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Original ledger classification: Inflow level + embedding rigidity [E]/[C]/[O]: the a^3 coefficient = the COMPUTED bulk Chern |C| = 1 of the v367 collar (TKNN/ASS quantisation + Callan-Harvey inflow + APS/Witten eta=CS), replacing v435's minimality; the seam F-normalisation = the affine embedding index k_Y = 5/3 (Ginsparg 1987, (3/5)(41/6) = 41/10 = b1) -- zero independent content, a face of SEAM.EQUIV.01; one phase, two responses (c_- = 8, C = 1) feeding both named targets; ALPHA.QUILLEN.EXACT.01 stays [O]

Recorded statement: THE ALPHA^3 LEVEL = THE COMPUTED BULK CHERN INVARIANT + THE SEAM F-NORMALISATION = THE AFFINE EMBEDDING INDEX (2026-07-02, closure-route round) -- upgrades BOTH residuals left after v433/v434/v435; does NOT close ALPHA.QUILLEN.EXACT.01.…

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Paper context: Architecture, E8 and the electromagnetic fixed point

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ALPHA.QUILLEN.INFLOW.01

Original claim text

THE ALPHA^3 LEVEL = THE COMPUTED BULK CHERN INVARIANT + THE SEAM F-NORMALISATION = THE AFFINE EMBEDDING INDEX (2026-07-02, closure-route round) -- upgrades BOTH residuals left after v433/v434/v435; does NOT close ALPHA.QUILLEN.EXACT.01. [E] TARGET RE-VERIFIED: alpha^-1 = 137.0359992168 unique root; Quillen split a^3 - 2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at the root < 1e-35 (v341/v382). [E] THE INVARIANT COMPUTED: FHS Chern C(M=1) = 1, C(M=3) = 0 -- the same integers as v367/v461, here read as the quantised U(1) response of the collar phase. [C] k0 = |C| = 1 UNDER THE EM1 READING: the a^3 coefficient EQUALS the computed bulk invariant of the SAME p+ip model that realises S3 -- 'why exactly 1' is answered by COMPUTATION plus the cited inflow identification (TKNN PRL 49 (1982) / Avron-Seiler-Simon PRL 51 (1983): integer quantisation of the U(1) response; Callan-Harvey NPB 250 (1985): boundary level = bulk response; APS 1975 / Alvarez-Gaume-Della Pietra-Moore Ann. Phys. 163 (1985) / Witten RMP 88 (2016): the metric-independent part of delta log det is the eta/CS level; Quillen 1985 / Bismut-Freed CMP 106 (1986): determinant-line curvature integrality) -- REPLACING v435's minimality assumption ('1 is the unit level') by a measurement on the model. [E] k_Y = 5/3 EXACT: tr_5bar(Y^2)/tr(T3^2) = (5/6)/(1/2) = 5/3 (Ginsparg PLB 197 (1987), the string-GUT hypercharge level); (3/5) x (41/6) = 41/10 = b1 -- the 'GUT 3/5' in the closure IS the inverse embedding index, not an imported convention; carrier decomposition 41/6 = 20/3 (3 generations x 16-content) + 1/6 (Higgs doublet). [C] F-NORMALISATION RETYPED: once the seam net is (E8)_1 at level 1 (SEAM.EQUIV.01), the U(1)_Y current sits inside a level-1 current algebra and <J(z)J(w)> = k/(z-w)^2 is FIXED by the embedding index -- level-1 current-algebra rigidity, no continuous dial; the v434 [C] ('how does the gauge curvature couple to the boundary measure') has ZERO independent content: it is a face of SEAM.EQUIV.01 (consistent with the v382 typing), its value the embedding index. [E] THE STRUCTURAL UNIFICATION: the two named targets share ONE object -- the collar phase carries TWO quantised responses, gravitational c_- = 16 x 1/2 x |C| = 8 = g_car + N_fam (feeds SEAM.EQUIV.01/S3, v456) and U(1) C = 1 (feeds the a^3 level, EM1); 'the one theorem and the one number' are two faces of the SAME invertible phase, and closing R1' (v469) feeds both residuals simultaneously. [O] NOT CLOSED: the bridge lemma 'delta log det_zeta(seam) = the inflow response' stays the named cited step; the EM1 reading stays [C]; ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 = 137.0359992 stays [E] regardless. UPDATED 2026-08-27 (v974): the inflow step 'boundary level = bulk response' is upgraded from cited to COMPUTED on the same model (probe laughlin_pump_em1_probe 20/20, PUMP_QUANTIZED_MATCHES_CHERN): Laughlin pump +1 per flux quantum (teleport bookkeeping with exact period closure), edge spectral flow -1/+1 antisymmetric as inflow demands, Streda dN/dPhi = C/(2 pi) exact integers over n_phi = 0..3 -- all equal to the FHS Chern, with the M = 3 controls all zero. The identification of the complex-copy U(1) with the physical Majorana collar stays part of the cited EM1 reading [C]; ALPHA.QUILLEN.EXACT.01 stays [O].

Original classification detail

Dependencies as recorded

ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.PROGRESS.02; ALPHA.QUILLEN.PROGRESS.03; ALPHA.QUILLEN.PROGRESS.04; EM.WARD.01; SEAM.EQUIV.CROSSEDPRODUCT.01; v341; v367; v382; v433; v434; v435; v456; v461

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Architecture, E8 and the electromagnetic fixed point
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Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
2026-07-02 ( v469/ v470 — the closure-route round on the two named targets: the SEAM.EQUIV.01 extension leg re-founded on 1995–2001 peer-reviewed crossed-product theorems [formula in source] the realisation axiom reduced to invariant level; the [formula in source] level [formula in source] the computed bulk Chern invariant [formula in source] the seam [formula in source] -normalisation [formula in source] the embedding index [formula in source] ; both targets stay )Research changelog; TFPT — Introduction and reading guide

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v470_alpha_inflow_level.pyRegistered verification module · frontier
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ALPHA.QUILLEN.INFLOW.01: the alpha^3 level = the COMPUTED bulk Chern invariant + the seam F-normalisation = the affine embedding index k_Y = 5/3. [E] target re-verified: alpha^-1 = 137.0359992168 unique root; Quillen split < 1e-35 (v341/v382). [E] the invariant computed: FHS Chern C(M=1)=1, C(M=3)=0 (the v367/v461 integers, read as the U(1) response). [C] k0 = |C| = 1 under the EM1 reading: the a^3 coefficient EQUALS the computed bulk invariant -- 'why exactly 1' answered by computation + cited inflow (TKNN 1982/ASS 1983 quantisation; Callan-Harvey 1985 inflow; APS/AGDPM/Witten RMP 88 eta=CS part of delta log det; Quillen/Bismut-Freed integrality) -- replacing v435's minimality assumption. [E] k_Y = tr(Y^2)/tr(T3^2) = (5/6)/(1/2) = 5/3 (Ginsparg 1987); (3/5)(41/6) = 41/10 = b1; 41/6 = 20/3 + 1/6. [C] F-normalisation retyped: <JJ> fixed by k_Y once the seam net is (E8)_1 level 1 -- no dial, ZERO independent content (a face of SEAM.EQUIV.01). [E] UNIFICATION: one invertible phase, two responses -- c_- = 8 (feeds SEAM.EQUIV.01/S3) and C = 1 (feeds EM1); closing R1' (v469) feeds both. [O] the bridge lemma stays cited; ALPHA.QUILLEN.EXACT.01 stays [O]; alpha^-1 stays [E]. Mixed exact (Wolfram-mirrored) + numerical (root/Chern).

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