A registered result in its source context

CELEST.WP4.01

Original ledger classification: Formal/Identity [E] (exact counting/character/theta arithmetic, integer/Fraction/sympy; mismatch localisation with exact cross-multiplication) + [C] (literature gradings CP Table 1 + Sym^2(248) decomposition; the boundary-limit reading, verdict B(ii)) + [O] (the constructive limit -> WP5; SEAM.EQUIV.01 unchanged)

Original claim text

WP4 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 4, verdict B(ii); the 'type mismatch, head-on'): the (E8)_1 character E4/eta^8 against the glue-equivariant E8[C^2] S-algebra. Exact integer/Fraction/sympy arithmetic (25 checks). [E] BASE: v492 replication (240 = 52+64+60+64, dims with Cartan (60,64,60,64)); equivariant tower D[d] = 60(d+1) even / 64(d+1) odd; cumulative generator count QUADRATIC -- 31 s^2 + 92 s + 60 (even) / 31 s^2 + 94 s + 63 (odd) exactly, leading 31 = k + h_vee (arithmetic note); [C] CP Table 1 (arXiv:2201.02595): J[m,n] carries 2d spin 1 - (m+n)/2, UNBOUNDED BELOW along the tower => no rational VOA (L0 bounded below) contains the towers as graded pieces. [E] EQUIVARIANT HILBERT SERIES: chi_sp(t) = (60 + 128t + 60t^2)/(1-t^2)^2 (palindromic numerator, value 248 at t = 1); ALL mu4 collapses w in {1,-1,i,-i} and the SU(2)_+ primary slice (60+64t)/(1-t^2) fail to start (1, 248, ...) -- the 'row sum' route fails at the first coefficient. [E] CHARACTER SIDE: chi = E4/eta^8 = (1, 248, 4124, 34752, 213126, 1057504, 4530744) exact through q^6; machinery cross-check q j(tau) = E4^3/eta^24 = (1, 744, 196884, 21493760); theta split (shell 1 = (52,64,60,64)), sector currents (60,64,60,64), glue-diagonal h = (0,1,1,1) INTEGER (v125 isotropy = locality); Sugawara c = 248/31 = 8 = rank = 2|mu4| = g_car + N_fam. [E] BRIDGES: zero-mode slice D[0] = 60 = the C0-sector current count EXACTLY; the single-line jet slice z1^d cycles (60,64,60,64) sector by sector; BOUNDARY/PERIOD LIMIT: one full mu4 period of loop energies carries dim g_(n mod 4) = (64,60,64,60), sum = 248 EXACTLY -- the single-particle level of the 248q stage; 248 is NOT a jet-tower dimension (d <= 100) -- only the period collapse produces it; BUT the loop Fock at integer level 1 counts 897266 >> 248: the rational truncation (null ideal) is NOT in the jet/loop algebra -- it must come from a limit. [E] MISMATCH LOCALISED THREE WAYS: (a) GRADING (spin unbounded below; 248 no tower dimension); (b) GROWTH: jet Fock f = (1, 128, 8436, 381056) vs chi = (1, 248, 4124, 34752) -- TWO-SIDED failure (f1 < chi1 undercount, f2 > chi2 overcount), f_n/chi_n strictly increasing n = 1..12 (exact cross-multiplication); Meinardus poles s = 2 vs s = 1 => growth exponents n^{2/3} vs n^{1/2}, asymptotically DISJOINT counting types; (c) TRUNCATION: free-current level-2 count 31124 = 248 + Sym^2(248); the character keeps 4124 = 1 + 248 + 3875; the deficit is EXACTLY 27000 = 30^3 = h_vee^3 -- the (E8)_1 null ideal deletes the 27000 in Sym^2(248) = 1 + 3875 + 27000 ([C] standard) with NO analogue in the jet algebra (level 1 agrees: 248 = 248). [E] KILL-TEST K4 EVALUATED, three-way: (i) exact realisation NO; (ii) boundary-limit realisation PARTIAL YES -- current stratum, period sum 248, integer glue weights and theta split hold exactly; the missing piece is precisely the rational truncation, which must come from a scaling/compactification limit -- the SAME structure as the MMST scaling-limit route of SEAM.EQUIV.01 (v336/v449: the rational net exists in the LIMIT of the lattice collar, not in the pre-limit algebra); (iii) the hard mismatch is exactly located; K4 does NOT degrade the contract to 'E8 is an admissible celestial gauge algebra' (the sector arithmetic holds exactly). [C] VERDICT B(ii): the (E8)_1 character is a BOUNDARY/LIMIT SHADOW of the equivariant S-algebra, not a conformal block of it in the bulk (jet) grading -- WP4 sharpens into a statement of SEAM.EQUIV.01 type (rational structure emerges only in a limit). [O] the constructive boundary limit itself (which limit, which topology, convergence) stays OPEN and belongs to WP5/SEAM.EQUIV.01; SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.

Original classification detail

WP4 [E]/[C]/[O]: the (E8)_1 character is NOT a conformal block of the E8[C^2] S-algebra in its own jet grading -- mismatch localised three ways (spin unbounded; growth n^{2/3} vs n^{1/2}; level-2 null ideal 27000 = h_vee^3 absent from the jets); BUT the boundary/period reading holds exactly at the current stratum (zero modes 60, jet slice (60,64,60,64), mu4 period sum 248, glue weights (0,1,1,1) integer) -- the MMST scaling-limit shape of SEAM.EQUIV.01; K4 fires only against the exact reading; verdict B(ii); constructive limit -> WP5; SEAM.EQUIV.01 stays [O]

Dependencies as recorded

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