Autoconsistency as an Organizing Principle: A Framework Proposal from Primordial Indeterminacy to Lepton-Sector Predictions

We propose a framework in which physical structure emerges from a single foundational schema. Three premises drive the derivation: the pre-structural state is one of primordial indeterminacy; persistence requires consistency; the system is closed, with no external referent. From these, the unique mechanism by which consistency can arise is self-distinction, leading to autoconsistency — structure self-supporting through mutual determination of its constituents. The Principle of Maximal Mutual Determination (PMMD) articulates this drive operationally at each categorial level. Continuum geometry emerges from the discrete relational structure via two complementary mechanisms. Compact phase structures (U(1) holonomies, finite SO(3) rotation angles) generate periodic quantities involving π. Non-compact path-integral selection of foam configurations weighted as exp(−Φ) generates logarithmic and exponential quantities involving e. The two transcendentals appear as topologically distinct outputs of the same discrete-to-continuum correspondence. The derivation chain is forced step by step: from the oriented three-cycle C3 (carrying the Z3 symmetry that propagates throughout), through the tetrahedron K4 in 3D, to the irreducible angular deficit Δ = 2π − 5·arccos(1/3) ≈ 0.128 rad concentrated on edges as Reggean discrete curvature, to the icosahedral lift with binary cover 2I ⊂ SU(2), to the E8 lattice via the McKay correspondence and icosian doubling, and to the branching chain E8 ⊃ E6 × SU(3) ⊃ SO(10) ⊃ SU(5) ⊃ Standard Model. The first branching is the unique route in which Z3 appears as the exact centre of a factor of the gauge group (both E6 and SU(3) have centre exactly Z3), preserving the elementary phase quantum 2π/3 of C3 as the most primitive cyclic phase of the chain; the alternative Z3-containing maximal subgroup SU(9) has centre Z9 and is excluded because it would require a more fundamental phase quantum 2π/9 not generated by the primordial structure. Principal quantitative outputs: α−1(E8) = 4π/(30Δ2) ≈ 25.4 at the GUT scale (structural unified coupling) Q = 2/3, the Koide ratio, as structural prediction of Z3 symmetry θalg = 2/9, the Koide angle, as algebraic invariant of SU(3) representation theory cθ = c2/(4π) ≈ 0.0805, the leading-order Koide-angle prefactor, derived from explicit Friedan one- and two-loop computation on CP2 with SU(3) fundamental weight ω1 in the Kähler convention, supported by three consistency checks (icosian doubling motivating CP2 as the natural target via E8 ⊃ E6 × SU(3), H-spinorial period confirming the 4π factor as standard SU(2) double-cover periodicity, and Atiyah–Singer/APS index theorem confirming the geometric normalization). The three checks share underlying CP2 geometry with the Friedan computation; their convergence reflects internal coherence of the geometric choices rather than four independent structural derivations. Empirical compatibility with PDG 2024 lepton-mass data at 0.91σ — a non-exclusion at current precision rather than positive empirical discrimination (the same data are also compatible with cθ = 0; discrimination of any specific structural form requires a factor-of-ten reduction in σ(mτ)) The framework's Beyond-the-Standard-Model (BSM) completion is read from the stratified spectrum of the 248 of E8. Three thematic extensions are articulated via a cellular reading of the foam: Photons and QED: Maxwell's equations, the photon Fock space, canonical commutators, and Feynman vertices emerge from foam dynamics. Mass and spin: relativistic dispersion, spin-1/2 from the topology of the rotation group, Fermi–Dirac statistics from quaternionic periodicity, and the Dirac equation as the cellular equation of motion of an H-loop. Extreme curvature: a structural curvature bound replaces classical singularities by cellular cores with stiff-fluid equation of state; the Hawking spectrum and a Planck-cutoff anomaly emerge from canonical hinge statistics; the Page time follows from cellular conservation; the cosmological constant is reformulated structurally as Λ·ℓP2 = 3/NdS2/3, expressing the smallness of Λ via the largeness of NdS (the latter being a cosmological boundary condition not derived ab initio); the universal −(3/2)log(A/ℓP2) logarithmic correction to the Bekenstein–Hawking entropy emerges from the three zero modes of the horizon Laplacian. Standard quantum-mechanical features — Hilbert-space structure, Born rule, Schrödinger evolution, and the Tsirelson bound 2√2 on Bell–CHSH correlations — are derived as conditional consequences of the SU(2) structure emerging from the icosahedral lift, via standard representation theory, Gleason's theorem, the non-relativistic limit of the cellular Dirac equation, and a geometric argument on the singlet state forced by shared primordial uncertainty. Notes on this version (v2): This version moderates the rhetorical strength of three claims relative to v1, bringing the framing into alignment with what the underlying mathematics establishes: (i) the cosmological-constant relation Λ·ℓP2 = 3/NdS2/3 is now described as a structural reformulation rather than an ab-initio resolution of the 122-orders-of-magnitude problem; (ii) the 0.91σ empirical compatibility is described as non-exclusion at current precision rather than positive discrimination; (iii) the structural prefactor cθ = c2/(4π) is described as derived from one explicit Friedan one-loop computation with three consistency checks, rather than from four independent converging arguments. The structural justification for Z3 as the exact factor-centre (preservation of the elementary phase quantum 2π/3) is articulated explicitly. The mathematical content and quantitative predictions of the framework are unchanged.