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

Version 7.0 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, v5.2, and v6.0; all on Zenodo). Its principal departure from v6.0 is that the discrete relational action is made operative—with stationarity δS = 0 as the working principle—and the flavour sector is closed through it: the full quark mixing from the electroweak gateway with every input derived at leading order, the neutrino oscillation sector at zero parameters from derived constants, the colour-flux census laws measured on 1.46×1010 lattice loops, and three exact flavour–gauge bridges, all recorded under an explicit adjudication ledger (forty-four falsification tests, eleven self-inflicted). The framework carries no free continuous parameter at the foundational level (sector-level effective models may carry measured coordinates, each recorded with its epistemic status and its named decider). 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, and no fields: the perpendicular “phason” content is a per-facet share of each relation, and the familiar fields are the continuum shadows of relation densities). The system’s stationary patterns form a chain of progressive stages, generated by stochastic vertex-by-vertex margin growth from the primordial C3 and terminating at percolation criticality on the E8 foam: C3 → K4 → cluster → 600-cell → 2I → E8-roots → E8-lattice → foam, with derivation rather than postulation at each step. The continuum description (a configuration φ: M → CP1 with action Sφ) is recovered as an emergent limit, and the Standard Model is recognised as the framework’s continuum shadow, with universality supplying the flow coefficients in the continuum window and the golden self-similarity of the lattice (parallel ×φ, perpendicular ×1/φ per inflation step) acting as its renormalisation group. Principal structural results System-primary foundation; no free continuous parameter at the foundational level: the dynamics is a discrete relational action, both couplings fixed (θ = π by spin-½; β → ∞ is maximal mutual determination itself), so λ = pc is the structural percolation threshold of the E8 foam graph. In v7.0 the action is operative in the equivalent sector form S = −Σrelations M + Σtransports(leak share), with δS = 0 as the classical principle and quantum-occupancy weighting forced by granularity. The qubit value-space CP1 derived, not posited: relationality forces projectivity, the equitable triadic balance forces the complex structure, minimality forces CP1. Three generations of matter from the W(E8) representation V112’s ℤ3-cyclic decomposition (Coxeter element of an A2 ⊂ E8, GAP-verified). Koide relation as a theorem of the ℤ3-cyclic structure—angle θ = 2/9 a derived SU(3) Casimir invariant, value Q = 2/3 re-grounded in v7.0 as maximal shared indetermination between the generation loop and its two neighbour-chains (the √2 from the cyclic channel’s dimension)—charged-lepton mass ratios at 10−5 precision. The flavour sector closed (new in v7.0). Quark mixing: the full CKM—|Vus|, |Vcb|, |Vub|, and the Jarlskog invariant—from the spurion’s exact weak–generation gateway (the phason-shadow adjacency, with the w-root as the sole lattice bridge), the connector’s magnitude and phase derived (transport universality, promoted by granularity-forced factorisation; winding minus spinor half-angle), and the two virtual-gap coordinates delivered at leading order by the action’s excursion curvature Δp = 3·My(ρp) (unique double landing among eight pre-declared candidates; Δ5 = 4/3 exact in form): a zero-free-parameter CKM at the 3–15% level, χ²/dof = 0.55 at the data-pinned refinements, CP violation structurally necessary and the CP branch data-sealed. Neutrino oscillations: closed at zero parameters from three derived constants—the ℤ3 trisection of the free-sector angle (holonomy locking), r² = D = 3/4 (granularity-forced self-counting stationarity, unique fixed point), and the Majorana core κ = −5/12 (gateway slot-service counting, cyclic protection by the exact Σω2g = 0 theorem)—χ² = 1.42 with sinδCP = −1 exact. Colour-flux census laws (new in v7.0), from 1.46×1010 exhaustively enumerated lattice loops: exact flux quantisation Φ ∈ (2/√3)ℤ; the closed-form conversion κ = 4π/(3√3) with the plaquette identity κq = 8π/9 = 2π·pinv(1u); quantum universality across loop lengths; exact zero-flux controls (W = +1.00000); and the unit-diffusivity law ⟨q²⟩ ≈ A at 29.6σ, with the persistent zero-flux backbone quantified as the obstruction the determination-cost weight must remove. The strong-coupling boundary and the confinement dictionary (new in v7.0): the bare string tension in closed form, σ̂UV = 4√3π²/(81φ6); the strong coupling at the cell scale landing at the protection scale (1/αs(E*) = 49.6 at two loops against 147/3 = 49); and the calibrated two-way dictionary c = 0.583 ⇔ (αs(MZ), √σ) jointly, the native-scheme matching constant c being the single named decider of the strong-sector story. Foam at percolation criticality: Monte-Carlo through L = 12, finite-size scaling rejecting the algebraic conjecture at >8σ, branching-criticality origin of λ = pc, and the macroscopic dimensionality dBS = 4 verified on the ideal lattice and the growth ensemble. Cosmological constant: the ∼120-order smallness as a structural output of the foam area-law, wDE = −1 exact. Baryon asymmetry: ηB/JCKM = pc·μ9cosmo, now rendered fully input-free by the zero-parameter Jarlskog of the closed flavour sector, bracketing the measured ηB. Matter–antimatter, dark matter, and primordial black holes: dark matter as Standard-Model matter propagating in the temporally opposite direction; τ-demarcation surfaces as event horizons; the heavy-tailed ΩDM/ΩB distribution. Gravitational sector: Einstein–Hilbert effective action, graviton spin-2 from foam homotopy, the strong equivalence principle, and the Bekenstein–Hawking 1/4 coefficient, all foam-emergent; ℓ*/ℓP = 2φ5. The continuum as emergent: rigorous in d = 2 (Γ-convergence to the SU(2)1 WZW fixed point), the physical d = 4 framed as asymptotic safety with the E8 foam as finite ultraviolet completion—in v7.0 sharpened by the inflation-invariance theorem: the flavour shapes are fixed-point quantities of the golden renormalisation group, with scheme covariance verified (χ² 5118 → 19.6 at 2 GeV) and φ-suppressed deviations from Standard-Model running predicted, in principle, near the cell scale. Three emergent facets: space, time, and quantum mechanics as three structurally distinct facets of the qubit system. v7.0 release content (v6.0 → v7.0) The principal additions of this release (the intermediate working label v6.1 was never archived; its four sharpenings are items 1–4): Self-duality derived via Mordell’s theorem, removing an axiom-level input of the chain. The intrinsic Stage 4–5 reformulation, stating the 600-cell → 2I passage in system-primary terms. The two coherence thresholds, articulating the growth dynamics’ ordering. The Koide value re-grounded as maximal shared indetermination (joint coherence over the two V112 relation-chains), robust to the measure. The electroweak gateway and the CKM closure (see above), including the lattice-level reading of the gateway, the exclusion ladder of falsified alternatives, the data-sealed configuration and CP branch, scheme covariance by the per-unit prescription, and the excursion-curvature derivation of the gaps. The neutrino mechanism: trisection by holonomy locking, D = 3/4 by self-counting stationarity (with the budget and node walls formalised), and the −5/12 Majorana core by slot-service counting with the exact cyclic-protection theorem. The relational action made operative: δS = 0; the D = 3/4 derivation; the closed-form bare tension; the golden inflation as renormalisation group; the inheritance of Standard-Model running re-founded as a limit theorem (universality), the native language remaining primary. The colour-flux census (n = 5, 6; 1.46×1010 loops): quantisation, closed-form κ, universality, exact controls, unit diffusivity. Three exact flavour–gauge bridges and the strong-coupling boundary, with the two-loop adjudication and the c-dictionary for (αs(MZ), √σ). The honest ledger and publication passes: forty-four adjudications (eleven self-inflicted, including the multiplicity demotion of the golden-gap labels and the two-loop demotion of the one-loop αs exactness), every claim carrying its epistemic status with supersessions recorded in place, and a closing Predictions and named deciders summary. All numerical results of v6.0 are preserved except where explicitly superseded in the recorded ledger; each supersession is stated in place with its adjudicating test. Document structure and counts The framework is articulated in 8 Parts plus the changelog Part, with 61 theorems, 12 lemmas, 28 propositions, 25 corollaries, 7 conjectures, 3 foundational axioms, 18 definitions, 164 remarks, and 167 bibliographic references (~408 pages): Foundations; the chain of stages; Standard-Model phenomenology (now including the closed flavour sector); the relational ontology and the gravitational sector; epistemic stratification and open problems; the chain-uniqueness programme; and the v6.0 → v7.0 changelog with the campaign ledger and the predictions summary. Computational reproducibility is provided by GAP and Python scripts in the proj