The integer alphabet of the theory falls out of (g_car, N_fam) = (5,3): the E₈ rank, the sheet and glue counts, and the Pythagorean mass volume as a difference of squares.

E₈ is the exceptional top of the McKay tower of finite SU(2) subgroups (2T→Ê₆, 2O→Ê₇, 2I→Ê₈), so choosing E₈ is choosing the icosahedron. The McKay graph is built from the group: the 120 icosians close to the binary icosahedral group 2I (element orders containing the 2,3,5 axes), and its nine irreducible-representation degrees are exactly the affine-E₈ Kac marks. A backward certificate of the closed E₈, not a P2 proof. The same exceptional geometry has two complex-multiplication readings: the square modulus (j=1728) gives the EM index 41 as a Gaussian norm, the hexagonal partner (j=0) gives the scalaron 7 as an Eisenstein norm.

The seam denominator is fixed three independent ways. If the seam is a horizon, the gravitational 8π forces c₃; it must then coincide with the geometric 2|μ₄| (Gauss–Bonnet) and the lattice rank E₈ — all three give 8. The gravity route no longer rests on arithmetic alignment alone: the temperature normalisation is now MEASURED on the seam itself (v526, SEAM.THERMAL.KMS.01) — the reconstructed free OS quotient admits exactly one detailed-balance thermal representation, β = N clock steps at both levels, β_angle = 2π exact, hence T_seam = 4c₃ = the Bisognano–Wichmann/Hawking normalisation; temperature joins geometry and anomaly as the THIRD LEG of c₃ ([C] reading: seam euclidean circle = thermal circle of the reconstructed horizon dynamics; the entropy-fraction bridge honestly does not close; 'the seam IS a horizon' stays [C]). The Seam–Horizon gate (SEAM.THEOREM.01) stays [O]: v150–v152 closed the mechanism and merged the normalisation into the one anchor, and v471 now exercises the replica chain numerically on the discretized collar with the seam's own kernel (real replica sheets n=2,3; BFK/Calderón conically clean on the kernel; the attractor mode's IR divergence regulated by the recovery gap) — the residual retypes to the cited continuum scaling limit (MMST class, the same single residual as SEAM.EQUIV.01) plus the one dimensionful anchor.

The boundary transport spectrum {1, (2/3)⁶, (1/3)⁶} has a sub-leading eigenvalue that appears in both sectors: the SM flavor gap and the horizon Page recovery are the same number.

The transport gap is positive, so by Perron–Frobenius the operator has a unique dominant eigenvector and iterating from any start converges to the same fixed direction. Parameter-freeness is an attractor, not a tuning.

The bridge between the static (lattice/spectrum) data and the dynamic (recovery) data is a clock — the order-30 Coxeter element — which factorizes into two coprime hands: a static carrier ring ℤ/5 = g_car (golden √5, no rate) and a dynamic family ring ℤ/6 = 2N_fam (the recovery rate (2/3)⁶, exponent 6 = 2N_fam). The dynamic hand runs 0,1,2,3,4,5 with position 0 the conserved law (the attractor, rate 0) and 1..5 the live phases; the static hand runs 1,2,3,4,5. So '0,1,2,3,4,5' is the law-inclusive reading and '1,2,3,4,5' the live-only reading of the same clock. The arithmetic is [E]; 'the bridge is one clock' is [C] (v319).

Collecting the arithmetic arc: the affine-E₈ network spectrum carries the atoms {2,3,5} as the angles 2cos(π/k), with the golden ratio φ = 2cos(π/5) the g_car = 5 signature (v313). The static (carrier) and dynamic (recovery) data split by number field — ℚ(√5) for the 5-fold carrier vs ℚ for the rational family rates (v314) — and the order-30 Coxeter element couples them as the cyclotomic compositum ℚ(ζ₃₀), whose Galois group is exactly μ₄ × ℤ₂ of degree 8 = rank E₈ (v315). The whole SM structural sector lives there: the three generations are the μ₃ cube-root orbit (Galois-refined 1+2, the fixed one the attractor, v317) and the two CP phases the ζ₆ family-factor data (v316). The magnitude seed φ₀ itself reduces to a pure function of π, so there are zero dimensionless free parameters — {a, π, v_geo} is the complete input (v318). [E] arithmetic / [C] the raw-seam realisation closed modulo cited theorems (residual [O] = the cited MMST continuum existence only, v336; extension leg on the peer-reviewed crossed-product package, realisation at invariant level, v469; stays [O]).

The arithmetic is not only descriptive — it makes a testable cross-prediction. Both leading CP phases are powers of the one hexagonal unit ρ = ζ₆ of the family factor: δ_CKM,lead = arg(ρ) = π/3 (60°) and δ_PMNS = arg(ρ⁴) = 4π/3 (240°), and since ρ⁴ = −ρ they are locked, δ_PMNS = δ_CKM,lead + π. This upgrades the previously assigned δ_PMNS = 240° to a Galois-forced relation to the leading (π/3) component of the quark phase: the quark and lepton leading CP phases are not independent. (The lock is to the structural π/3, not the full measured γ = δ_CKM,lead + 3λ² ≈ 68.7° that carries the quark transport correction — so δ_PMNS = 240°, not 248.7°.) Sharpened (v322): the selected node is fixed by the deck order, δ_PMNS = |μ₄|·δ_CKM,lead = 4·(π/3); the sub-leading correction is bounded by the quark analogue 3λ² ≈ 8.7°, so the prediction is the band 240° ± ~9°, currently +1.08σ vs NuFIT 6.0 (NO, δ_CP = 212°⁺²⁶₋₄₁) — and the nearest wrong hexagonal node (180° / 300°) is 60° away, cleanly discriminated at DUNE/Hyper-K (~5–15°). Kill test: a δ_PMNS robustly incompatible with 240° (>3σ at DUNE/Hyper-K/JUNO) falsifies the whole Galois-CP organisation (v320/v322). [E] relation / [C] phase identification / [X] kill test.

One step links the two open bedrock items. The μ₄ clock is literally a geometric rotation, ρ = diag(iⁿ) = exp(i(π/2)L) with L = diag(n) the seam rotation generator, so μ₄ ⊂ U(1)_rot. For a rotation-covariant seam covariance C = f(L) the modular Hamiltonian K = log((1−C)/C) = g(L) is itself a function of L, so the modular flow commutes with all rotations and the μ₄ clock is a modular symmetry for free — the discriminator: a mere period-4 curvature preserves the clock but its flow is not geometric ([K,L] ≠ 0). This is the Bisognano–Wichmann content: given the seam is the (E₈)₁ chiral net (v308), BW/Hislop–Longo make the vacuum modular flow geometric, so QGEO.SYM.01 (ω∘ρ = ω) is downstream of SEAM.EQUIV.01 + a rotation-invariant vacuum — not an independent axiom (v323). On the exactly solvable four-interval realisation of the four marks the invariance is manifest at the state level (ρCρ⁻¹ = C at machine precision, with the fermionic clock the order-8 double-cover lift ρ⁴ = −1; v480) — the mechanism in the cited multilocal free-fermion class, the raw-collar premise unchanged. And the rotation-invariant vacuum is itself a conformal-NET AXIOM (a chiral net's Möbius-covariant vacuum is the unique invariant positive-energy vector), so QGEO.SYM.01 is in fact a COROLLARY of SEAM.EQUIV.01 with no extra premise (v335, Lean qgeoSymIsCorollary) — the two open bedrock items collapse to ONE, its MMST route SEAM.EQUIV.MMST.01 now closed modulo cited theorems (lattice v367/v368 + S3 stack v376–v379, ground-state witnesses v489/v490). [E] construction / [C] linkage / [O] residual (= the cited MMST continuum scaling-limit existence only, v336; extension leg on the peer-reviewed crossed-product package, realisation at invariant level, v469; stays [O]).

The rewrite question made concrete. The pure (2,3,5) network gives only the Coxeter skeleton and the golden 5-fold angle; the recovery rate (2/3)⁶ is provably not in its adjacency spectrum (v312). But the smallest substrate that carries everything is a product: the carrier network T_net = (A+2I)/4 (attractor = Kac marks = the E₈ skeleton) fibred by a 3-node family cusp T_cusp = diag((1−w)⁶), w ∈ {0,1/3,2/3} (spectrum {1,(2/3)⁶,(1/3)⁶}). The fibred T_net ⊗ T_cusp carries both at once: top eigenvalue 1 with eigenvector marks ⊗ (w=0) (skeleton × democratic cusp), and (2/3)⁶ as a genuine eigenvalue (network attractor × cusp subleading). This is the cyclotomic split made dynamical — substrate = carrier × family = the 30 = 5×6 of v315, with the recovery gap living entirely in the family factor (v324). [E] construction / [C] reading.

The three hypergraph modules (v299 carrier growth, v327 branching rule M, v324 fiber product) merge into a single rewrite on a 9×3 labelled grid. One micro-step = network lazy diffusion on each cusp column plus M on each node's family vector (T_micro = T_net ⊗ M, purely local on 27 cells). The joint attractor is marks ⊗ (w=0); one clock hand (2N_fam = 6 family steps) carries recovery (2/3)⁶; v324's T_net ⊗ T_cusp emerges as the cusp-readout basis; v299's growth E₆→E₇→E₈→Ê₈ is unchanged with the fiber attached. The cusp weight 2/3 = |ℤ₂|/N_fam is derived from the rule arity (v327); what remains non-graph-spectral for a full-structure rewrite is the analytic seed φ₀ alone (v312). [E] mechanism unified / [O] seed + φ₀.

The uniform rule generates only λ₂ — its zero mode persists under every power, so no iterate reaches the verified third transfer mode (1/3)⁶. The gap closes with a uniqueness statement (v486, HYP.REWRITE.02): for the symmetric rule M(s,h) (one absorbing family channel + ℤ₂ pair) the survival spectrum is {s+h, s−h}, and demanding the physical pair {2/3, 1/3} forces uniquely (stay, hop, leak) = (1/2, 1/6, 1/3) = (1/|ℤ₂|, 1/(|ℤ₂|N_fam), 1/N_fam) — the lazy ℤ₂-pair walk, every rate an atom expression; over the order-6 hand eig(B⁶) = {(2/3)⁶, (1/3)⁶} exactly, so both decay gaps (6ln(3/2) recovery, 6ln3 subdominant) have one recursive generator. And the split selection is not a choice (v487, HYP.REWRITE.03): the lazy rule's one-step spectrum is exactly the complete resummed-clock ladder below the wall (v124: {1−n/3 : n = 0,1,2}), while the uniform rule collapses its odd mode onto the wall (0 = 1−3/3); deck parity IS the rung index ([B,σ] = 0, σ-even → rung 1, σ-odd → rung 2 — the structural home of the parity assignment the FRB comb searches test), and ladder faithfulness + ℤ₂ equivariance + rates ≥ 0 force both the split and the assignment uniquely (the swap needs hop = −1/6 < 0). Corollary: ω₁/ω₂ = rate(2)/rate(1) = log_{3/2}3 — the two comb tones of the empirical program are one bend apart. [E] uniqueness + generation + forcing / [O] the arity {2,3} (anchor input) and the clock's semiclassical derivation (R1).

The tree-level retained seed φ₀^tree = 1/(6π) equals F/(4hπ) on the icosahedral hypergraph (F = 20 triangular hyperedges, h = 30, |Aut| = 120): equivalently (F/(g_car·N_fam))·c₃, with F/h = |ℤ₂|/N_fam = 2/3 (the same survival ratio as v327). Gauss–Bonnet consistent with c₃ = 1/(|ℤ₂|·4π). The puncture 48c₃⁴ remains analytic, not a graph fraction (v396). Its geometric side is now EXACT (v483, HYP.PHI0.GEOM.01): every twisted heat trace of the flat τ=i pillowcase is a t-independent rational (σ/ρ traces = 1 = the Atiyah–Bott fixed-point counts; contact term exactly 1/2; clock-equivariant trace exactly 1), so the π⁻⁴ in 48c₃⁴ = 3/(256π⁴) cannot come from flat orbifold geometry at any order — it must sit in the per-mark coupling weight (4 = |μ₄| insertions of weight c₃; the bare 1/(2π) 4-cycle differs by the exact rational 3/16). The puncture target narrows from 'derive the term' to that one rule — and that rule is itself not a new unknown (v484, SEAM.CONTACT.UNIT.01): 'c₃ per insertion' IS the KMS seam unit 2π = 1/(4c₃) with 1/4 = 1/|μ₄| (one bare boundary propagator orbit-averaged over the four marks), derived on the seam circle for the finite cycle sector; the puncture target and ALPHA.QUILLEN.EXACT.01 merge into one remaining analytic step (diagonal ζ-renormalisation + multiplicity matching, 48 = Ω_adm / 41 = 10b₁) — settled at the computable level by v485: the renormalised diagonal vanishes exactly at the KMS seam circumference, the mark determinant resums closed-form (det(I−uC) = (1−4u)(1+2u)², BFK route), and 48/41 are one state set under two response weights; the remaining [O] is the abstract-seam ζ-det identification, a face of SEAM.EQUIV.01 alone. [E] leading term + geometric side + contact unit + diagonal/resummation / [C] reading / [O] the keystone face.