A registered result in its source context

CELEST.WP5B.01

Original ledger classification: Formal/Identity [E] (exact construction/singularity/control arithmetic, integer/Fraction, deterministic; Chevalley basis + affine PBW engine unit-tested on all 61504 pairs) + [C] (the named identifications: Weyl complete reducibility beyond BFS depth 4; Kac mechanics beyond level 2; per-period quarter dictionary) + [O] (the GNS/limit-state question WP5c-e -- the twisted-slot moding tension is the precise WP5c question; SEAM.EQUIV.01 unchanged)

Original claim text

WP5b OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5b, the second constructive milestone of WP5 -- 'the singular vector as an operator'; WP5a/v497 fixed the ideal's size and location, WP5b constructs the deleting object): exact integer/Fraction arithmetic throughout (53 checks), no floats, deterministic. [E] CONSTRUCTION: |s> = (E^theta_{-1})^2|0>, theta the highest root, weight 2 theta (the top of the 27000), level 2 = 8 quarter units = q^2 (an INTEGER level); the Chevalley/Frenkel-Kac basis is MACHINE-built from the v497 root data (bimultiplicative 2-cocycle eps(a,b) = (-1)^(m_a^T B m_b), asymmetry verified on all 57600 ordered root pairs; the [e_a,e_{-a}] = SGN*H(a) sign FORCED by the Jacobi identity, SGN = -1, not chosen; the invariant form kappa DERIVED from invariance, kappa(theta_vee,theta_vee) = 2; Jacobi exact on ALL 57600 sl2-slice triples + 6000 seeded triples + all 496 theta-adjacent triples; all root-sum structure constants N_ab = +-1). [E] SINGULARITY (main theorem, machine): J^a_1|s> = 0 for ALL 248 basis elements, 0 violations; case classification of the J_1 condition: 190 vanish at the first bracket, 57 at the second, exactly 1 (a = F^theta) by the central-term cancellation -- the ONLY case that sees the level k; plus J^a_2|s> = 0 (all 248), E^a_0|s> = 0 (all 120 positive roots), h_i-eigenvalues = <2 theta, a_i_vee> exactly, Shapovalov norm 0 at k = 1 (4 at k = 2); the affine PBW engine ([J^a_m,J^b_n] = [a,b]_{m+n} + m k kappa(a,b) delta_{m+n,0}) unit-tested on all 61504 basis pairs. [E] LEVEL DIAL: the F^theta_1 coefficient is 2(1-k) -- 2 at k = 0, 0 EXACTLY at k = 1, -2 at k = 2: in the centerless loop algebra (k = 0) the deletion object does NOT exist -- the central extension is load-bearing. [E] MU4 DATA (glue frame, v492/v496): theta-grading dims (1,56,134,56,1), g_0 = e7 + u(1); the h-adapted Weyl chamber MACHINE-constructed (8 simple roots re-derived, Cartan det 1, E8 Dynkin diagram with branch lengths [1,2,4]); theta_glue the unique dominant root, height 29 = h_vee - 1, <theta,h> = 5 => glue class j(theta) = 1; clock phase on |s> = i^(2j) = -1; class(2 theta) = 2 sheet-EVEN (consistent with WP5a S6.8); the deleting theta-sl2 crosses the sheet-ODD classes (1,3) while its square is sheet-even; E^theta expressible in the sector-C_1 equivariant variables (spatial degree m = -j mod 4, 2 monomial slots) and the square is mu4-INVARIANT; quarter bookkeeping (v496 per-period dictionary): E^theta_{-1} = 4 quarter units = one full mu4 period (248 currents), |s> = 8 quarters = q^2; cross-table theta-grading x glue class consistent (rows (1,56,126,56,1), columns (52,64,60,64)). [E] LEVEL-2 MATCH: weight 2 theta has multiplicity 1 in the level-2 Fock => U(g)|s> is THE 27000; dim V(2 theta) = 27000 = 30^3 = h_vee^3 (Freudenthal + Weyl-orbit sums, v497 replication); the direct g_0-lowering orbit BFS from |s> reproduces the Freudenthal weight multiplicities of V(2 theta) EXACTLY, weight by weight, through depth 4; quotient 31124 - 27000 = 4124 = theta-split sector sum (1036,1024,1040,1024); LATTICE KERNEL READING: (2theta,2theta)/2 = 4 > 2, so the quotient (E8)_1 theory has NO momentum-2theta state at q^2 -- |s> IS the kernel generator of Fock -> E4/eta^8. [E] NEGATIVE CONTROLS (honest separation): (a) E^theta_{-1}|0> (level 1) NOT singular -- exactly one witness (a = F^theta, -> -k|0>), the 248-current state that must REMAIN (chi_1 = 248); (b) 4 seeded generic level-2 states all non-singular (14-33 witnesses each); (c) at k = 2 the square is not singular but the CUBE (E^theta_{-1})^3|0> IS -- the (E^theta)^{k+1} pattern is GENERIC Kac level mechanics, not E8-specific; (d) D8/SO(16)_1 through the SAME pipeline (Chevalley + Jacobi on all 12544 sl2-slice triples): (E^theta_{-1})^2|0> IS singular there too (the singular-vector mechanism is level-1 GENERIC), BUT dim V(2 theta_D8) = 5304 != 14^3 and D8 keeps 3 extra level-1 primaries (comark-1 weights, h = (1/2,1,1), h = 1/2 NON-integer breaking one-block fusion) vs E8 (comarks all >= 2, vacuum the ONLY level-1 primary, glue weights (0,1,1,1) integer) -- the ONE-BLOCK closure is the E8/mu4-specific part; the singular vector itself is generic level-1 Kac mechanics, honestly separated. [C] NAMED IDENTIFICATIONS: Weyl complete reducibility of the finite-dim level-2 Fock extends the depth-4 orbit agreement to the full dim U(g)|s> = 27000; 'the singular vector generates the maximal submodule' (Kac) used only beyond level 2; the per-period quarter dictionary (v496 S4) for the u-unit bookkeeping. [O] HONEST LIMIT (= the precise WP5c question): in the TWISTED quarter-SLOT moding of the chi_w limit Fock two sector-C_1 modes never sum to 8 quarters (0 solutions, minimum total 6 = q^{3/2}) -- the per-PERIOD dictionary (v496), not the per-slot identification, carries |s> to q^2; exactly the GNS/limit-state seam (kernel containing the ideal) that WP5c must construct; local normality/Haag duality (WP5d) and Costello-Li (WP5e) stay OPEN. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.

Original classification detail

WP5b [E]/[C]/[O]: the deletion operator EXISTS and is explicit -- |s> = (E^theta_{-1})^2|0> in a machine-built Chevalley/Frenkel-Kac basis, J^a_1|s> = 0 on all 248 generators (case tally 190/57/1, exactly one case sees k), level dial 2(1-k): only k = 1 deletes, no deletion object in the centerless loop algebra; mu4-compatible: j(theta) = 1, clock phase i^2 = -1, class(2 theta) = 2 sheet-even, 8 quarters = q^2, expressible in the sector-C_1 equivariant variables (KILL branch not triggered); U(g)|s> = V(2 theta) = 27000 = h_vee^3 (orbit BFS = Freudenthal through depth 4), quotient 4124 = sector sum; controls: the 248-current state and generic states not singular, cube at k = 2 (generic Kac mechanics), D8 contrast (5304 != 14^3, h = 1/2 breaks fusion) -- one-block closure E8/mu4-specific; open: the twisted-slot moding carries |s> only via the per-period dictionary -- the precise WP5c GNS/limit-state question; WP5c-e stay [O]; SEAM.EQUIV.01 stays [O]

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5A.01; CELEST.WP4.01; SEAM.EQUIV.01; v128; v492; v496; v497