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

Version 6. 0 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 organising principle, maximal mutual determination, applied to a minimal qubit system, rather than being separately postulated. This version refines and improves on earlier ones (v2, v2. 1, v3. 0, v4. 0, v5. 0, v5. 1, and v5. 2; all on Zenodo). Its principal departure from v5. 2 is a system-primary relational foundation: the fundamental object is no longer a continuous field on a background manifold but a closed, minimal qubit system whose dynamics is a discrete relational action, from which the continuum description is recovered as an emergent limit; correspondingly, the framework now carries no free continuous parameter. Framework summary The framework articulates physics as the structural unfolding of a single organising principle—maximal mutual determination—on a closed, minimal qubit system, with dynamics given by a discrete relational action on nodes, edges, and triangular faces (no manifold, no metric, no integral). The system’s stationary patterns form a chain of nine progressive stages, generated by stochastic vertex-by-vertex margin growth from the primordial C3 and 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. The familiar continuum description (a configuration φ: M → CP1 with action Sφ) is recovered as an emergent limit. Because the percolation threshold pc is a structural constant of the E8 foam graph—fixed by its branching criticality, not a tunable coupling—the framework has no free continuous parameter. Principal structural results System-primary foundation and no free parameter: the dynamics is a discrete relational action Sdisc = β·ΣE (1 − |⟨ψi|ψj⟩|2) + θ·Q (with Q the Berg–Lüscher topological charge). Both couplings are fixed—θ = π by the qubit being spin-½ (Haldane θ = 2πS, Wess–Zumino–Witten level k = 1), and β → ∞ is maximal mutual determination itself—so the coupling λ = pc is the structural percolation threshold of the E8 foam graph and the framework carries no free continuous parameter. The qubit value-space CP1 derived, not posited: relationality forces projectivity (only relative phases are physical), the equitable triadic balance forces the complex structure (an order-three phase lives in U (1) but in neither the real nor the quaternionic alternative), and minimality of the primitive forces CP1. The qubit is upgraded from a foundational axiom to a forced consequence (Proposition prop: cp1-forced). Three generations of matter from the W (E8) irreducible representation V112’s ℤ3-cyclic decomposition, with the ℤ3 subgroup identified as the cyclic group generated by the Coxeter element of an A2 ⊂ E8 (Proposition prop: Z3-A2-Coxeter-identification, GAP-verified). Koide relation as a theorem of the ℤ3-cyclic structure: angle θ = 2/9 and value Q = 2/3; 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). Finite-size-scaling 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). The identity λ = pc is given a branching-criticality origin—a single spanning cluster appears when the branching ratio reaches unity—with the exact triangle correction 1/pc = (z−1) − Te = 183 (z = 240, Te = 56), covering 84% of the gap to the measured 1/172. 55. The macroscopic dimensionality d = 4 is verified numerically (dBS = 4) on the ideal E8 lattice cut-and-project and on the framework’s own growth-dynamics ensemble. Cosmological constant: the ∼120-order-of-magnitude smallness is a structural output of the foam area-law Λ = cW·φ−8N. The framework derives the suppression form and its magnitude, while the precise value is observation-fixed—the round-trip Λobs → N → Λ is the identity (the V112-only baseline reads Λ ≈ 4. 17 × 10−123). Equation of state wDE = −1 exact (no phantom crossing). Baryon asymmetry from the 3-fold cosmological merger algebra: the genuine prediction is the ratio ηB/JCKM = pc·μ9cosmo = 1. 99 × 10−5 (a foam quantity independent of the measured CP input), giving ηB ≈ (6. 0–6. 2) × 10−10 and bracketing the latest determination (6. 120 ± 0. 038) × 10−10; the three Sakharov criteria are automatically satisfied. Matter–antimatter, dark matter, and primordial black holes (structural unification): dark matter is identified with ordinary Standard-Model matter propagating in the temporally opposite direction within the same percolating foam (Theorem thm: dm-time-direction). The per-vertex heterogeneity of the temporal allocation τ (v) forces, by elementary topology, τ-demarcation surfaces within every cosmologically extended cluster; these are event horizons (Theorem thm: tau-demarcation-PBH). Consequences: a foam-scale mass spectrum dominantly intermediate-mass (102–105 M⊙, consistent with GW190521) ; ΩDM/ΩB identified as the τ-volume ratio, with v6. 0 deriving its heavy-tailed distribution (Theorem thm: omega-DM-heavy-tailed) ; and Bekenstein–Hawking microstates as τ-allocation configurations on horizon foam-cells. v6. 0 adds a causal-consistency refinement of the τ-demarcation and a three-tier cosmological emergence (Corollary cor: three-tier-cosmological-emergence). Gravitational sector: Einstein–Hilbert effective action, graviton wave equation, graviton spin-2 from foam homotopy, strong equivalence principle from c-universality, and the Bekenstein–Hawking 1/4 coefficient—all foam-emergent. The closing relation ℓ*/ℓP = 2φ5 cross-validates the foam-cellular scale. The continuum as emergent: a bridge theorem makes the continuum a derived limit—rigorous in d = 2 (Γ-convergence of the Dirichlet energy, the θ = π CP1 model flowing to the non-trivial SU (2) 1 Wess–Zumino–Witten fixed point), with the physical d = 4 continuum framed as the asymptotic-safety scenario and the E8 foam supplying a finite ultraviolet completion. Three emergent facets: space, time, and quantum mechanics emerge as three structurally distinct facets of the qubit system—space from the chain progression, time from the system’s directionality together with the asymmetric initial condition, and quantum mechanics from the relational (Kähler-form) structure via Pancharatnam phase accumulation. v6. 0 release content (v5. 2 → v6. 0) The principal additions and reformulations of this release: System-primary relational foundation. The fundamental object is reformulated as a discrete relational action on a graph (no manifold, no metric, no integral) ; the continuum functional of v2–v5 is recovered as an emergent limit, not posited. “Substrate” is replaced by “system” throughout, and “field” is reserved for the effective mathematics and the recovered Standard-Model and gravitational fields. Both couplings fixed; no free continuous parameter. θ = π is forced by the qubit being spin-½; β → ∞ is maximal mutual determination itself. Consequently λ = pc is derived as the structural percolation threshold of the E8 foam graph. Emergent-continuum bridge theorem (Theorems thm: v6-continuum-2d, thm: v6-continuum-4d), rigorous in d = 2; the physical d = 4 continuum framed as the asymptotic-safety scenario with the E8 foam as ultraviolet completion. CP1 derived rather than posited (Proposition prop: cp1-forced): relationality forces projectivity, the equitable triadic balance forces the complex structure, and minimality forces CP1; the qubit value-space is upgraded from axiom (C1) to a forced consequence. Discrete chain uniqueness via the SIC. Determination is formalised as informational completeness, identifying K4 as the qubit SIC-POVM; the d2 = 4 cap forces the tetrahedral-aggregation route to E8. The internal SIC (four states of one qubit) is distinguished from the relational tetrahedron (four distinct qubit-nodes of the foam), and growth (accretion) and fission (differentiation) are articulated as dual narrations of the same balance-forced chain. Branching-criticality origin of λ = pc: the exact triangle correction 1/pc = (z−1) − Te = 183 covers 84% of the gap to the measured 1/172. 55. Numerical verification of the cut-and-project dimensionality dBS = 4 on the ideal E8 lattice and on the framework’s own growth-dynamics ensemble. Cosmological-sector refinements: a causal-consistency refinement of the τ-demarcation, the heavy-tailed ΩDM/ΩB distribution (Theorem thm: omega-DM-heavy-tailed), and a three-tier cosmological emergence (Corollary cor: three-tier-cosmological-emergence). Cosmological-constant status made explicit: the foam area-law derives the suppression form and its ∼120-order magnitude, while the specific value is observation-fixed (the Λobs → N → Λ round-trip is the identity) ; the foam’s self-similarity is correspondingly clarified as a structural property of the E8 cut-and-project, not a growth law. All numerical predictions of v5. 1–v5. 2 (Koide, charged-lepton mass ratios, three generations, gauge couplings, cosmological constant, baryon asymmetry, the Bekenstein–Hawking 1/4 coefficient, and the trans-Planckian phenomenology) are preserved without modification; the v6. 0 refinements add structural articulation and sharpened predictions but do not invalidate any prior numerical result. Document structure and counts The framework is articulated in 8 Parts with 61 theorems, 10 lemmas, 28 propositions, 25 corollaries, 7 conjectures, 3 foundational axioms, 18 definitions, 144 remarks, and 165 bibliographic references (~373 pages): Part I Foundations (system-primary relational foundation, discrete relational action, and the emergent-continuum bridge) Parts II–IV the chain of stages, stage-by-stage with chirality propagation (relational extension, categorial pass