A registered result in its source context

CELEST.WP5D.01

Original ledger classification: Formal [E] (the exact arithmetic: Cartan dets, KLM/Longo-Rehren quotients, Sigma d^2, theta_v at nu = 16, machine identities) + Lattice/Numerical [L]/[N] (the Gaussian lattice witnesses, ED-validated to 1e-15: extensivity, the ln 2 plateau, the duality-asymmetry witness) + [C] (the continuum dictionary offset = ln index -- KLM/Longo-Xu/CHMP -- and the interacting condensation step) + [O] (continuum Haag duality / local normality / split of the quotient net = WP5d-beta; Costello-Li = WP5e; SEAM.EQUIV.01 unchanged)

Original claim text

WP5d-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5d, alpha stage -- 'the two-interval index from the lattice'; the local-net question): numpy/scipy Gaussian machinery + exact Fractions (39 checks). Roadmap discriminator: (E8)_1 => mu = 1 (Haag-dual two-interval net, ONE sector); SO(16)_1 => mu = 4; preregistered KILL = 'the two-interval mu-offset does NOT vanish after the condensation'. [E-machine]/[L] SETUP: the v367 seam edge as 16 chiral Majorana layers (c_- = 8 = g_car + N_fam, v489), faithful 1d reduction = the critical Majorana ring (v450 style, NS closure, c = 1/2 per layer); Peschel-exact; three two-interval prescriptions (fermionic rho / orbifold sector sum {1, P_A, P_B, P_AB} / dual algebra {1, P_AB}); the superselection identity [rho, P_AB] = 0 machine-exact; EVERY Gaussian formula (covariance round trip, Peschel entropies, trace formula, interval parity = Pfaffian, disjoint Renyi-2 deficit) validated against exact diagonalisation to <= 1.5e-15. [L/N] FERMIONIC MI EXTENSIVE (the mu = 1 reference): I(x) = -(c/3) ln(1-x) with c fitted 0.5000; max |F(x)| = 5.97e-5 at N = 512 over the full x grid, |F| falling monotonically to 1.3e-6 at N = 1024 -- the lattice witness of Casini-Huerta extensivity. [L/N] THE MU-OFFSET (core): Renyi-2 exactly Delta_2 = -ln[(1+R)/2] with R = Tr(rho P rho P)/Tr rho^2; the 16-layer seam carrier (diagonal Z2) saturates ln 2 to MACHINE PRECISION (|Delta_2 - ln 2| = 1.1e-15 at N = 512, 0 at N = 1024/2048; single-layer R falls as N^-0.500, Ising disorder decay); ED (N = 12..24) confirms vN and Renyi-3/4 deficits track it (not a Renyi-2 artefact); index reading [F : F_even] = exp(Delta_inf) = 2.000000 => mu_gauged = exp(2 Delta_inf) = 4.000000 = v490's mu_pre (TWO independent lattice witnesses: parity census there, entropic two-interval offset here); gauging every layer separately costs 16 ln 2 -- the offset measures ln[index of the subnet], not a universal blob. [L/N] HAAG-DUALITY INDICATOR: fermionic complementarity S(E) = S(E') to 8.2e-12 (graded-duality proxy); the orbifold BREAKS it (D(E) - D(E') = 0.427 at a = 64 -- the direct lattice witness that Even(E)-prime != Even(E-prime)); the complementary-pair sum is exchange-symmetric (dev 4.3e-14) and bounded by ln 4 = ln mu(SO(16)_1), max 1.2637 at the self-dual x = 1/2, -> ln 2 at the squeezed ends: the ln 4 KLM budget is a DOUBLE-LIMIT statement (sum of the two one-sided ln 2 saturations), NOT a fixed-x identity -- the first naive fixed-x check was WRONG and was replaced by the honest budget + asymmetry pair; the dual-algebra prescription {1, P_AB} is machine-identically defect-free. [E] CONDENSATION ARITHMETIC, LATTICE-ANCHORED: det Cartan(D5) * det Cartan(A3) = 4 * 4 = 16 = mu(carrier); KLM/Longo-Rehren quotients mu/|L|^2: 16/4^2 = 1 (one-step mu4 glue) and 16/2^2 = 4 -> 4/2^2 = 1 (via SO(16)_1), both routes exact Fractions; Sigma d_i^2 cross-checks (Ising 1+1+2 = 4, SO(16)_1 four abelian sectors = 4, (E8)_1 one sector = 1); theta_v = e^{2 pi i nu/16} = 1 EXACTLY at nu = 2 c_- = 16 (bosonic, condensable, v490), rivals nu = 1, 2, 8 give theta_v != 1. THE ANCHORED CHAIN: mu = 1 (graded fermion, MEASURED offset 0) -> 4 (gauged, MEASURED ln 2 offset + v490 parity census) -> 1 (theta_v = 1 extension, ARITHMETIC [C]: the condensed (E8)_1 net is interacting, not free-fermion computable -- v490's honest fence): the lattice pins BOTH measurable ends, the arithmetic closes the middle. [E/N] NEGATIVE CONTROLS: (a) nu = 1 single Majorana: offset non-trivial (Delta_2 = 0.6364 at N = 2048, rising toward ln 2) and theta_v = e^{i pi/8} != 1 -- NO bosonic condensation channel, the offset NOT removable: the KILL discriminator has teeth; (b) trivial dimerised phase: c = 0.00000, MI <= 1.3e-7, 16-layer deficit <= 2.6e-8 -- the ln 2 is a property of the CRITICAL seam edge, not of the prescription; (c) wrong sector sum ({1, P_AB} only, 2 of 4 structures): offset EXACTLY 0 instead of ln 2 -- measurably wrong bookkeeping, as preregistered. [C] the continuum dictionary 'offset = ln index' (KLM two-interval index; Longo-Xu; Casini-Huerta-Magan-Pontello entropic order parameters) and the condensation step itself (theta_v = 1 simple-current extension). [O] continuum Haag duality, local normality, split/strong additivity of the quotient net (WP5d-beta, running as exploration), Costello-Li (WP5e). VERDICT: the mu = 1 indicator chain PASSES; the preregistered KILL is NOT triggered. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.

Original classification detail

WP5d-alpha [E]/[L]/[N]/[C]/[O]: the two-interval mu-index measured entropically on the seam lattice edge -- fermionic MI extensive (mu = 1 reference, c fit 0.5000, |F| -> 0), the sector-summed orbifold pays exactly one classical bit (16-layer carrier ln 2 plateau to 1.1e-15 at N = 512), [F : F_even] = 2, mu_gauged = 4 = the v490 census (two independent lattice witnesses); complementary-pair budget <= ln 4 (double-limit statement) + the orbifold duality-failure witness; condensation arithmetic anchored at both measured ends: 16 -> 4 -> 1 (KLM/Longo-Rehren, Sigma d^2, theta_v = 1 exactly at nu = 16, rivals != 1) => mu = 1 after condensation, preregistered KILL not triggered; controls: nu = 1 non-removable, trivial phase zero, wrong sector sum loses the full ln 2; open: WP5d-beta (continuum Haag duality / local normality / split, exploration running) + WP5e (Costello-Li); SEAM.EQUIV.01 stays [O]

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5C.01; SEAM.CCC.01; SEAM.MU.01; SEAM.EQUIV.01; v367; v378; v450; v489; v490