The even-Hamming code on five slots is the D₅ half-spinor: its dimension is the Pascal sum 1 + 5 + 10 = 16, which forces g_car = 5 uniquely. The E₈ root count is then a pure carrier trace. The current reduction state of P2 (2026-07-22, markers unchanged): the compiler's dimensionless inputs reduce to four marks (derived from P1-side topology, v216), one discrete symmetry-lift bit (flag transitivity of the four marks, V₄ → D₄, ⟺ τ = i — bare mark-transitivity is automatic on the free circle and no local jet sees the side bit, v491/v499/v506/v507/v510/v512) and π; AX.P2.01 stays the declared axiom.

The seam owns exactly one measuring device — a single scalar two-point kernel — and four theorems pin what it can do. A scalar kernel exists iff it pairs the two sheets (exactly 2 = |ℤ₂| kernels = the glue ambiguity, v110); the certified channel counts the code by itself — one neutral kernel per code state, graded (1,5,10), so the Pascal closure is two countings of one set, not a condition (v112); pair transport is minimally complete — degree ≤ 1 generates nothing, degree 2 generates every code operation (v111); and the carrier net is 16 free Majorana fermions whose tower carrier → SO(16)₁ → E8₁ never changes the field content — only the certificate grows, and the central charge is the rank of the one kernel: 5 on the carrier block, 8 on the seam hull (v113). The interior is free; the structure is the certificate. Honest residue: the premise 'the seam is the free c=8 net' is the G_net gate itself — one theorem now closes both the metric story and the carrier choice — and that premise is itself no longer free-standing: it is a fixed-point theorem whose only residual factors into the already-open A2 (net existence) and GATE.QGEO, with the irreducible core {π, v_geo} a theorem (v160–v165).

D₅ = so(10) (spinor 16) and A₃ = su(4) (the four-puncture family geometry ℙ¹∖μ₄) have the same discriminant group ℤ₄. Their discriminant-form norms are two TFPT constants that add to the E₈ root norm, so the glue closes as a lattice theorem — not a posited 248.

The Coxeter number of E₈ is h = 30 = 2·3·5 — exactly the three discrete atoms (sheet ℤ₂, families ℤ₃, carrier g_car = 5). The rank is the count of live phases of the order-30 cycle.

The fine-structure constant is the unique positive root of a parameter-free cubic built only from c₃, the abelian coefficient (Σ L + N_Φ = 41 = 10 b₁) and the exact seam generating function. Existence and uniqueness are proved; the value lands 1.9σ from CODATA-2022. The abelian coefficient is pinned three independent ways — carrier algebra 10 b₁ = g_car·2^(g_car−2)+1 = 41, the U(1) hypercharge index, and the external RGE generator PyR@TE 3, which reproduces β_g₁ = (41/10)g₁³ verbatim (v159) — so the EM input is not a free knob. The three terms reassemble as the stationarity of a U(1) determinant line (Maxwell α³ + Calderón −2c₃³α² + transport), every coefficient a named index/heat-kernel/discriminant atom (v341/v342). The one residual — the from-first-principles proof that this IS the exact ζ-regularised Quillen functional — is the tracked external target ALPHA.QUILLEN.EXACT.01 (v382), never the value. Four honest steps narrow it without closing it: a solvable 4D model reaches the a₄ heat-kernel order (v433); the matter factor b₁ is the U(1)_Y a₄ coefficient via the β = a₄ theorem, collapsing the three residuals to one [C] (the seam F-normalisation) + one [O] (v434); and a π-power test isolates the cubic α³ as the unique metric-independent (π⁰) topological rung, whose coefficient is a conditional integer Chern-Simons level (v435). A fifth step (v470) upgrades both leftovers: the α³ level equals the computed bulk Chern invariant |C| = 1 of the same collar model that realises S3 (TKNN/Avron–Seiler–Simon quantisation + Callan–Harvey inflow + the APS/Witten η=CS reading of δ log det), replacing v435's minimality assumption; and the seam F-normalisation is the affine embedding index k_Y = tr(Y²)/tr(T₃²) = 5/3 (Ginsparg 1987; (3/5)·(41/6) = 41/10 = b₁ exactly) — zero independent content, a face of SEAM.EQUIV.01. One invertible phase, two quantised responses (c₋ = 8 gravitational, C = 1 U(1)). A sixth step (v472) exhibits the bridge lemma at the finite level: the determinant line of the occupied frame over the U(1)-twist moduli of the same collar — the moduli space of flat U(1) connections, the Quillen-shaped object the target names — carries FHS curvature = 1 = the inflow level, exactly and size-independently, with clean controls and the twist-moduli integer equal to the Bloch-BZ integer (Niu–Thouless–Wu); what stays [O] is the continuum ζ-det identification on the abstract seam (= the SEAM.EQUIV.01 face). A seventh step (v484, SEAM.CONTACT.UNIT.01) unifies this target with the φ₀-puncture target: the shared 'c₃ per boundary insertion' rule (the {0,3,6} ladder here, the per-mark weight there) 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 bare Green function takes integer multiples of c₃·ln2 at the μ₄ separations; the log-det contact expansion is exactly graded in c₃ per insertion; the Λ prefactor 3/(4π²) = 48c₃² carries the same Ω_adm = 48 at two insertions). The two [O] targets merge into one remaining analytic step (diagonal ζ-renormalisation + multiplicity matching) — and an eighth step (v485, SEAM.CONTACT.UNIT.02) settles that step at the computable level: the renormalised diagonal vanishes EXACTLY at the KMS seam circumference (G_reg(0;ℓ) = (1/π)ln(ℓ/2π), zero iff ℓ = 2π = 1/(4c₃)), the mark determinant resums in closed form (det(I−uC) = (1−4u)(1+2u)², BFK route v151; linear term absent because Tr C = 0), and the 48/41 multiplicities are ONE state set under two response weights (flat = Ω_adm vs Y²/Ginsparg = 10b₁ = 40+1, Tr₁₆Y² = 10/3 exact). Every finite piece of the merged target is proven; the single remaining [O] is the abstract-seam ζ-det identification — a face of SEAM.EQUIV.01, the v382 typing now substantiated computationally. ALPHA.QUILLEN.EXACT.01 stays [O]; α⁻¹ stays [E].

The same α⁻¹ ≈ 137 generates the electroweak scale (divided by the carrier 5), the cosmological constant (times 2) and the Hubble scale (via the square root) — the action ladder 1 : 5 : 10 is the Pascal row of the carrier.