A registered result in its source context

CELEST.WP5DB.01

Original ledger classification: Formal [E] (the exact algebra spans/indices, the PP identities with integer attainment, the Lambda^2-compound and parity machine identities, the index consistency) + Lattice/Numerical [L]/[N] (the deficit tables, the nome ladder, the power laws, the Klich-Levitov slope; Gaussian-exact, ED-validated) + [C] (the KLM dictionary: Longo-Xu bounded-deficit dichotomy, Longo split heredity, CHMP order parameters) + [O] (the continuum proof for the seam quotient net and the condensed (E8)_1 net -- Xu's orbifold theorem cited, not claimed; WP5e/Costello-Li; SEAM.EQUIV.01 unchanged)

Original claim text

WP5d-BETA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5d, beta stage -- 'split + strong additivity from the lattice'; the two remaining Kawahigashi-Longo-Mueger legs of complete rationality = split property + strong additivity + finite two-interval mu-index, the index being WP5d-alpha/v501): numpy/scipy Gaussian machinery + exact GF2/integer algebra (37 checks, deterministic, ~12 s); the U(1)/Dirac number-conserving subnet is the preregistered negative control -- the current-net analogue where strong additivity famously FAILS in the continuum (Buchholz-Schulz-Mirbach). [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 on four sizes; literal matrix rank 32/32); DISJOINT intervals give exactly HALF (rank 16/32, index 2 -- the missing sector is odd(x)odd, the single ln-2 bit of v501, localised at the split point); the U(1)/neutral algebras do NOT generate the union even WITH the shared site (gaps 2/10/52, GROWING -- the charge-transfer tower is missing): the exact lattice shadow of the current-net strong-additivity failure (infinite index). [N] ENTROPIC WITNESS: the fermionic touching MI diverges with exact (c/3) ln 2 increments per doubling (0.115525, dev 1.2e-6 -- pure UV point-gap, present for EVERY net, hence NOT the discriminator); the honest discriminator is BOUNDED vs DIVERGENT sector-sum deficit: the Z2/orbifold deficit rises 0.4045 -> 0.5899 (N = 64 -> 4096), stays < ln 2, with (ln 2 - Delta_2) ~ N^(-p), p = 0.2444 vs 2 Delta_mu = 1/4 (the Ising disorder-operator dimension h_mu = 1/16 per chirality); split-point scan exactly reflection symmetric (3.9e-14); ED vN deficits same direction; gap continuity to the v501 x -> 1 endpoint. HONEST NOTE: the naive 'defect -> 0' expectation is FALSE at the sharp lattice split -- bounded <=> finite index (Longo-Xu) is the correct statement. U(1) CONTRAST: D_U1(touch) BURSTS the finite-group ceiling ln 2 from L = 128 on and keeps growing as (1/2) ln Var Q_A with Var Q_A = (1/pi^2) ln L + const (Klich-Levitov slope 0.10134 vs 1/pi^2 = 0.10132): infinite index, no saturation. [N] SPLIT PROPERTY (nuclearity witness): the singular values of the off-diagonal covariance block for DISJOINT intervals fall exponentially in k with rate = the elliptic nome pi K(1-x)/K(x) to 1.3-2.0% (d = 16/64/256/512 at N = 4096: rates 2.29/3.24/4.82/5.81); sigma_1 ~ x^0.5044 vs 1/2 (the dimension-1/2 Majorana two-point law); trace norm summable and falling (Buchholz-Wichmann nuclearity proxy); the dimerised control decays exponentially per site (a gapped seam splits trivially). [E] ORBIFOLD INHERITANCE, EXACT AT MACHINE LEVEL: P_A maps the coupling block C -> -C (sigma_k identical, 8.3e-17) and the even-bilinear coupling Gram is the second compound Lambda^2 C with singular values {sigma_a sigma_b} (2.8e-14) -- the SAME ladder at doubled rate; the orbifold Renyi-2 MI is < 3% of the fermionic one and falls with x (split heredity under finite index, Longo, [C]). [E] PIMSNER-POPA / LONGO-REHREN, EXACT: E(a) - a/2 = PaP/2 holds IDENTICALLY (the PP inequality with lambda = 1/2 = 1/[F : F_even], one line, exact); ATTAINMENT in exact integers: all 16384 (even, odd) monomial pairs at N = 8 Majoranas -- 8192 extremal with M^2 = 4, Tr M = 0, PMP = -M exactly, 0 violations; all 2048 hermitianised odd monomials at N = 12 (2048/2048); random pencil sweep min lambda = 0.5000000000; the two-interval group {1, P_A, P_B, P_AB} gives lambda = 1/4 = 1/mu exactly; INDEX CONSISTENCY, TWO ROUTES: exp(Delta_inf) = 2.000000000 = 1/lambda_PP (the v501 entropic route, |Delta - ln 2| = 1.1e-15 at N = 512) and exp(2 Delta_inf) = 4 = 1/lambda_E4 = mu -- the Longo-Rehren index witnessed entropically AND operator-algebraically, independently; U(1) control: lambda = 1/(m+1) -> 0 (infinite index, the operator-algebraic twin of the entropic divergence). [E/N] NEGATIVE CONTROLS + FINITE-SIZE HONESTY: dimerised/trivial phase -- every indicator trivial (touching deficit N-independent to 3.6e-15, MI bounded, sigma_1 exponential); U(1) small-L honesty (non-monotone below L ~ 16, documented); Z2 finite-size honesty: at N = 4096 the touching deficit still sits 14.9% below ln 2 (the N^{-1/4} Ising approach is SLOW -- the verdict rests on the bound + fitted exponent, stated as witness, not proof); wrong prescription {1, P_AB} on the touching pair: deficit EXACTLY 0 (machine identity) -- the bookkeeping is falsifiable. [C] the lattice -> continuum KLM dictionary: bounded deficit = finite index = complete rationality (Longo-Xu); split heredity under finite index (Longo); the entropic order parameters (CHMP); the nome ladder = the free-fermion two-interval modular package. [O] the continuum proof: the theorem 'finite-group orbifolds of completely rational nets are completely rational (hence split + strongly additive)' is XU'S THEOREM (algebraic orbifold CFTs) -- CITED, NOT CLAIMED; the continuum proof for THE seam quotient net itself and for the interacting condensed (E8)_1 net stays open (WP5e / Costello-Li); SEAM.EQUIV.01 stays [O], untouched; no marker moves. VERDICT: with v501, all THREE KLM ingredients of complete rationality -- split, strong additivity, finite mu -- are witnessed on the lattice for the orbifold prescription; the U(1) control fails both new legs exactly as the continuum current net does.

Original classification detail

WP5d-beta [E]/[L]/[N]/[C]/[O]: split + strong additivity witnessed on the seam lattice edge for the orbifold prescription -- strong additivity EXACT with the shared boundary Majorana (GF2 spans full, rank 32/32; disjoint index exactly 2 = the v501 ln-2 bit), the entropic touching defect BOUNDED < ln 2 with the Ising 1/4-exponent approach (p = 0.2444) while the preregistered U(1) control diverges ((1/2) ln Var Q, Klich-Levitov pinned): bounded-vs-divergent is the lattice discriminator; split via the elliptic-nome sigma_k ladder (1.3-2.0%, sigma_1 ~ x^{1/2}, trace norm summable) with EXACT orbifold inheritance (C -> -C + Lambda^2 C compound, Longo heredity); Pimsner-Popa E(a) - a/2 = PaP/2 identically (lambda = 1/2 = 1/[F:F_even], integer attainment 16384 + 2048 with 0 violations), lambda_E4 = 1/4 = 1/mu, index consistency exp(Delta) = 2 = 1/lambda_PP two routes; controls: dimerised trivial, wrong prescription exactly 0, finite-size honesty documented; with v501 all THREE KLM ingredients of complete rationality witnessed on the lattice; continuum uplift stays [O] (Xu's theorem cited, not claimed; WP5e/Costello-Li); SEAM.EQUIV.01 stays [O]

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5D.01; SEAM.CCC.01; SEAM.MU.01; SEAM.EQUIV.01; v367; v450; v489; v490; v501