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Original ledger classification: Formal/Identity [E] (exact prefactor/level arithmetic: Hurwitz-zeta vacuum energies, inner/shift twist distribution, discriminant-form = Casimir two-route identity, current/embedding/central-charge k-fixing, singular-vector dial -- sympy/Fraction exact, Wolfram-mirrored) + [C] (Kac generation for k >= 3) + [O] (the BCOV/Costello-Li twistor inflow = WP5e proper; SEAM.EQUIV.01 unchanged)

Recorded statement: WP5e-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, alpha stage -- 'prefactor and level bookkeeping', the CFT-side anchor of the BCOV/Costello-Li twistor uplift): exact sympy/Fraction…

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Paper context: tfpt research contracts

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CELEST.WP5E.ALPHA.01

Original ledger classification: Formal/Identity [E] (exact prefactor/level arithmetic: Hurwitz-zeta vacuum energies, inner/shift twist distribution, discriminant-form = Casimir two-route identity, current/embedding/central-charge k-fixing, singular-vector dial -- sympy/Fraction exact, Wolfram-mirrored) + [C] (Kac generation for k >= 3) + [O] (the BCOV/Costello-Li twistor inflow = WP5e proper; SEAM.EQUIV.01 unchanged)

Original claim text

WP5e-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, alpha stage -- 'prefactor and level bookkeeping', the CFT-side anchor of the BCOV/Costello-Li twistor uplift): exact sympy/Fraction arithmetic, no floats (33 checks). Roadmap target: boundary partition function = E4/eta^8 INCLUDING the q^{-1/3} prefactor; preregistered KILL = 'the anomaly inflow demands a level != 1'. [E] PART A -- THE PREFACTOR IS EXACT mu4 VACUUM-ENERGY BOOKKEEPING: Hurwitz-zeta derivation (zeta(-1,theta) = -B2(theta)/2 symbolic; E_b(theta) = -1/24 + theta(1-theta)/4 exact; fermion counterpart E_f = -E_b, Ising h_sigma = 1/16 per Majorana); the clock is INNER -- h = (2,2,2,2,2;0^4) and the A3 detector h' = (0^5;1,1,1,-3) BOTH read the glue class mod 4 on all 240 roots ((h,h) = 20, (h',h') = 12, sum 32 = 2*4^2) -- so ALL 8 Cartan/torus-boson directions sit at clock eigenvalue 1: twist distribution theta = 0^8 in EVERY sector C_j, a SHIFT orbifold (lattice cosets, integer-moded oscillators), NOT a rotation orbifold; E_osc(C_j) = 8*(-1/24) = -1/3 = -c/24 SECTOR-INDEPENDENT (c = 8 = rank = Sugawara 248/31 -- the central-charge route and the boson-count route agree); sector ground exponents -1/3 + (0,1,1,1). [E] DISCRIMINANT FORM = CASIMIR ENERGY, TWO ROUTES: coset minima h_D5 = (0,5/8,1/2,5/8), h_A3 = (0,3/8,1/2,3/8), glue diagonal (0,1,1,1) INTEGER; route 1 = SPECTRAL FLOW (s_j = j h/4 costs j^2(h,h)/32: 5j^2/8, 3j^2/8, sum j^2 = 0 mod 1 = the isotropy/Lagrangian condition); route 2 = FREE FERMIONS, exact not just mod 1 (D5 = 10 Majorana -> 5/8, A3 = D3 = 6 -> 3/8, glue diagonal = 16 = 2^(g_car-1) = the WP5d seam carrier -> 1; R-NS shift 1/16 per Majorana); HONEST SHARPENING (i): the ROTATION reading fails -- the geometric A3 deck diag(i,i^-1) gives twisted Casimir 3/16 != 3/8 (the v492 'clock != deck' lesson at the Casimir level, factor 2). [E] CHARACTER RECONSTRUCTION: all glue-coset norms even (0 odd cross-terms: isotropic diagonal <=> even extended lattice = E8); sum of the four Theta_Cj = E4 = (1,240,2160,6720); sector characters have leading exponents -1/3 + h_j, leading coefficients (1,64,60,64), currents (60,64,60,64), and sum EXACTLY to E4/eta^8 = q^{-1/3}(1,248,4124,34752). [E] PART B -- k = 1 FORCED THREE INDEPENDENT WAYS: (i) the CURRENT CONDITION -- simple-current weights are exact closed forms (D5: h(k om_v) = k/2, h(k om_{s,c}) = 5k/8; A3: h(k om_j) = k j(4-j)/8), the Z4 extension closes onto the E8 ADJOINT (60+64+60+64 = 248 spin-1 currents) only if h(J) = 1, and h(J; k) = k => k = 1 uniquely; HONEST SHARPENING (ii): 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: D5+A3 in E8 reduces to 47k^2 + 219k - 266 = 47(k-1)(k+266/47) = 0 (unique positive root k = 1), D8 in E8 to 128k(1-k) = 0 (128 = spinor dim); (iii) CENTRAL CHARGE = THE PREFACTOR ITSELF: 248k/(k+30) = 8 <=> 240(k-1) = 0 (240 = #roots), -c/24 = -1/3 only at k = 1; PLUS the WP5b singular-vector dial: PBW engine ||(E^theta_{-1})^m 0||^2 = m! k(k-1)...(k-m+1), singular vector EXACTLY at grade k+1, Weyl dims from root data (248, 3875, 27000; Sym^2 248 = 30876), k = 1 deletes V(2 theta): 31124 - 27000 = 4124 = chi_2 (singular-vector consistency); k = 2: <s|s> = +4 = the WP5b/WP5c value. [E] NEGATIVE CONTROLS: (a) D8/SO(16)_1 -- same rank 8, same c = 8, SAME q^{-1/3} prefactor, but coset weights (0,1/2,1,1): the prefactor does NOT discriminate E8 vs SO(16), the h-INTEGRALITY of the glue diagonal does; Theta_D8 + Theta_s = E4 (the {1,s} Lagrangian extension IS E8); (b) WRONG TWIST: Z2 rotation gives E = +1/6 (h = 1/2 -- no mu4-shift sector there); a mu4 rotation makes E_j = (-1/3, 1/24, 1/6, 1/24) sector-DEPENDENT (the common prefactor and the single eta^8 break); (c) WRONG k in {2,3,4}: ALL FIVE dials fail coherently (h(J) = k != 1; embedding residuals 360/814/1362; c != 8; prefactor != q^{-1/3}; <s|s> = 4/12/24 so 27000 survives, level-2 dim stays 31124 != 4124); at k = 1 all five pass. [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] HONEST FENCE -- WP5e is NOT closed: the BCOV/Costello-Li construction itself (the boundary partition function from the twistor side, the actual anomaly-inflow computation on PT, Costello's one-loop axion coupling lambda_g/(8 pi sqrt 3), the Costello-Paquette-Sharma level-from-flux mechanism) stays OPEN; v495 anchors the Lie-algebra arithmetic of the axion coefficient only; the KILL branch is NOT triggered on the CFT side -- all computable dials say k = 1 -- but its twistor side is untested. v502 anchors the CFT side of WP5e (alpha stage). SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.

Original classification detail

WP5e-alpha [E]/[C]/[O]: the q^{-1/3} prefactor of E4/eta^8 derived as exact mu4 vacuum-energy bookkeeping (inner clock => shift orbifold theta = 0^8, common -c/24 = -1/3 at c = 8; discriminant form = Casimir via spectral flow AND 16-Majorana free fermions; rotation reading fails 3/16 != 3/8) and k = 1 forced three independent ways (current condition h(J) = k = 1; conformal embedding 47(k-1)(k+266/47) = 0 resp. 128(k-1) = 0; c = 8 <=> 240(k-1) = 0; plus the WP5b singular-vector dial 31124 - 27000 = 4124) with the honest sharpening that glue-h integrality holds for ALL k and fixes nothing; controls: D8 same prefactor (discriminator = h-integrality), rotation twists break the common prefactor, wrong k fails all five dials; the BCOV/twistor inflow itself stays [O] -- WP5e not closed, SEAM.EQUIV.01 untouched

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP4.01; CELEST.WP5A.01; CELEST.WP5B.01; CELEST.WP5C.01; CELEST.WP5D.01; SEAM.EQUIV.01; v492; v496; v497; v498; v500; v501

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Registered verification modules

  • v502_celestial_wp5e_prefactor_level.pyRegistered verification module · 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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