A registered result in its source context

CELEST.WP5E.EPS2.01

Original ledger classification: Formal/Identity [E] (exact CPS-skeleton/pairing/lockstep/sector arithmetic: S^3 period (2 pi i)^2 N, exceptional flux 2 pi N, contraction 2N, pairing enumeration 48 -> 2, fluxes (64, 60, 64), current count (240, 0, 0, 0), #primaries (1, 3, 5, 10, 15, 27), lockstep theorem det(A-1) = -4 with the 0/24 negative probe, k = 2 / SO(16) / A2 controls -- sympy/Fraction exact, Wolfram-mirrored) + [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 -- typed as an analogy until the backreaction is computed) + [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)

Original claim text

WP5e-EPSILON-2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, epsilon-2 stage -- 'the CPS level-from-flux dial', verdict B; exact sympy/Fraction arithmetic, no floats, 28 checks, <1 s): 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). [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 exactly; their eq. 3.33 -- large-gauge invariance of hCS quantises N in Z); the exceptional-sphere Kahler flux int_{P^1} omega = 2 pi N exactly (K = |u|^2 + N log|u|^2, the flat part dies at t -> 0; their eq. 3.37); the boundary level magnitude 2N via the symplectic-boson double contraction (exact for N = 1, 2, 3; their eq. 9.4), Dynkin normalisation T(so8 vector) = 1, T(adj) = 6 -- 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 the full form condition 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 = sum_m dim(g_m) P_mi = (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, in CPS language the brane matter content); 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), both factors). 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 is isotropic and 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 (level from sector count, independent of the CPS transplantation). [E] THE 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 sharply, the four branch points are ONE mu4 orbit i^k t, the periods Pi_j = t(i-1) i^{j-1} have |Pi_j| = sqrt(2) t EQUAL on all three spheres, the monodromy acts as the pure phase i on the period vector and det(A - 1) = -4 != 0: no invariant flux vector, no sphere-dependent level assignment possible -- with the CPS flux-level dictionary, k1 = k2 = k3 IS A THEOREM of clock invariance. NEGATIVE PROBE (new falsifier): the clock-forbidden family Z^4 - Z (smooth, disc = -27) has branch points {0, 1, w, w^2}: NOT a mu4 orbit, and over ALL 24 orderings ZERO chains give three equal period moduli (1 vs sqrt 3). HONEST LIMIT: uniform flux doubling keeps the lockstep -- the theorem proves EQUALITY of the three k, not the value; the value 1 comes from the current/sector counters and, CPS-faithfully, from the minimal quantum N = 1. [E] NEGATIVE CONTROLS: k = 2 (current count 0, embedding residual 360, c = 31/2 => prefactor -31/48 != -1/3, #primaries 3 != 1 -- while the lockstep dial honestly does NOT fire); SO(16) glue (classes (60, 0, 60, 0) => fluxes (0, 60, 0): spheres 1/3 FLUX-DARK, ch2 defect -30 != -78, condensation stops at 16/2^2 = 4 = det Cartan(D8): four sectors survive); A2 intersection form (det 3 != 4 = |Z4|: McKay bijection broken, glue diagonal 23/24 not integer, mu = 12 not a perfect square: NO Lagrangian subgroup exists, Dedekind 2/3 != 5/4, Coxeter order 3 != 4)

Original classification detail

WP5e-epsilon-2 [E]/[C]/[O] (verdict B): k = 1 from flux quantisation -- partially derived. DERIVED exact: one level is a THEOREM (lockstep from clock invariance: one mu4 orbit, |Pi_j| = sqrt(2) t equal, monodromy = pure phase i, det(A-1) = -4 no invariant flux vector; falsifier Z^4 - Z: 0/24 orderings lockstep), flux-per-current = 1 uniformly (pairing matrix = identity up to diagram flip, pinned by complete enumeration from the v505 ch2 ledger; fluxes (64, 60, 64) = flavour multiplicity, NOT the level -- the naive reading killed by the lockstep test itself), the sector-counter dial pins the value (16 = ord^2 fractional sectors condense through the Lagrangian mu4 glue to 1; #primaries((E8)_k) = 1 iff k = 1), and the CPS skeleton is exact (level linear in the flux quantum, minimal quantum N = 1 <-> k = 1). ANALOGY [C]: the CPS dictionary on PT/Z4. OPEN [O]: the type-I B-model backreaction on PT/Z4. The v505 'one scale => one level' sentence is lifted to theorem status (text move, no gate move); WP5e stays [O]; SEAM.EQUIV.01 untouched; no marker moves

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; SEAM.EQUIV.01; v492; v493; v502; v505; v508