A registered result in its source context

CELEST.WP5C.01

Original ledger classification: Formal/Identity [E] (exact construction/Gram/control arithmetic, integer/Fraction, deterministic; the compact anti-involution machine-determined; the full 9361-block level-2 Gram exact) + [C] (the named identifications: quasi-free extension; Kac positivity beyond level 2; module identification in the component route) + [O] (local normality, net structure/Haag duality, Costello-Li -- WP5d/e; SEAM.EQUIV.01 unchanged)

Original claim text

WP5c OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5c, the third constructive milestone of WP5 -- 'the GNS limit state'; WP5a/v497 made the boundary limit precise, WP5b/v498 built the deleting operator, WP5c constructs the STATE whose GNS kernel contains the null ideal): exact integer/Fraction arithmetic throughout (35 checks), no floats, deterministic. [E] CONSTRUCTION: the quasi-free family omega_w -- loop sector (r = 0) = the affine k=1 vacuum n-point functions via the v498 PBW engine and the MACHINE-DETERMINED compact anti-involution theta(e_a) = -e_{-a}, theta(h) = h (the UNIQUE sign that is an anti-automorphism on ALL 61504 basis pairs; contravariance <J^x_{-1} u, v> = <u, J^{theta x}_1 v> exact); radial sector (r >= 1) = oscillator pairings x^{w r} (the exact Gibbs regulator e^{-beta E_w}, kept x-adic; rational sample x = 1/2). [E] CONVERGENCE SHARP: loop entries w-independent; radial entries x^{w r} have x-adic order w r >= w, so omega_w = omega_inf mod x^{N+1} for ALL w >= N+1 (N = 8) and NOT yet at w = N (the r = 1 modes sit at order N exactly) -- the WP5a stabilisation threshold w = n+1 holds IDENTICALLY at the state level; positivity of every omega_w checked at the one-particle level (all 36 radial modes). [E] THE KERNEL (core result, fully machine): level-2 Fock = 31124 monomials in 9361 weight blocks (dim profile 164:1, 37:240, 7:2160, 1:6960); the FULL exact Gram computed block by block: level-1 rank 248 POSITIVE DEFINITE (240 root modes at norm k, Cartan block = the E8 Cartan matrix, det 1); level-2 rank 4124 EXACTLY (the preregistered target check), kernel 27000, every one of the 9361 blocks PSD; rank per Weyl orbit (norm 8/6/4/2/0) = (0,0,1,8,44), and (dim, rank, inertia) CONSTANT along every Weyl orbit (verified on all blocks, no orbit assumption); kernel = V(2 theta) WEIGHT BY WEIGHT (rank_lambda = dim_lambda - mult_lambda(V(2 theta)) for all 9361 weights, Freudenthal cross-check); component route: unique highest-weight vectors at (0, omega, theta, 2theta) with norms (3844, 49, 2, 0); direct ideal checks F_i|s> and F_j F_i|s> norm 0 and orthogonal to their full blocks; GNS graded dims (1, 248, 4124) = E4/eta^8 through q^2. [E] MU4 EQUIVARIANCE: frame map glue -> standard via the UNIQUE E8 diagram isomorphism; omega_w o clock = omega_w for ALL w (nonzero vacuum values only at total weight 0 = clock class 0); the clock unitary DESCENDS to GNS(omega_inf); GNS level-2 rank split by clock class = (1036, 1024, 1040, 1024) = Theta_Cj/eta^8 at q^2, level-1 split (60,64,60,64) -- the WP5a two-routes identity now holds at the STATE level. RESOLUTION OF WP5B S4.10: |s> IS the ZERO VECTOR of GNS(omega_inf) -- nothing needs to 'carry' it to q^2; the twisted two-C1-mode floor (6 quarters = q^{3/2}) concerns only pre-limit objects (GNS has no state at 6 quarters: 6 mod 4 != 0). [E] NEGATIVE CONTROLS: (a) k = 2: <s|s> = +4 (WP5b sign confirmed), all 240 norm-8 and all 6720 norm-6 monomials nonzero, dominant blocks FULL rank -- the GNS kernel contains NO level-2 ideal (the deletion is a k = 1 state-level fact); (b) k = 0: level-1 Gram identically 0 -- no current layer without the central extension; (c) D8/SO(16)_1 same pipeline: level-1 rank 120 PD, FULL level-2 Gram over 7380 monomials in 2705 blocks: rank 2076 = 7380 - 5304, kernel = V(2 theta_D8) weight by weight, PSD, lattice route Theta_D8/eta^8 = (1, 120, 2076) agrees; but h = (0, 1/2, 1, 1): the vacuum GNS is 1 of 4 blocks vs E8 = the FULL character -- ONE-BLOCK completeness of the limit state is E8/mu4-specific; (d) wrong family E'_w = m w + r: level-1 limit rank 0 (the 248 layer ERASED), u^4 survivor content 710955 != 248 -- the kill has teeth; (e) no damping: the w-uniform undamped vacuum keeps the full jet Fock, 897266 != 248; (f) CCR OBSTRUCTION: on a FIXED radial pairing c > 0 every state has omega(b b*) >= c (positivity) -- NO w-uniform state damps the radial modes: the contract's family formulation is NECESSARY, and the family EXISTS => the preregistered KILL is probed and NOT triggered. [C] NAMED IDENTIFICATIONS: the quasi-free/Gaussian extension of the positive one-particle data to the full *-algebra (standard); Kac positivity beyond level 2 (levels <= 2 machine-checked here); complete reducibility / multiplicity-freeness in the component route (the rank is [E], the module identification is the [C] part); k = 2 non-dominant blocks by Weyl-orbit rank constancy. [O] HONEST LIMIT: local normality of omega_inf, the net structure / Haag duality (WP5d), Costello-Li (WP5e), and the operator-algebraic closure of the radial contraction beyond the graded/x-adic topology stay OPEN. VERDICT: SUCCESS on the preregistered criteria at the graded-state level (target check dim GNS level-2 = 4124 met exactly). SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.

Original classification detail

WP5c [E]/[C]/[O]: the GNS limit state EXISTS and is explicit -- the quasi-free family omega_w (affine k=1 vacuum + x^{wr}-contracted radial oscillators; compact adjoint machine-determined on all 61504 pairs) is positive for every finite w, stabilises exactly at the WP5a threshold w = N+1, and its limit has GNS levels (1, 248, 4124): full 9361-block exact Gram, level-2 rank 4124 = chi_2, kernel 27000 = V(2 theta) weight by weight, every block PSD, rank table (0,0,1,8,44); the 248 layer survives (level-1 rank 248 PD); the clock descends to GNS with rank split (1036,1024,1040,1024) = the per-period theta-split; |s> is the GNS zero vector (WP5b S4.10 resolved at the state level); CCR obstruction: no w-uniform damping state -- the family is necessary AND exists (KILL not triggered); controls: k = 2 keeps everything, k = 0 has no current layer, D8 = one block of four, wrong family/no damping caught (710955/897266 != 248); open: local normality, net/Haag duality (WP5d), Costello-Li (WP5e); SEAM.EQUIV.01 stays [O]

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5A.01; CELEST.WP5B.01; SEAM.EQUIV.01; v128; v492; v496; v497; v498