A registered result in its source context

CELEST.WP5E.DELTA2.01

Original ledger classification: Formal/Identity [E] (exact Hom-table/certificate arithmetic: g0 = d5 (+) a3 without u(1), sector factorisation into minuscule orbits, innerness theorem at arities 2 and 3, bilinear table (2, 1, 1, 1), trilinear neutral dims (3, 2, 2, 1, 1), unique symmetric survivor = su(4) d-symbol, Q_dd = (1/60)(T3 - P3/4) two routes, rank-3/4 certificate psi(A_fix) = 64, relaxed solution (-1, 8, 2; 1920), SO(16)/false-g0/false-sector controls -- sympy/Fraction exact, Wolfram-mirrored) + [C] (the number fences 64 = dim g1 = 2^6, 1920 = |W(D5)| = 8 x 240; the |W(D5)| reading typed look-elsewhere-loaded and convention-contingent by v513) + [O] (the arity >= 4 ledger, the physical justification of the cubic GS term, ch2-naturalness for cubic vertices UNDECIDED; delta-1 UNDECIDED at exploration level at promotion time; since DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, tension with v516 stated); SEAM.EQUIV.01 unchanged; CELEST.SEAM.01 stays Open)

Original claim text

WP5e-DELTA-2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, delta-2 stage -- 'the full-tensor ledger', the named escape route of the v508 exchange no-go, intermediate verdict, exact; integer-scaled weight multisets + Kostant/Weyl alternating sums + sympy/Fraction, no floats, 41 checks, ~4 s): does the FULL adjoint tensor structure of e8 (non-Cartan external legs, higher-arity vertices) open a sphere-axion channel to the rigid 32 T3 residual of the Atiyah-Bott ledger (v505 S3.7: A_fix = (9, -30, -15, 0, 32) in (P1, P2, P3, T5, T3))? The collapse is CONFIRMED full-tensorially, but one cubic door opens. [E] STRUCTURE: g0 = d5 (+) a3 is SEMISIMPLE with NO u(1) factor (the 52 class-0 roots split 40 + 12 into pure D5 + pure A3 roots, root span rank 8 = rank E8); the twisted sectors factorise multiplicity-free into minuscule Weyl orbits g1 = (16_s, 4), g2 = (10, 6), g3 = (16_c, 4bar) = (g1)^*, each an irreducible g0-module (64, 60, 64); INNERNESS THEOREM: the grading element h = (2,2,2,2,2;0^4) lies in the CARTAN OF g0, so t = exp(2 pi i h/4) is in the torus of G0 and every g0-invariant tensor automatically carries total sector charge 0 mod 4 -- the v508 collapse is a full-tensor statement AND holds across arities (all 15 non-neutral sector triples: Hom = 0; the sphere axions eta_1,2,3 have no invariant partner operator at any arity <= 3). [E] HOM TABLES: bilinear dim Hom_{g0}(g_j x g_j') nonzero ONLY at j' = -j with dims (2, 1, 1, 1) (Killing blocks; all 12 charged pairs vanish); invariant symmetric bilinear forms on e8 = 4-dim (2 + 1 + 1), the extra blocks B13/B22 restrict to zero on the Cartan (no new quadratic exchange content); trilinear neutral multisets {0,0,0}: 3 (= 1 sym + 2 antisym), {0,1,3}: 2, {0,2,2}: 2, {1,1,2}: 1, {2,3,3}: 1 -- ALL cross-sector trilinears are pure brackets (antisymmetric in the repeated legs, they vanish with two identical external legs); the ONLY totally symmetric survivor on all of e8 is the su(4) d-symbol (so(10) has no cubic Casimir: Inv(S^3 45) = 0 protects T5, Inv(S^3 a3) = 1). [E] THE NEW CHANNEL: the quartic projection of the d-symbol exchange is Q_dd = (0, 0, -1/240, 0, 1/60) = (1/60)(T3 - P3/4), computed by TWO independent routes (Killing-propagator contraction with 2 h_vee = 60 verified from the roots; hand route |x^2 - (tr x^2/4) 1|^2/60); Phi_T3(Q_dd) = 1/60 != 0 -- v508's master-kill functional Phi_T3 does NOT annihilate the trilinear exchange (the product theorem covers quadratic vertices only); bracket vertices, tadpoles and mixed d x f die exactly; no T5 analogue exists (Phi_T5 stays a universal annihilator, exactly as A_fix requires). [E] NEW CERTIFICATE: M = [E13, E22, E00, Q_dd] has rank 3, augmented rank 4 -- the pairing in the charge (generous) reading remains OBSTRUCTED, but with the WEAKER certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 72 - 8 = 64 (the T3 obstruction 32 is traded for a combined (P3, T3) obstruction: the d-channel produces T3 only with the fixed shadow P3 = -T3/4, and the leftover P-block violates the charge-0 span -- the K0 || K2 collapse of v508); the RELAXED reading (charge bookkeeping dropped on the bilinear P-block) becomes EXACTLY solvable: A_fix = -u + 8v + 2w + 1920 Q_dd (residual 0) -- where v508 found T5 = T3 = 0 persisting even relaxed, the d-channel closes the T3 gap with small integer coefficients. [C] NUMBER FENCES (typed [C] only, look-elsewhere discipline): 64 = dim g1 = 2^6; 1920 = |W(D5)| = 2^7 x 3 x 5 = 8 x 240 -- FENCE SHARPENED (2026-07-22, v513/CELEST.DTERM.NONDERIV.01): the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513; the convention-stable [E] content is c_d = Phi_T3(A_fix) x 2 h_vee(E8) = 32 x 60. [O] OPEN AFTER DELTA-2: the arity >= 4 ledger; the physical justification of the cubic GS term (c_d = 1920 under the e8-Killing propagator); ch2-naturalness for CUBIC vertices UNDECIDED (no canonical geometric datum exists for a cubic vertex); the burden on delta-1 (twisted BCOV contact terms) is SHARPENED: it must supply either the psi = 64 slice or the selection-rule relaxation -- and the delta-1 exploration run (probe, 27/27, exploration-level, v-numbered on demand) stands UNDECIDED: the strict holomorphic q^0 reading is REFUTED at the Z2/Eguchi-Hanson anchor (method boundary, not a kill), the forced leading (T5,T3) ratio is 4:3 (sector 2 : sectors 1+3), the SL(2,Z) orbit decomposition 15 = 12 + 3 identifies the contact term as the MODULAR COMPLETION of the AB data (Harvey-Moore tau-integral, not a free number), and massive coset levels n >= 2 are mandatory. [E] NEGATIVE CONTROLS: (a) Killing control = v508 replication (A_fix = (9, -30, -15, 0, 32), E13 = (16, -96, 144, 0, 0), E22 = (16, 32, 16, 0, 0), E00 = 225 E22); (b) SO(16)/D8 glue: bilinears (1, 0, 1), trilinears (1, 0, 1, 0), Inv(S^3 adj_d8) = 0 -- no symmetric cubic, no odd sectors, no A3 block: the delta-2 channel cannot even be posed (E8/mu4 doubly special); (c) deliberately false g0 = d5 (+) a2 (+) u(1): bilinear entries (5, 2, 2) != (2, 1, 1) and Inv(S^3 g0) = 6 != 1; false sector content (16_s, 4bar) kills every pairing -- the computation SEES the exact structure. SEAM.EQUIV.01 stays [O]; NO marker moves anywhere -- CELEST.SEAM.01 stays Open, the O-fence is only SHARPENED.

Original classification detail

WP5e-delta-2 [E]/[C]/[O] (intermediate verdict): the full-tensor ledger -- the v508 collapse is CONFIRMED full-tensorially and arity-crossing (innerness theorem: h in the g0 Cartan, every invariant tensor charge-0 mod 4; all 15 non-neutral trilinear triples Hom = 0), and the charge-0 ledger contains exactly ONE structure invisible to the quadratic ansatz: the su(4) d-symbol (unique symmetric trilinear on e8, so(10) has no cubic Casimir), whose exchange quartic Q_dd = (1/60)(T3 - P3/4) breaks the Phi_T3 master kill (1/60 != 0) -- the T3 channel OPENS; the pairing stays obstructed in the charge reading with the weaker certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 64 = dim g1 [C], and becomes exactly solvable relaxed: A_fix = -u + 8v + 2w + 1920 Q_dd with c_d = 1920 = |W(D5)| [C] (the |W(D5)| reading look-elsewhere-loaded and convention-contingent -- v513); SO(16)/D8 has no symmetric cubic and no odd sectors (E8/mu4 doubly special), false g0/sector controls separate; the burden on delta-1 is sharpened to the psi = 64 slice or the selection-rule relaxation (delta-1 itself UNDECIDED at exploration level at promotion time, since DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, tension with v516 stated; the pre-v518 state: strict holomorphic reading refuted at the Z2 anchor, contact term = modular completion / Harvey-Moore tau-integral, forced (T5,T3) ratio 4:3, massive coset levels n >= 2 mandatory); SEAM.EQUIV.01 stays [O], no marker moves

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; SEAM.EQUIV.01; v492; v493; v502; v505; v508; v509