Celestial wp5e prefactor level
v502_celestial_wp5e_prefactor_level.py · frontier
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CELEST.WP5E.ALPHA.01: WP5e-alpha of the celestial-holographic research contract CELEST.SEAM.01 -- 'prefactor and level bookkeeping', the CFT-side anchor stage of WP5e (the BCOV/Costello-Li twistor uplift: boundary partition function = E4/eta^8 INCLUDING the q^{-1/3} prefactor; preregistered kill: the anomaly inflow demands a level != 1). Exact sympy/Fraction, no floats. [E] PART A -- THE q^{-1/3} PREFACTOR IS EXACT mu4 VACUUM-ENERGY BOOKKEEPING: Hurwitz-zeta derivation zeta(-1,theta) = -B2(theta)/2, E_b(theta) = -1/24 + theta(1-theta)/4 exact; the clock is INNER (h = (2,2,2,2,2;0^4) and h' = (0^5;1,1,1,-3) BOTH read the class mod 4 on all 240 roots; (h,h) = 20, (h',h') = 12, sum 32) => twist distribution theta = 0^8 in ALL FOUR sectors -- a SHIFT orbifold (lattice cosets, integer-moded oscillators), NOT a rotation orbifold; E_osc = -1/3 = -c/24 sector-INDEPENDENT (c = 8 = rank = Sugawara 248/31); sector ground exponents -1/3 + (0,1,1,1). [E] DISCRIMINANT FORM = CASIMIR ENERGY, TWO ROUTES: spectral flow j^2 (h,h)/32 (5j^2/8, 3j^2/8, sum j^2 = 0 mod 1 = isotropy/Lagrangian) and EXACTLY via free fermions (D5 = 10 Majorana, A3 = D3 = 6, glue diagonal = 16 = the WP5d seam carrier 2^(g_car-1); R-NS shift n/16 = 5/8, 3/8, 1); the ROTATION reading FAILS (the geometric A3 deck gives 3/16 != 3/8: the v492 'clock != deck' lesson at the Casimir level). [E] CHARACTER: sum of the four Theta_Cj = E4 = (1,240,2160,6720); the four sector characters (leading coefficients (1,64,60,64), currents (60,64,60,64)) sum EXACTLY to E4/eta^8 = q^{-1/3}(1, 248, 4124, 34752). [E] PART B -- k = 1 FORCED THREE WAYS: (i) CURRENT CONDITION h(J) = k = 1 (the Z4 extension closes onto the E8 adjoint 60+64+60+64 = 248 only at spin 1); HONEST SHARPENING: glue-h integrality h(J^j; k) = k(0,1,1,1) holds for ALL k = 1..8 -- the originally conjectured integrality route fixes NOTHING, the naive dial is dead, the three other dials carry; (ii) CONFORMAL EMBEDDING 47k^2 + 219k - 266 = 47(k-1)(k+266/47) = 0 resp. D8-in-E8: 128k(1-k) = 0; (iii) CENTRAL CHARGE 248k/(k+30) = 8 <=> 240(k-1) = 0 (= the prefactor itself); plus the WP5b singular-vector dial: PBW norms m! k(k-1)...(k-m+1), singular vector exactly at grade k+1, k = 1 deletes V(2 theta): 31124 - 27000 = 4124 (Weyl dims 248/3875/27000 from root data). [E] NEGATIVE CONTROLS: D8/SO(16)_1 -- same rank, same c = 8, SAME q^{-1/3} prefactor, but h = (0,1/2,1,1): the discriminator is glue-h INTEGRALITY, not the prefactor (Theta_D8 + Theta_s = E4); Z2 ROTATION gives E = +1/6 (h = 1/2, no mu4-shift sector); mu4 ROTATION makes E_j = (-1/3, 1/24, 1/6, 1/24) sector-DEPENDENT (the common prefactor breaks); wrong k = 2, 3, 4 fail ALL FIVE dials coherently (residuals 360/814/1362; <s|s> = 4/12/24; level-2 dim stays 31124 != 4124). [C] Kac generation of the maximal submodule by the grade-(k+1) singular vector for k >= 3 (k = 1, 2 machine-anchored by WP5b/WP5c). [O] the BCOV/Costello-Li twistor inflow itself (Costello's one-loop axion lambda_g/(8 pi sqrt 3) on PT, the Costello-Paquette-Sharma level-from-flux mechanism) stays OPEN -- v502 anchors the CFT side of WP5e only (alpha stage), the KILL branch ('inflow demands a level != 1') is NOT triggered on the CFT side but its twistor side is untested; WP5e stays [O], SEAM.EQUIV.01 untouched, no marker moves. Exact (sympy/Fraction).
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