Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""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)

58b8eb3621fcb2705f5a2d8345c31340fd1d279fd5a8d0848e354624f3713396

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.