Celestial wp5d klm completeness
v504_celestial_wp5d_klm_completeness.py · frontier
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CELEST.WP5DB.01: WP5d-beta of the celestial-holographic research contract CELEST.SEAM.01 -- 'split + strong additivity from the lattice', the two remaining KLM legs of complete rationality (split + strong additivity + finite mu-index; the index is v501). numpy/scipy Gaussian machinery + exact GF2/integer algebra; the U(1)/Dirac number-conserving subnet = the preregistered negative control. [E] STRONG ADDITIVITY ALGEBRAICALLY EXACT: 'touching' = shared boundary Majorana => Even(A) v Even(B) = Even(A u B) EXACTLY (GF2 spans full 64/64, 256/256, 512/512, 1024/1024; matrix rank 32/32); disjoint exactly half (rank 16/32, index 2 -- the missing sector odd(x)odd = the ln-2 bit of v501, localised at the split point); U(1)/neutral algebras do NOT generate the union even WITH the shared site (gaps 2/10/52 growing). [N] ENTROPIC WITNESS: fermionic touching MI diverges with exact (c/3)ln2 increments (0.115525, dev 1.2e-6 -- pure UV reference); the Z2/orbifold sector-sum deficit is BOUNDED < ln 2, monotone, (ln2 - Delta_2) ~ N^(-p) with p = 0.2444 vs 2 Delta_mu = 1/4 (Ising disorder exponent); split-point scan reflection-symmetric 3.9e-14; ED vN same direction; gap continuity to the v501 x -> 1 endpoint; honest note: 'defect -> 0' would be FALSE at the sharp lattice split -- bounded <=> finite index (Longo-Xu) is the correct statement; U(1) CONTRAST: D_U1 bursts ln 2 from L = 128 and grows as (1/2) ln Var Q_A (Klich-Levitov 0.10134 vs 1/pi^2 = 0.10132). [N] SPLIT/NUCLEARITY: sigma_k of the off-diagonal block fall exponentially with rate = elliptic nome pi K(1-x)/K(x) to 1.3-2.0% (d = 16/64/256/512: rates 2.29/3.24/4.82/5.81); sigma_1 ~ x^0.5044 (vs 1/2, the dimension-1/2 two-point law); trace norm summable, falling (Buchholz-Wichmann proxy). [E] ORBIFOLD INHERITANCE EXACT: P_A: C -> -C (sigma_k identical 8.3e-17); the even-bilinear coupling Gram = the second compound Lambda^2 C with singular values {sigma_a sigma_b} (2.8e-14) -- same ladder, doubled rate; orbifold MI < 3% of the fermionic, falling (Longo heredity [C]). [E] PIMSNER-POPA: E(a) - a/2 = PaP/2 IDENTICALLY => lambda = 1/2 = 1/[F:F_even] EXACT; attainment in integers (all 16384 monomial pairs N = 8: 8192 extremal with M^2 = 4/Tr M = 0/PMP = -M, 0 violations; N = 12: 2048/2048; random pencil min lambda = 0.5000000000); two-interval group lambda = 1/4 = 1/mu exact; INDEX CONSISTENCY of two routes (exp(Delta) = 2.000000000 = 1/lambda_PP; exp(2 Delta) = 4 = 1/lambda_E4 = mu). [E/N] NEGATIVE CONTROLS: U(1) PP lambda = 1/(m+1) -> 0 (infinite index); dimerised trivial (deficit N-independent 3.6e-15, sigma_1 exponential); U(1) small-L honesty; Z2 finite-size honesty (14.9% below ln 2 at N = 4096 -- witness, not proof); wrong prescription {1, P_AB}: deficit exactly 0. [C] the lattice -> continuum KLM dictionary (Longo-Xu, CHMP, Longo heredity). [O] the continuum theorem 'finite-group orbifolds of completely rational nets are completely rational' is XU'S THEOREM -- cited, not claimed; the proof for the concrete seam quotient net and the interacting condensed (E8)_1 net stays WP5e/Costello-Li; SEAM.EQUIV.01 stays [O]. VERDICT: with v501, all THREE KLM ingredients of complete rationality are witnessed on the lattice.
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