A registered result in its source context

CELEST.WP5A.01

Original ledger classification: Formal/Identity [E] (exact stabilisation/derivation/control arithmetic, integer/Fraction; Freudenthal + Weyl + peeling machinery unit-tested) + [C] (the named identifications: radial decoupling = collar limit; Kac singular vector via counting consequence) + [O] (the operator question WP5b-e; SEAM.EQUIV.01 unchanged)

Original claim text

WP5a OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5a, the first constructive milestone of WP5 -- 'the null ideal from the limit'; the v496 boundary-limit shadow made precise): exact integer/Fraction arithmetic throughout (34 checks), no floats. [E] STABILISATION THEOREM: a one-parameter family chi_w of graded Fock characters on the CHIRAL jet generators (loop energy m = p-q >= 1, radial number r = min(p,q); the m = 0, r >= 1 invariant-ring tower carries g_0), generator energy E_w = m + w*r in quarter units (u = q^{1/4}). (i) w = 2 IS the chiral jet grading (E_2 = p + q identically; generator counts 64,120,128,180,192,240,256,300 per degree); (ii) the u^n coefficient of chi_w equals the quarter-moded loop Fock for ALL w >= n+1 and is STRICTLY larger for every w <= n (verified n <= 8, w <= 10) -- the v496 'boundary-limit shadow' is the precise coefficientwise limit epsilon = 1/w -> 0 of a family CONTAINING the jet grading, with explicit threshold w = n+1; (iii) the limit does NOT produce the character (897266 >> 248 at integer level 1): limit and quotient are two separate, honestly typed steps; convention clarified: 897266 = quarter-moded loop Fock at u^4 = INTEGER level 1 (v496 S4.5). [E] NULL IDEAL DERIVED FROM ROOT DATA (not cited): Freudenthal multiplicity machinery (unit-tested on the adjoint: zero-weight mult 8, dim 248 by Weyl = orbit sum), dominant-weight tally of Sym^2(248) (30876 pairs, zero-weight mult 156) and generic character peeling DERIVE Sym^2(248) = V(2 theta) + V(omega) + V(0) = 27000 + 3875 + 1 (peeling residual exactly zero; independent Weyl-orbit cross-check); 27000 = 30^3 = h_vee^3 EXACTLY; C2 = (124, 96, 0) with 124 = C2(27000), Dynkin indices 6750/750 integers; level-1 integrability <2 theta,theta_vee> = 4 > 1 and <omega,theta_vee> = 2 > 1 => the only level-1 primary is the vacuum ([C] Kac); the maximal submodule of the level-1 vacuum Fock is generated by the level-2 singular vector of weight 2 theta ([C] Kac, used only through its counting consequence): level-2 quotient 31124 - 27000 = 4124 = 1 + 248 + 3875 = chi_2 EXACTLY. [E] TWO ROUTES, ONE NUMBER: the mu4 theta-split sector characters Theta_Cj/eta^8 (v492 S4 machinery, the boundary/twisted presentation) summed at q^2 = (1036,1024,1040,1024) give the SAME 4124 -- the untwisted null-ideal quotient and the mu4-sector lattice sum count the same level-2 space; at q^3 the sum is 34752 = chi_3; ideal bookkeeping beyond level 2 typed [C] (ideal series (0,0,27000,2600000,...) nonnegative, strictly increasing through q^12). [E] NEGATIVE CONTROLS: (a) SO(16)_1 through the SAME pipeline (D8 frame): Sym^2(120) = 5304 + 1820 + 135 + 1 derived (FOUR components vs E8's three); 5304 != h_vee(so16)^3 = 14^3 = 2744 -- the h_vee^3 identity is NOT generic; quotient 7380 - 5304 = 2076 = Theta_D8/eta^8 at q^2 (independent lattice route: the pipeline is the correct general level-1 vacuum recipe, validated on a second algebra); BUT the D8 block weights are (0, 1/2, 1, 1) -- h = 1/2 non-integer, the four blocks can NOT fuse into one local character (four-block theory) vs E8's integer (0,1,1,1) fusing into E4/eta^8: the procedure is not trivially projective; (b) false periodisation: period sums over P = 1, 2, 8 quarter-steps give 64, 124, 496 -- only P = 4 = |mu4| gives 248; (c) swapped glue (60/64 exchanged): the zero-mode anchor breaks (D[0] = 64 != 60 = lattice C0 currents). [C] NAMED IDENTIFICATIONS: 'radial decoupling = MMST collar thickness -> 0' (the v336/v449 lattice-collar route has the SAME two-step shape: pre-limit overcounting algebra, rational net only in the limit) is a named identification, not a computed isomorphism; the Kac singular-vector statement enters only through its counting consequence. [O] HONEST LIMIT (= the rest of WP5): the limit does NOT generate the truncation -- the singular vector as an OPERATOR in the limit net stays open (WP5b); the GNS limit state (kernel containing the ideal) and local normality / Haag duality of the limit net stay open (WP5c/WP5d); convergence topology beyond coefficientwise stays open; the celestial route now has the SAME two-step shape as MMST (limit + maximal ideal), with the ideal quantitatively fixed. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.

Original classification detail

WP5a [E]/[C]/[O]: the boundary limit is a precise coefficientwise limit -- chi_w family (E_w = m + w r, quarter units) contains the chiral jet grading at w = 2 and stabilises to the quarter-moded loop Fock at threshold w = n+1 (exact, n <= 8, w <= 10); the limit does NOT produce the truncation (897266 >> 248) -- but the ideal SIZE is DERIVED from root data (Sym^2(248) = 27000 + 3875 + 1, 27000 = h_vee^3, Freudenthal + Weyl + peeling) and the quotient 31124 - 27000 = 4124 = the independent mu4 theta-split sector sum; SO(16)_1 negative control (four blocks, 5304 != 14^3, h = 1/2 breaks fusion); only P = |mu4| = 4 gives 248; open: singular vector as operator (WP5b), GNS limit state (WP5c), local normality/Haag duality (WP5d), Costello-Li uplift (WP5e); SEAM.EQUIV.01 stays [O]

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP4.01; SEAM.EQUIV.01; v125; v128; v336; v377; v449; v492; v496