediate Λ refinement cW = 17/24 is now structurally justified by the new W(E6)-module decomposition (Corollary cor:V112-W6-decomposition). Document structure and counts The framework is articulated in 8 Parts with 57 theorems, 10 lemmas, 19 propositions, 21 corollaries, 7 conjectures, 3 foundational axioms, 16 definitions, 102 remarks, and 148 bibliographic references (~305 pages): Part I Foundations Parts II–IV chain stage-by-stage with chirality propagation Part V Standard Model phenomenology (Koide, three generations, gauge coupling, sector-precision hierarchy) Part VI relational ontology and complete gravitational sector Part VII epistemic stratification (five-stratum classification of all results) Part VIII chain uniqueness rigorisation programme (seven sprints)