Celestial wp5e eps2 level from flux
v509_celestial_wp5e_eps2_level_from_flux.py · frontier
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CELEST.WP5E.EPS2.01: WP5e-epsilon-2 of the celestial-holographic research contract CELEST.SEAM.01 -- 'the CPS level-from-flux dial', executed (verdict B): derive the boundary level k = 1 from flux quantisation on the lockstep spheres, following the Costello-Paquette-Sharma (CPS) Burns-holography mechanism (PRL 130, 061602; arXiv:2306.00940). Exact sympy/Fraction, no floats (28 checks, <1 s). [E] CPS SKELETON REPLICATED EXACTLY: the S^3 period of the brane backreaction 3-form int Omega_N = (2 pi i)^2 N (pullback density -sin(2 theta), phases cancel; their eq. 3.33); the exceptional-sphere Kahler flux int_{P^1} omega = 2 pi N (K = |u|^2 + N log|u|^2, flat part dies at t -> 0); boundary level magnitude 2N via the symplectic-boson double contraction (N = 1, 2, 3), normalisation Dynkin index T(so8 vector) = 1 -- the CPS formula 'level = flux quantum x Dynkin index of the boundary matter', level LINEAR in the flux. [E] PAIRING MATRIX PINNED, NOT POSTULATED: the v505 ch2 ledger (f(m) = -(C^-1)_mm/2 = (-3/8, -1/2, -3/8)) + integrality + unimodularity + effectivity pin P_mi = c1(T_m).[Sigma_i] = delta_mi by COMPLETE enumeration (row candidates 8/6/8; 48 unimodular solutions of P C^-1 P^T = C^-1; with P >= 0 exactly 2 = identity + A3 diagram flip); full c1 matrix -(C^-1)_mn; glue FLUXES F_i = (64, 60, 64) = dim g_i, total 188 = 248 - 60, flip-invariant; flux per charged root current uniformly ONE quantum through exactly one sphere. [E] LEVEL FORMULA, honest: 'level = total open-string flux' is KILLED by the lockstep test itself (k_i = (64, 60, 64) unequal -- F_i is the FLAVOUR multiplicity); what works: k = flux per adjoint current = F_i/dim g_i = (1, 1, 1), anchored non-circularly by the lattice current count (240, 0, 0, 0) for k = 1..4 (v505 S4.8) and the embedding index j(D5 in E8) = j(A3 in E8) = 1 (Killing coefficient 60 = 2 h_vee(E8)). ORD-4-VS-LEVEL-1 RESOLVED: the clock order counts FRACTIONAL flux sectors, not the level -- mu = det(D5) det(A3) = 16 = ord(monodromy)^2, the mu4 glue diagonal isotropic + Lagrangian (order 4 = sqrt 16), KLM condensation 16 -> 16/4^2 = 1 = zero residual fractional flux; NEW DIAL: #primaries((E8)_k) = (1, 3, 5, 10, 15, 27) for k = 1..6 from the affine marks -- exactly ONE sector iff k = 1 (independent of the CPS transplantation). [E] LOCKSTEP THEOREM (the lifting of v505 S4.10 'one scale => one level' to a THEOREM): clock invariance P(iZ) = P(Z) forces a3 = a2 = a1 = 0, the four branch points are ONE mu4 orbit, |Pi_j| = sqrt(2) t EQUAL on all three spheres, monodromy = pure phase i, det(A - 1) = -4 => no invariant flux vector => k1 = k2 = k3 IS A THEOREM of clock invariance; NEGATIVE PROBE (new falsifier): the clock-forbidden family Z^4 - Z (smooth, disc -27, branch points {0, 1, w, w^2} not a mu4 orbit): over ALL 24 orderings ZERO chains with three equal period moduli (1 vs sqrt 3); HONEST LIMIT: uniform flux doubling keeps the lockstep -- the theorem proves EQUALITY, not the value; the value 1 comes from the current/sector counters and the CPS minimal quantum N = 1. [E] NEGATIVE CONTROLS: k = 2 (current count 0, embedding residual 360, prefactor -31/48 != -1/3, #primaries 3 != 1 -- while the lockstep dial honestly does NOT fire); SO(16) glue (fluxes (0, 60, 0): spheres 1/3 FLUX-DARK, ch2 defect -30 != -78, condensation stops at 16/2^2 = 4 = det Cartan(D8)); A2 intersection form (det 3 != 4, glue diagonal 23/24 not integer, mu = 12 not a perfect square: NO Lagrangian subgroup, Dedekind 2/3 != 5/4, Coxeter 3 != 4). [C] the CPS dictionary itself on PT/Z4 (their geometry is the C^2 blow-up with ONE (-1)-sphere and so(8) at negative level -- not the A3 ALE). [O] the type-I B-model backreaction on PT/Z4 (twisted Kodaira-Spencer measure, the A3 Omega_N) -- the same fence as v502 S5.7/v505; delta-1/delta-2 and epsilon-1 stay open; SEAM.EQUIV.01 untouched; no marker moves. Exact (sympy/Fraction).
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