"""v470 -- 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 (the 2026-07-02 closure-route round, promoted from experiments/tfpt-discovery/ closure_route_alpha_inflow_level.py). It does NOT close ALPHA.QUILLEN.EXACT.01. After v433/v434/v435 the EM-Ward residual was exactly ONE [O] (residual 2: the origin of the cubic alpha^3 Chern/Maxwell moment, where v435 could only argue "the level is an integer IF the EM1 reading holds, and 1 is the MINIMAL nonzero level") plus ONE [C] (residuals 1&3: the exact seam F-normalisation). This module upgrades both: (a) THE LEVEL IS COMPUTED, NOT MINIMAL. The a^3 coefficient k0 = 1 EQUALS the bulk Chern invariant |C| = 1 of the SAME p+ip collar model that realises S3 of SEAM.EQUIV.01 (v367/v461). The quantisation and the bulk-boundary identification are published theorems: TKNN PRL 49 (1982) / Avron-Seiler-Simon PRL 51 (1983) (the U(1) response of a gapped 2d fermion system is an integer Chern number); Callan-Harvey Nucl. Phys. B250 (1985) (anomaly inflow: the boundary CS/anomaly level equals the bulk response); APS 1975 / Alvarez-Gaume-Della Pietra-Moore Ann. Phys. 163 (1985) / Witten Rev. Mod. Phys. 88 (2016) (the metric-independent part of delta log det is the eta-invariant = the CS level); Quillen 1985 / Bismut-Freed CMP 106 (1986) (determinant-line curvature = 2 pi i x an integral class). v435's minimality assumption is replaced by a measurement on the model. (b) THE F-NORMALISATION IS THE EMBEDDING INDEX. 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 its 2-point normalisation = k/(z-w)^2 is FIXED by the affine embedding index -- no continuous dial. The hypercharge index is the string-GUT k_Y = tr(Y^2)/tr(T3^2) = (5/6)/(1/2) = 5/3 (Ginsparg, Phys. Lett. B197 (1987)): EXACTLY the "GUT 3/5" that converts the SM 41/6 into b1 = 41/10 (v434). So the one [C] has ZERO independent content -- it is a face of SEAM.EQUIV.01, its value the embedding index. (c) THE STRUCTURAL UNIFICATION. The two named targets share ONE object: the collar phase has two quantised responses -- gravitational c_- = 8 (=> the seam net, SEAM.EQUIV.01/S3, v456) and U(1) C = 1 (=> the Maxwell a^3 level, EM1). "The one theorem and the one number" are two faces of the SAME invertible phase; closing the realisation R1' (v469) feeds both residuals simultaneously. [E] 1. 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 to < 1e-35 (v341/v382). [E] 2. THE INVARIANT COMPUTED: FHS Chern C(M=1) = 1, C(M=3) = 0 (control) -- the same integers as v367/v461, here read as the U(1) response. [C] 3. k0 = |C| = 1 UNDER THE EM1 READING: the a^3 coefficient equals the computed bulk invariant (inflow + quantisation cited); replaces v435's minimality. [E] 4. k_Y = 5/3 EXACT: tr_5bar(Y^2) = 3(1/3)^2 + 2(1/2)^2 = 5/6, tr(T3^2) = 1/2, ratio 5/3; (3/5) x (41/6) = 41/10 = b1; carrier decomposition 41/6 = 20/3 (3 generations x 16-content) + 1/6 (Higgs doublet). [C] 5. F-NORMALISATION RETYPED: = the embedding index inside the level-1 seam net, conditional ONLY on SEAM.EQUIV.01 (consistent with the v382 "face of" typing). [E] 6. UNIFICATION: c_- = 16 x (1/2) x |C| = 8 (gravitational) and C = 1 (U(1)) -- one phase, two responses, feeding both named targets. [O] 7. 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 stays [E] regardless. Mixed: exact (k_Y = 5/3, 41/6 = 20/3 + 1/6, (3/5)(41/6) = 41/10 -- Wolfram-mirrored) + numerical (root/split re-verification mpmath, FHS Chern numpy, Python-only).""" import mpmath as mp import numpy as np import sympy as sp from tfpt_constants import (check, summary, reset, c3, b1, phi0, phibase, dtop, g_car, N_fam, dim_Splus) SX = np.array([[0, 1], [1, 0]], complex) SY = np.array([[0, -1j], [1j, 0]], complex) SZ = np.array([[1, 0], [0, -1]], complex) def _occ_vec(kx, ky, M): d = np.array([np.sin(kx), np.sin(ky), M - np.cos(kx) - np.cos(ky)]) w, v = np.linalg.eigh(d[0] * SX + d[1] * SY + d[2] * SZ) return v[:, 0] def _chern(M, N=24): """Fukui-Hatsugai-Suzuki plaquette Chern number (same convention as v367).""" ks = np.linspace(0, 2 * np.pi, N, endpoint=False) u = [[_occ_vec(kx, ky, M) for ky in ks] for kx in ks] F = 0.0 for i in range(N): for j in range(N): ip, jp = (i + 1) % N, (j + 1) % N u00, u10, u01, u11 = u[i][j], u[ip][j], u[i][jp], u[ip][jp] Ux = np.vdot(u00, u10); Ux /= abs(Ux) Uy = np.vdot(u10, u11); Uy /= abs(Uy) Ux2 = np.vdot(u01, u11); Ux2 /= abs(Ux2) Uy2 = np.vdot(u00, u01); Uy2 /= abs(Uy2) F += np.angle(Ux * Uy * np.conj(Ux2) * np.conj(Uy2)) return F / (2 * np.pi) def _phi_seam(a): Q = dtop * mp.e ** (-2 * a) return phibase + Q * (1 - Q) ** (mp.mpf(-5) / 4) def _F_U1(a): return a ** 3 - 2 * c3 ** 3 * a ** 2 - 8 * b1 * c3 ** 6 * mp.log(1 / _phi_seam(a)) def run(): reset() print("v470 ALPHA.QUILLEN.INFLOW.01: the alpha^3 level = the computed bulk Chern " "invariant; the F-normalisation = the embedding index k_Y = 5/3; does NOT " "close ALPHA.QUILLEN.EXACT.01") # 1. target re-verified (root + split) a = mp.findroot(_F_U1, mp.mpf("0.0073")) ainv = 1 / a lhs = a ** 3 - 2 * c3 ** 3 * a ** 2 rhs = 8 * b1 * c3 ** 6 * mp.log(1 / _phi_seam(a)) check("TARGET RE-VERIFIED [E]: alpha^-1 = %.10f unique root of F_U(1); the " "Quillen split a^3 - 2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) holds at the " "root to < 1e-35 (v341/v382)" % float(ainv), abs(ainv - mp.mpf("137.0359992168407")) < mp.mpf("1e-9") and abs(lhs - rhs) < mp.mpf("1e-35")) # 2. the invariant computed on the collar model C_topo = round(_chern(1.0)) C_triv = round(_chern(3.0)) check("THE INVARIANT COMPUTED [E]: FHS Chern C(M=1) = %d (topological), " "C(M=3) = %d (control) -- the same integers as v367/v461, here read as the " "quantised U(1) response of the collar phase (TKNN 1982; Avron-Seiler-" "Simon 1983)" % (C_topo, C_triv), abs(C_topo) == 1 and C_triv == 0) # 3. k0 = |C| under the EM1 reading [C] k0 = 1 check("k0 = |C| = %d UNDER THE EM1 READING [C]: the a^3 coefficient EQUALS the " "computed bulk invariant of the SAME model that realises S3 -- 'why exactly " "1' is answered by computation + the cited inflow identification (Callan-" "Harvey 1985; eta/CS part of delta log det: APS 1975, Alvarez-Gaume-Della " "Pietra-Moore 1985, Witten RMP 88 (2016); determinant-line integrality: " "Quillen 1985, Bismut-Freed 1986), replacing v435's minimality assumption" % abs(C_topo), k0 == abs(C_topo)) # 4. k_Y = 5/3 exact + the b1 conversion + the carrier decomposition trY2 = 3 * sp.Rational(1, 3) ** 2 + 2 * sp.Rational(1, 2) ** 2 trT32 = 2 * sp.Rational(1, 2) ** 2 kY = trY2 / trT32 b1_SM = sp.Rational(41, 6) ferm = sp.Rational(2, 3) * 3 * (6 * sp.Rational(1, 6) ** 2 + 3 * sp.Rational(2, 3) ** 2 + 3 * sp.Rational(1, 3) ** 2 + 2 * sp.Rational(1, 2) ** 2 + 1) higgs = sp.Rational(1, 3) * 2 * sp.Rational(1, 2) ** 2 check("k_Y = 5/3 EXACT [E]: tr_5bar(Y^2)/tr(T3^2) = (5/6)/(1/2) = %s (Ginsparg " "1987); (1/k_Y) x 41/6 = 3/5 x 41/6 = %s = b1 -- the 'GUT 3/5' IS the " "inverse embedding index, not an imported convention; carrier decomposition " "41/6 = %s (fermions, 3 x 16-content) + %s (Higgs doublet)" % (kY, b1_SM / kY, ferm, higgs), kY == sp.Rational(5, 3) and b1_SM / kY == sp.Rational(41, 10) and ferm + higgs == b1_SM and abs(float(b1) - float(sp.Rational(41, 10))) < 1e-15) # 5. F-normalisation retyped [C] check("F-NORMALISATION RETYPED [C]: once the seam net is (E8)_1 level 1 " "(SEAM.EQUIV.01), the U(1)_Y 2-point normalisation = k/(z-w)^2 is " "FIXED by the embedding index k_Y = 5/3 -- level-1 current-algebra " "rigidity, no continuous dial; the v434 [C] has ZERO independent content " "(a face of SEAM.EQUIV.01, consistent with the v382 typing)", True) # 6. the unification: one phase, two responses c_minus = sp.Rational(dim_Splus, 2) * abs(C_topo) check("UNIFICATION [E]: the collar phase carries TWO quantised responses -- " "gravitational c_- = 16 x 1/2 x |C| = %s = 8 = g_car + N_fam (feeds " "SEAM.EQUIV.01/S3, v456) and U(1) C = %d (feeds the a^3 level, EM1) -- " "'the one theorem and the one number' are two faces of ONE invertible " "phase; closing R1' (v469) feeds both" % (c_minus, abs(C_topo)), c_minus == 8 == g_car + N_fam and abs(C_topo) == 1) # 7. honest scope [O] check("NOT CLOSED [O]: 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", True) return summary("v470 ALPHA.QUILLEN.INFLOW.01: the alpha^3 level upgraded from " "'unit level by minimality' (v435) to '= the computed bulk Chern " "invariant |C| = 1' (TKNN/ASS + Callan-Harvey + APS/Witten, " "computed on the v367 collar); the seam F-normalisation retyped " "as 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]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)