Follow the source behind the result.

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

"""v463 -- SEAM.EQUIV.UNIQ.01: the c=8 HOLOMORPHIC-UNIQUENESS pin -- the positive lever
that makes the IDENTIFICATION half of SEAM.EQUIV.01 classification-forced rather than
assumed (B of the A+B "actually close it" round).

The S3 closure (v458/v459) shows the continuum-EXISTENCE of the seam's scaling limit is
citable (MMST for the free SO(16)_1 part + AGT/AMT lattice-VOA for the 128-spinor
extension).  What stayed implicit was the IDENTIFICATION: "why is a chiral c=8 holomorphic
limit necessarily (E8)_1, and not some same-c rival?"  This module supplies the missing
positive lever: among ALL level-1 current algebras c=8 is NOT unique, but adding HOLOMORPHY
(a single sector, an SL(2,Z)-covariant character) forces the (E8)_1 character E4/eta^8 and
hence dim V_1 = 248, which selects E8 uniquely.  So once the limit is pinned as "c=8 +
holomorphic" (v376/v379 + the det K=1 discriminator v457), it IS (E8)_1 by CLASSIFICATION,
with no residual choice.

  [E] 1. c=8 IS NOT UNIQUE AT LEVEL 1.  the level-1 central charge is c = dim g/(1+h^v);
         c=8 has THREE simple solutions -- A_8 (su(9), dim 80), D_8 (so(16), dim 120) and
         E8 (dim 248).  So c=8 alone is necessary but NOT sufficient: two genuine gleich-c
         rivals exist (this is exactly why v457's det K=4 SO(16) rival is not a strawman).
  [E] 2. HOLOMORPHY FORCES dim V_1 = 248.  a holomorphic (single-sector) chiral CFT at
         c=8 has an SL(2,Z)-covariant character; the only weight-0 form with leading
         q^{-c/24}=q^{-1/3} and integer coefficients is j^{1/3}=E4/eta^8, whose q^1
         coefficient is 248.  The candidate character space is 1-dimensional (the vacuum
         a_0=1 fixes it), so dim V_1=248 is FORCED, not chosen.
  [E] 3. dim V_1 = 248 + LEVEL-1 c=8 => E8 UNIQUELY.  among simple Lie algebras only E8 has
         dim 248 (and 248/(1+30)=8 confirms level-1 c=8); the gleich-c rivals A_8 (80) and
         D_8 (120) are excluded by the forced 248.  So holomorphy promotes "c=8" to "E8".
  [E] 4. THE (E8)_1 TOWER = E4/eta^8.  the graded dimensions {1,248,4124,34752,213126,...}
         are exactly the E4/eta^8 q-coefficients (matches v377/v462), the unique holomorphic
         c=8 partition function.
  [C] 5. CLASSIFICATION CLOSES THE IDENTIFICATION.  Dong-Mason / Schellekens: the
         holomorphic VOA at c=8 is UNIQUE and equals the E8 lattice VOA V_{E8}; the even
         unimodular rank-8 lattice is unique = E8 (Mordell-Witt).  So "c=8 holomorphic
         limit = (E8)_1" is classification-forced.  Combined with v458 (free part citable)
         + v459 (spinor extension), the continuum/identification HALF of SEAM.EQUIV.01 is
         [C] conditional on named, hypothesis-audited published theorems -- leaving the
         REALISATION R1 (v464) as the sole structural input.  SEAM.EQUIV.01 stays [O] until
         R1 is discharged.

Exact (sympy/integer q-series + the Lie-algebra scan; mpmath cross-check E4/eta^8=j^{1/3});
Wolfram-mirrored.  Does NOT by itself close SEAM.EQUIV.01 (it closes the identification half).
"""
from math import comb

import mpmath as mp
import sympy as sp

from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8


def _e4_over_eta8_coeffs(ncoef=6):
    """Integer q-coefficients a_n of q^{1/3}*E4/eta^8 = sum a_n q^n (the (E8)_1 tower).

    E4 = 1 + 240 sum_{n>=1} sigma_3(n) q^n ;  eta^8 = q^{1/3} prod (1-q^n)^8.
    => E4/eta^8 = q^{-1/3} * [ E4 * prod (1-q^n)^{-8} ].
    """
    M = ncoef + 1
    sig3 = [0] + [sum(d ** 3 for d in range(1, n + 1) if n % d == 0) for n in range(1, M)]
    e4 = [1] + [240 * sig3[n] for n in range(1, M)]
    inv = [0] * M
    inv[0] = 1
    for n in range(1, M):                         # multiply by (1-q^n)^{-8}
        factor = [0] * M
        k = 0
        while n * k < M:
            factor[n * k] = comb(k + 7, 7)
            k += 1
        new = [0] * M
        for i in range(M):
            if inv[i]:
                for j in range(M - i):
                    new[i + j] += inv[i] * factor[j]
        inv = new
    out = [0] * M
    for i in range(M):
        for j in range(M - i):
            out[i + j] += e4[i] * inv[j]
    return out[:ncoef + 1]


def _simple_lie_data():
    """(label, rank, dim, dual_Coxeter h^v) for the simple Lie algebras."""
    data = []
    for n in range(1, 13):
        data.append((f"A{n}", n, n * (n + 2), n + 1))
    for n in range(2, 9):
        data.append((f"B{n}", n, n * (2 * n + 1), 2 * n - 1))
    for n in range(2, 9):
        data.append((f"C{n}", n, n * (2 * n + 1), n + 1))
    for n in range(3, 11):
        data.append((f"D{n}", n, n * (2 * n - 1), 2 * n - 2))
    data += [("G2", 2, 14, 4), ("F4", 4, 52, 9), ("E6", 6, 78, 12),
             ("E7", 7, 133, 18), ("E8", 8, 248, 30)]
    return data


def run():
    reset()
    print("v463 SEAM.EQUIV.UNIQ: the c=8 holomorphic-uniqueness pin -- holomorphy forces "
          "dim V_1=248 => E8, isolating it from its gleich-c rivals A_8/D_8")

    c_target = g_car + N_fam                                   # 8

    # ---- 1. c=8 is not unique at level 1 ----
    lie = _simple_lie_data()
    c8 = [(lab, dim) for (lab, rk, dim, hv) in lie
          if sp.Rational(dim, 1 + hv) == c_target]
    labels = {lab for lab, _ in c8}
    check("c=8 NOT UNIQUE AT LEVEL 1 [E]: c=dim/(1+h^v)=8 has THREE simple solutions -- "
          "A8 (dim 80), D8=so(16) (dim 120), E8 (dim 248); c=8 is necessary but NOT "
          "sufficient (the v457 det K=4 SO(16) rival is a real gleich-c competitor): %s"
          % sorted(c8),
          labels == {"A8", "D8", "E8"})

    # ---- 2. holomorphy forces dim V_1 = 248 ----
    coeffs = _e4_over_eta8_coeffs(6)
    check("HOLOMORPHY FORCES dim V_1=248 [E]: the only weight-0 form with leading "
          "q^{-1/3} and integer coeffs is j^{1/3}=E4/eta^8; its q^1 coefficient (the "
          "1-dim candidate space, vacuum a_0=1) is %d -- dim V_1 FORCED, not chosen"
          % coeffs[1],
          coeffs[0] == 1 and coeffs[1] == 248)

    # ---- 3. dim 248 + level-1 c=8 => E8 uniquely ----
    dim248 = [(lab, rk, dim, hv) for (lab, rk, dim, hv) in lie if dim == 248]
    e8_only = (len(dim248) == 1 and dim248[0][0] == "E8"
               and sp.Rational(248, 1 + 30) == c_target and rankE8 == 8)
    check("dim V_1=248 + LEVEL-1 c=8 => E8 UNIQUELY [E]: among simple Lie algebras only "
          "E8 has dim 248 (248/(1+30)=8); the gleich-c rivals A8 (80) and D8 (120) are "
          "excluded by the forced 248 -- holomorphy promotes 'c=8' to 'E8': %s" % dim248,
          e8_only)

    # ---- 4. the (E8)_1 tower = E4/eta^8 ----
    tower = [1, 248, 4124, 34752, 213126]
    check("(E8)_1 TOWER = E4/eta^8 [E]: graded dims {1,248,4124,34752,213126} are exactly "
          "the E4/eta^8 q-coefficients (matches v377/v462), the unique holomorphic c=8 "
          "partition function: %s" % coeffs[:5],
          coeffs[:5] == tower)

    # numeric cross-check E4/eta^8 = j^{1/3}
    mp.mp.dps = 40
    tau = mp.mpc(0, mp.mpf('1.3'))
    q = mp.e ** (2j * mp.pi * tau)
    E4 = 1 + 240 * mp.nsum(lambda n: (n ** 3) * q ** n / (1 - q ** n), [1, mp.inf])
    eta = q ** (mp.mpf(1) / 24) * mp.nprod(lambda n: 1 - q ** n, [1, mp.inf])
    jfun = E4 ** 3 / eta ** 24
    ident = mp.almosteq(E4 / eta ** 8, jfun ** (mp.mpf(1) / 3), 1e-25)
    check("E4/eta^8 = j^{1/3} [E]: the holomorphic c=8 character is the unique cube root "
          "of the modular j-function (numeric, |.|<1e-25)", bool(ident))

    # ---- 5. classification closes the identification ----
    verdict = (labels == {"A8", "D8", "E8"} and coeffs[1] == 248 and e8_only
               and coeffs[:5] == tower and ident)
    check("CLASSIFICATION CLOSES THE IDENTIFICATION [C]: Dong-Mason/Schellekens -- the "
          "holomorphic VOA at c=8 is UNIQUE = V_{E8} (E8 lattice VOA), and the even "
          "unimodular rank-8 lattice is unique = E8 (Mordell-Witt); so 'c=8 holomorphic "
          "limit = (E8)_1' is classification-forced. With v458 (free part) + v459 (spinor "
          "extension) the continuum/identification HALF of SEAM.EQUIV.01 is [C] conditional "
          "on named audited theorems; the REALISATION R1 (v464) is the sole remaining "
          "input. SEAM.EQUIV.01 stays [O] until R1 is discharged", verdict)

    return summary("v463 SEAM.EQUIV.UNIQ: c=8 holomorphic-uniqueness pin -- c=8 has three "
                   "level-1 candidates (A8/80, D8/120, E8/248) so c=8 is not sufficient; "
                   "holomorphy forces the character E4/eta^8 => dim V_1=248 => E8 uniquely "
                   "(248/(1+30)=8), tower {1,248,4124,34752,213126}, E4/eta^8=j^{1/3}. "
                   "Dong-Mason/Schellekens: holomorphic c=8 VOA unique = V_{E8}; so the "
                   "identification half of SEAM.EQUIV.01 is [C] (classification-forced), "
                   "leaving the realisation R1 (v464). SEAM.EQUIV.01 stays [O]")


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

970dae3399b411d143b0e02ed052e0b9fe91ededf092f7c23cd4c79265d0c58a

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.