Celestial wp5b singular vector
v498_celestial_wp5b_singular_vector.py · frontier
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CELEST.WP5B.01: WP5b of the celestial-holographic research contract CELEST.SEAM.01 -- 'the singular vector as an operator', the second constructive milestone of WP5 (WP5a fixed the ideal's size and location; WP5b constructs the deleting object). [E] CONSTRUCTION: |s> = (E^theta_{-1})^2|0>, theta the highest root, weight 2 theta (top of the 27000), level 2 = 8 quarters = q^2 (INTEGER); Chevalley/Frenkel-Kac basis MACHINE-built (bimultiplicative 2-cocycle, asymmetry on all 57600 pairs; [e_a,e_{-a}] sign FORCED by Jacobi, SGN = -1; kappa derived, kappa(theta_vee,theta_vee) = 2; Jacobi exact on 57600 sl2-slice triples + 6000 seeded + all 496 theta-adjacent; all N_ab = +-1). [E] SINGULARITY: J^a_1|s> = 0 for ALL 248 basis elements, 0 violations; case tally 190 (first bracket) / 57 (second) / 1 (F^theta via central-term cancellation -- the ONLY case that sees k); plus J^a_2|s> = 0 (248), E^a_0|s> = 0 (120), Shapovalov norm 0 at k = 1 (4 at k = 2); affine PBW engine unit-tested on all 61504 basis pairs. [E] LEVEL DIAL: F^theta_1 coefficient = 2(1-k) -- 2 at k = 0, 0 EXACTLY at k = 1, -2 at k = 2: without the central extension the deletion object does not exist. [E] MU4 DATA: theta-grading dims (1,56,134,56,1), g0 = e7+u(1); h-adapted chamber machine-built (Cartan det 1, E8 Dynkin diagram, branch lengths [1,2,4]); <theta,h> = 5 => j(theta) = 1, height 29 = h_vee-1; |s> in class 2 (sheet-even, WP5a consistent); clock phase i^2 = -1; cross-table theta-grading x glue class consistent (rows (1,56,126,56,1), columns (52,64,60,64)); the deleting theta-sl2 sheet-odd (1,3), its square sheet-even. [E] LEVEL-2 MATCH: Mult(2 theta) = 1 in the level-2 Fock => U(g)|s> is THE 27000; direct g0-orbit BFS reproduces the Freudenthal multiplicities of V(2 theta) exactly through depth 4; quotient 31124 - 27000 = 4124; lattice reading (2theta,2theta)/2 = 4 > 2 => no 2theta state at q^2 in E4/eta^8 -- |s> IS the kernel generator of Fock -> E4/eta^8. [E] NEGATIVE CONTROLS: E^theta_{-1}|0> NOT singular (1 witness F^theta, -> -k|0>) -- the 248-current state that must remain; 4 generic level-2 states non-singular (14-33 witnesses); k-dial: at k = 2 only the CUBE is singular ((E^theta)^{k+1} mechanics = generic Kac); D8/SO(16)_1: the singular vector exists there TOO (level-1 generic) but the h = 1/2 blocks do not fuse (3 comark-1 weights, dim V(2 theta_D8) = 5304 != 14^3) vs E8 (comarks >= 2, vacuum the only level-1 primary, glue weights (0,1,1,1) integer) -- one-block closure is the E8/mu4-specific part, honestly separated. [C] Weyl complete reducibility beyond BFS depth 4; Kac mechanics beyond level 2; per-period quarter dictionary (v496). [O] HONEST LIMIT: in the twisted quarter-slot moding two C_1 modes never sum to 8 quarters (minimum 6 = q^{3/2}) -- the per-PERIOD dictionary (v496), not the per-slot identification, carries |s> to q^2; exactly the precise WP5c question (GNS limit state, kernel containing the ideal); WP5c/d/e stay [O]; SEAM.EQUIV.01 untouched. Exact (integer/Fraction).
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