"""v234 -- the Seam-Holomorphy selection certificate: ONE condition, three faces, E8. This is the arithmetic backbone of the proposed single closing theorem (Research Contract: 'a gapped RP seam bulk has a boundary net with no nontrivial abelian sector'). It certifies that the entire structural residual of TFPT reduces to ONE condition with THREE provably-equivalent faces, all of which force E8 -- so P2 / G_net / Target A / QGEO are not separate gates but one. THE CONDITION: 'the seam carries no nontrivial abelian sector.' Three faces: (i) AQFT: the boundary chiral net is HOLOMORPHIC (mu-index 1, single vacuum sector). (ii) geometry: the seam link is a HOMOLOGY 3-sphere <=> the binary group Gamma is PERFECT <=> Gamma = 2I (Poincare sphere). (iii) rep theory: exactly ONE 1-dim irrep (the trivial one). All three are the SAME number: for a finite SU(2) subgroup Gamma (an affine-ADE McKay group), #(1-dim irreps) = |Gamma^ab| = |H_1(S^3/Gamma)| = #(mark-1 nodes of the affine Dynkin diagram). It equals 1 ONLY for affine E8 (2I) among all ADE. [E] 1. #(mark-1 affine-Dynkin nodes) = |Gamma^ab| = |H_1(link)| for every ADE type: A_n -> n+1, D_n -> 4, E6 -> 3, E7 -> 2, E8 -> 1. Only E8 gives 1. [E] 2. So 2I is the UNIQUE nontrivial perfect finite SU(2) subgroup; S^3/2I is the unique homology-sphere spherical space form (the Poincare sphere); and 2I is the unique one with no nontrivial abelian sector. [E] 3. HOLOMORPHIC c=8: c = g_car + N_fam = 5 + 3 = 8, and a holomorphic chiral CFT has c in 8Z; at c=8 the unique even unimodular rank-8 lattice is E8 (Gram even, det 1) -> (E8)_1. So holomorphy + c=8 forces E8 too. [E] 4. CONVERGENCE: faces (i),(ii),(iii) are one condition; v83/v154 use (i), v232 uses (ii), v219 uses (iii) -- they are the SAME E8-selector, so Target A, G_net, the carrier P2 and the Kleinian seam are ONE gate. [O] 5. THE CLOSING THEOREM (stated, not proved): RP + gap + chirality => the seam bulk is quasi-free (v160) => invertible bulk => HOLOMORPHIC boundary net (no abelian sector) => E8. The single analytic step 'free bulk => holomorphic boundary' is the one residual; everything else then follows. Status: [E] for the selection arithmetic (the three faces coincide and force E8); [O] for the analytic closing theorem itself. Mirrored in wolfram/tfpt_readouts_extension.wl. """ import numpy as np from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8 # affine Dynkin Kac marks (the McKay irrep degrees of the binary group Gamma); # #(marks == 1) = |Gamma^ab| = |H_1(S^3/Gamma)| = #(1-dim irreps) AFFINE_MARKS = { "A4 (Z5)": [1, 1, 1, 1, 1], # cyclic Z5 -> 5 ones "D5 (Dic)": [1, 1, 2, 2, 1, 1], # binary dihedral -> 4 ones "E6 (2T)": [1, 1, 1, 2, 2, 2, 3], # binary tetrahedral -> 3 ones "E7 (2O)": [1, 2, 3, 4, 3, 2, 1, 2], # binary octahedral -> 2 ones "E8 (2I)": [1, 2, 3, 4, 5, 6, 4, 2, 3], # binary icosahedral -> 1 one } def e8_cartan(): # finite E8 from the affine E8 McKay graph minus the trivial (mark-1) node edges = [(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (5, 8)] A = np.zeros((8, 8), dtype=int) for i, j in edges: A[i - 1, j - 1] = A[j - 1, i - 1] = 1 return 2 * np.eye(8, dtype=int) - A def run(): reset() print("v234 Seam-Holomorphy selection: ONE condition (no abelian sector), three faces, E8") # face (ii)/(iii): #mark-1 = |abelianisation| = |H1(link)|; only E8 -> 1 ones = {name: marks.count(1) for name, marks in AFFINE_MARKS.items()} check("FACE (ii)/(iii) [E]: #(mark-1 affine-Dynkin nodes) = |Gamma^ab| = " "|H_1(S^3/Gamma)| = #(1-dim irreps): A4->5, D5->4, E6->3, E7->2, E8->1 " "(%s) -- ONLY E8 gives 1" % ones, ones == {"A4 (Z5)": 5, "D5 (Dic)": 4, "E6 (2T)": 3, "E7 (2O)": 2, "E8 (2I)": 1} and sum(1 for v in ones.values() if v == 1) == 1 and ones["E8 (2I)"] == 1) check("=> 2I is the UNIQUE nontrivial perfect finite SU(2) subgroup; S^3/2I is " "the unique homology-sphere spherical space form (Poincare sphere); the " "unique one with no nontrivial abelian sector", ones["E8 (2I)"] == 1 and min(v for k, v in ones.items() if k != "E8 (2I)") >= 2) # mark sums = Coxeter numbers (sanity: the marks are the genuine affine data) hh = {name: sum(marks) for name, marks in AFFINE_MARKS.items()} check("the marks are genuine affine data: sum(marks) = Coxeter number " "(E6=12, E7=18, E8=30=2*3*5); sum(E8 marks^2)=120=|2I|=|R^+(E8)|", hh["E6 (2T)"] == 12 and hh["E7 (2O)"] == 18 and hh["E8 (2I)"] == 30 and sum(m * m for m in AFFINE_MARKS["E8 (2I)"]) == 120) # face (i): holomorphic c=8 = g_car+N_fam -> unique even unimodular rank-8 = E8 C = e8_cartan() detC = round(np.linalg.det(C.astype(float))) even = all(C[i, i] == 2 for i in range(8)) check("FACE (i) [E]: c = g_car + N_fam = 5+3 = 8; a holomorphic chiral CFT has " "c in 8Z, and at c=8 the unique even unimodular rank-8 lattice is E8 " "(Gram even [diag 2], det 1) -> (E8)_1", (g_car + N_fam) == rankE8 == 8 and even and detC == 1 and np.all(np.linalg.eigvalsh(C.astype(float)) > 0)) # face convergence: the three faces are ONE selector check("CONVERGENCE [E]: faces (i) holomorphy [v83/v154], (ii) homology-sphere " "[v232], (iii) one 1-dim irrep [v219] are the SAME E8-selector -- so " "Target A, G_net, the carrier P2 and the Kleinian seam are ONE gate, not four", ones["E8 (2I)"] == 1 and detC == 1 and (g_car + N_fam) == 8) # the closing theorem, stated (not proved) check("CLOSING THEOREM [O] (stated): RP + gap + chirality => quasi-free bulk " "(v160) => invertible bulk => holomorphic boundary net (no abelian sector) " "=> E8. The single residual analytic step is 'free bulk => holomorphic " "boundary'; P2/QGEO/G_net then follow. NOT proved here -- this certifies " "only that the three faces coincide and force E8", True) return summary("v234 Seam-Holomorphy selection (one condition, three faces, E8)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)