Follow the source behind the result.

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

"""v396 -- the phi0 leading term from icosahedral combinatorics (sharpening v312/v395).

v312's honest negative: phi0 = 1/(6 pi) + 48 c3^4 is analytic/transcendental, not a rational
edge fraction.  v395 unified the carrier x family mechanism; the remaining non-graph-spectral
input is phi0 alone.  This module tests whether the LEADING term phibase = 1/(6 pi) has a
forced combinatorial reading on the icosahedral hypergraph / (2,3,5) skeleton -- without
claiming the puncture correction 48 c3^4 is graph-determined.

  [E] 1. ICOsahedral COUNTS ARE ANCHOR-FORCED: the icosahedron has V=12, E=30=h(E8), F=20
        triangular hyperedges, |Aut|=120=|2I|; and F = g_car(g_car-1), h = 2 N_fam g_car,
        4h = |Aut| (all from {g_car,N_fam} = {5,3}).
  [E] 2. THE LEADING TERM IS EXACT: phibase = 1/(6 pi) = (4/3) c3 (since c3 = 1/(8 pi));
        equivalently phibase = F/(4 h pi) = F/(|Aut| pi) = (F/(g_car N_fam)) c3.
  [E] 3. F/h = |Z2|/N_fam: the hyperedge-to-Coxeter ratio equals the family survival ratio
        (the same 2/3 as v327/v395); so phibase = (F/h)/(4 pi) = c3 (g_car-1)/N_fam.
  [E] 4. GAUSS-BONNET CONSISTENCY: on S^2 the icosahedron vertex-deficit sum is 12*(pi/3)
        = 4 pi (the sphere total curvature); c3 = 1/(|Z2| 4 pi) uses the same 4 pi seam
        datum (v216/v342), and phibase carries one extra N_fam in the denominator.
  [E] 5. HONEST NEGATIVE -- puncture term NOT combinatorial: dtop = 48 c3^4 = 3/(256 pi^4)
        is not a small rational hypergraph fraction; the full phi0 still has an analytic
        piece beyond the icosahedral leading term.
  [C] 6. READING: phibase is the tree-level seed phi0^ret at leading order; it is NOT an
        independent rewrite injection -- it is (hyperedge count)/(carrier x family) x c3,
        with c3 itself the boundary Gauss-Bonnet datum.  The puncture 48 c3^4 remains [O].

HONEST SCOPE: [E] the exact combinatorial identities + Gauss-Bonnet consistency + puncture
negative; [C] the seed-as-combinatorics reading (reinterpretation, not a new derivation of
P1).  Sharpens v312/v395; does NOT close phi0 entirely.  Python-only (sympy)."""
import itertools

import numpy as np
import sympy as sp

from tfpt_constants import check, summary, reset, c3, g_car, N_fam, phibase, phi0

PHI = (1 + np.sqrt(5)) / 2


def _icosahedron_counts():
    """Return (V, E, F, |Aut|) for the unit icosahedron graph."""
    V = []
    for a, b in itertools.product([1, -1], repeat=2):
        V += [(0, a, b * PHI), (a, b * PHI, 0), (b * PHI, 0, a)]
    V = np.array(sorted(set(map(tuple, np.round(V, 9)))))
    D = np.round(((V[:, None] - V[None]) ** 2).sum(-1), 6)
    el2 = np.min(D[D > 1e-9])
    A = (np.abs(D - el2) < 1e-6).astype(int)
    faces = [t for t in itertools.combinations(range(len(V)), 3)
             if A[t[0], t[1]] and A[t[0], t[2]] and A[t[1], t[2]]]
    nbr = [set(np.where(A[i])[0]) for i in range(len(V))]

    def _autos():
        autos = []
        def bt(perm, used):
            k = len(perm)
            if k == len(V):
                autos.append(tuple(perm))
                return
            for img in range(len(V)):
                if img in used:
                    continue
                if all((perm[j] in nbr[img]) == (j in nbr[k]) for j in range(k)):
                    bt(perm + [img], used | {img})
        bt([], set())
        return autos

    return len(V), int(A.sum() // 2), len(faces), len(_autos())


def run():
    reset()
    print("v396  phi0 leading term from icosahedral combinatorics (sharpening v312/v395)")

    pi = sp.pi
    Z2 = g_car - N_fam
    h_e8 = 2 * N_fam * g_car
    c3_sym = sp.Rational(1, 1) / (8 * pi)
    base_sym = sp.Rational(1, 1) / (6 * pi)
    dtop_sym = 48 * c3_sym ** 4

    V, E, F, naut = _icosahedron_counts()

    # 1. icosahedral counts are anchor-forced
    check("ICOSAHEDRAL COUNTS [E]: V=%d, E=%d=h(E8), F=%d triangular hyperedges, |Aut|=%d=|2I|; "
          "F=g_car(g_car-1)=%d, h=2 N_fam g_car=%d, 4h=|Aut|=%d (all from {g_car,N_fam}={5,3})"
          % (V, E, F, naut, g_car * (g_car - 1), h_e8, 4 * h_e8),
          V == 12 and E == 30 == h_e8 and F == 20 == g_car * (g_car - 1)
          and naut == 120 == 4 * h_e8)

    # 2. leading term exact identities
    via_c3 = sp.simplify(base_sym - sp.Rational(4, 3) * c3_sym)
    via_Fh = sp.simplify(base_sym - F / (4 * h_e8 * pi))
    via_aut = sp.simplify(base_sym - F / (naut * pi))
    via_ratio = sp.simplify(base_sym - (F / (g_car * N_fam)) * c3_sym)
    check("LEADING TERM EXACT [E]: phibase = 1/(6 pi) = (4/3) c3 = F/(4 h pi) = F/(|Aut| pi) "
          "= (F/(g_car N_fam)) c3 -- four equivalent forms, all 0 difference",
          via_c3 == 0 and via_Fh == 0 and via_aut == 0 and via_ratio == 0)

    # 3. F/h = |Z2|/N_fam and phibase = (F/h)/(4 pi) = c3 (g_car-1)/N_fam
    ratio_Fh = sp.Rational(F, h_e8)
    survival = sp.Rational(Z2, N_fam)
    via_Fh4pi = sp.simplify(base_sym - ratio_Fh / (4 * pi))
    via_gN = sp.simplify(base_sym - sp.Rational(g_car - 1, N_fam) * c3_sym)
    check("F/h = |Z2|/N_fam [E]: hyperedge/Coxeter ratio F/h = %s = |Z2|/N_fam = %s (the v327 "
          "survival ratio); phibase = (F/h)/(4 pi) = c3 (g_car-1)/N_fam"
          % (ratio_Fh, survival),
          ratio_Fh == survival and via_Fh4pi == 0 and via_gN == 0)

    # 4. Gauss-Bonnet consistency on S^2
    deficit = sp.Rational(V, 1) * (pi / 3)          # 12 vertices, 5 triangles => pi/3 each
    c3_gb = sp.Rational(1, 1) / (Z2 * 4 * pi)
    check("GAUSS-BONNET [E]: icosahedron vertex-deficit sum V*(2 pi - 5 pi/3) = 4 pi (sphere "
          "total curvature); c3 = 1/(|Z2| 4 pi) uses the same 4 pi boundary datum (v216/v342); "
          "phibase = c3 (g_car-1)/N_fam inserts one extra N_fam in the denominator vs c3",
          deficit == 4 * pi and sp.simplify(c3_gb - c3_sym) == 0
          and sp.simplify(base_sym - sp.Rational(g_car - 1, N_fam) * c3_sym) == 0)

    # 5. honest negative: puncture term
    p0 = float(phi0)
    pbase = float(phibase)
    dtop = float(dtop_sym)
    is_rational_edge = any(abs(dtop - a / b) < 1e-9 for b in range(1, 200) for a in range(1, b))
    check("HONEST NEGATIVE [E]: the puncture dtop = 48 c3^4 = %.2e is NOT a small rational "
          "hypergraph fraction; full phi0 = phibase + dtop = %.7f still has an analytic piece "
          "beyond the icosahedral leading term (v312 negative, puncture part unchanged)"
          % (dtop, p0),
          not is_rational_edge and abs(pbase + dtop - p0) < 1e-12 and dtop / pbase < 0.01)

    # 6. reading
    check("READING [C]: phibase is the tree-level retained seed at leading order -- NOT an "
          "independent rewrite injection but (hyperedge count F)/(g_car N_fam) x c3, with c3 "
          "the boundary Gauss-Bonnet datum; the puncture 48 c3^4 remains open [O]",
          via_ratio == 0 and via_gN == 0 and dtop_sym != 0)

    return summary("v396 phi0 leading term: phibase = F/(g_car N_fam) c3 = 1/(6 pi); F/h = |Z2|/N_fam; puncture 48 c3^4 still analytic")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

ddb5eb8f0a6fee2e6e60a51a095a37f05bda3a9d525bf27f99081dee28d842d0

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.