TFPT scientific source excerpt
Original repository path: verification/v983_simple_current_generator.py
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Inclusive original lines: 9-52
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000009 | verifies EXACTLY, at the lattice/VOA-arithmetic level, that the candidate
000010 | 
000011 |     lambda = (omega_s, omega_f)   (D5 spinor weight, A3 fundamental weight)
000012 | 
000013 | is that generator, in explicit coordinates (omega_s = (1/2)^5 in R^5,
000014 | omega_f = (3/4, -1/4, -1/4, -1/4) in R^4, L0 = D5 (+) A3):
000015 | 
000016 |   [E] 1. NORMS: |omega_s|^2 = 5/4, |omega_f|^2 = 3/4, |lambda|^2 = 2, so the
000017 |         conformal weight h_lambda = |lambda|^2/2 = 1 is an INTEGER -- the
000018 |         vertex operator Y(e^lambda, z) has trivial self-monodromy (the
000019 |         arithmetic precondition for a bosonic simple current).
000020 |   [E] 2. ORDER FOUR: 4*lambda in L0 while lambda, 2*lambda are NOT in L0 --
000021 |         the class [lambda] generates the diagonal Z4 in the discriminant
000022 |         group Z4 x Z4 of L0.
000023 |   [E] 3. EVEN + UNIMODULAR: lambda . L0 subset Z (checked on a generator
000024 |         set), so every coset L0 + k*lambda has even norms (isotropic glue
000025 |         group); det(L0) = 4*4 = 16 and the index-4 extension gives
000026 |         det L = 16/4^2 = 1.  Even + unimodular + rank 8 => L ~= E8
000027 |         (classification, [C] cited; the explicit E8 certificate is v1).
000028 |   [E] 4. COSET ROOT CENSUS (the fusion-power decomposition made exact):
000029 |         norm-2 vectors per coset = [52, 64, 60, 64] for k = 0..3 (total
000030 |         240); with the rank-8 Cartan sector this is exactly
000031 |         248 = (45 + 15) + (16, 4) + (10, 6) + (16bar, 4bar); the odd
000032 |         fusion powers k = 1, 3 carry 64 + 64 = 128 = the FULL missing
000033 |         spinor sector -- one field generates all four cosets.
000034 |   [E] 5. KILL TESTS (a wrong glue must not close): the vector-class glue
000035 |         (v, omega_f) has norm 7/4 and the mismatched glue (omega_s,
000036 |         omega_2) has norm 9/4 -- both NOT in 2Z, so the glued lattice is
000037 |         NOT even and cannot be E8.
000038 |   [E] 6. HOLOMORPHY SHADOW (sector-count arithmetic of the N4 residual):
000039 |         the glued lattice has det 1 => a SINGLE irreducible sector
000040 |         (mu-index 1 at the lattice level), while the rival D8 (SO(16)_1)
000041 |         has det(Cartan D8) = 4 => four sectors.  This is the lattice
000042 |         shadow of the holomorphy selector, NOT the operator-algebraic
000043 |         statement.
000044 | 
000045 | HONEST SCOPE (firewall): everything here is exact lattice arithmetic in
000046 | explicit rational coordinates -- the ARITHMETIC half of the contract.  The
000047 | analytic half stays [O]: strong convergence of the finite-collar
000048 | implementers Psi_{lambda,N} (Lemma S1/S2 type: trace-class covariance
000049 | convergence + Shale-Stinespring), the crossed-product/scaling-limit
000050 | exchange (Lemma S3), and the operator-algebraic holomorphy selector
000051 | (Lemma S4 / N4).  No net-level, no 4D and no status-marker claim is made;
000052 | 1+1D in and out per the contract.