A registered result in its source context

CELEST.WP5E.M1.01

Original ledger classification: Formal/Identity [E] (exact residue/family/period/kernel arithmetic: Poincare residue three routes, cover factor 4 = |mu4|, closed twistor family XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4(1 - la^4)^2, CY-compatible clock lift Omega -> +Omega, period lockstep 2 pi i t0 (i-1)(1, i, i^2) with covariance Pi -> i Pi, BM kernel (2 pi i)^2 anchor + dK = 0 + scale degree 0, lens forcing N in 4Z, 12 conifold nodes on Z8 -- sympy exact, no floats, Wolfram-mirrored) + [C] (global patching of the local BM kernels on the resolved 3-fold; Hitchin small resolution; the CPS 'exceptional-curve branes = glue currents' dictionary) + [O] (M2: the twisted KS measure per sector; M3: the a0 uplift; the quantised BCOV coefficient on PT/Z4)

Original claim text

MILESTONE M1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 WP5e back-reaction programme, milestone M1 -- 'the A3 Omega_N', SUCCESS on the preregistered v514 S8.1 criterion; exact sympy arithmetic, no floats, 30 checks, ~1 s): compute the A3 analogue of CPS eq. 3.5 on PT/Z4 -- the back-reacted holomorphic 3-form Omega_N sourced by the glue currents wrapping the three lockstep spheres -- and check its periods exactly (preregistered SUCCESS = a closed-form Omega-deformation whose S^3/Z4 periods are (2 pi i)^2 x (integer flux vector) reproducing the lockstep phase structure (i-1)(1, i, i^2); KILL = unequal sphere fluxes or a non-integer period). [E] RESIDUE FORM DERIVED: on F = XY - P(Z) all three ambient representatives satisfy dF ^ omega2 = -dX^dY^dZ and pull back to omega2 = dX^dZ/X (Poincare residue, derived not assumed); cover pullback at a0 = 0: omega2 = 4 dz1^dz2 = |mu4| x (flat form) -- the residue normalisation CARRIES the source charge 4; clock diag(i,1): omega2 -> i omega2 (v492 S5 replicated); weight ledger 4+4+2-8 = 2, Omega in O(4) = -deg K_PT; EH anchor 2 = |Z2|; wrong-weight control Z in O(4) breaks three ways (4 != 2, 6 != 4, 5 != 3). [E] THE TWISTOR FAMILY CLOSED: from the four O(2) centre sections q_p(la) = i^p t0 - i^-p t0 la^2 the family is EXACTLY XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4 (1 - la^4)^2 (e1 = e3 = 0 identically; la = 0 gives exactly the v493 clock family; a2 in O(4) opens only off-seam, a2(0) = a2(inf) = 0); clock lift gamma: (Z, la) -> (iZ, -i la), gamma^4 = 1, Omega -> +Omega (CY-compatible); Omega = omega2 ^ <la dla> in O(4), weight chain 4+4+2-8+2 = 4. [E] OMEGA_N CLOSED + QUANTISED PERIODS: Omega_N = Omega_0 + sum N_p K_p with CPS/Bochner-Martinelli kernels on the centre twistor lines L_p = {X = Y = 0, Z = q_p(la)}; dK = 0 away from the source, scale degree 0; PERIOD TABLE: boundary S^3/Z4 at the orbifold point (2 pi i)^2 x 1 (minimal invariant charge 4 = |mu4| -- the lens fundamental domain FORCES N in 4Z: quotient large-gauge quantisation (2 pi i)^2 N/4 integer only for N = 0 mod 4), deformed phase (2 pi i)^2 x 4, linking S^3 uniformly (2 pi i)^2 x N, per-sphere flux vector (2 pi i)^2 x N (1,1,1) (forced uniform twice over: clock orbit + K4 connectivity of the four centre lines, all 6 pairs meet), seam-fibre phase vector 2 pi i t0 (i-1)(1, i, i^2); the undeformed Omega_0 carries ZERO quantised 3-flux (all flux sourced); (2 pi i)^2 = (fibre residue) x (transverse residue) (density -sin 2 theta); orbifold-limit cover count 16 = |mu4|^2; minimal quantum 1 <-> k = 1 (v509 bridge). [E] CLOCK COVARIANCE: periods 2 pi i (q_{j+1} - q_j) generic-exact (profile and path drop identically), seam vector moduli all 2 sqrt(2) pi t0, Pi_{j+1}(-i la) = i Pi_j(la) for ALL la; the 12 collision nodes sit exactly on the 8 eighth roots of unity (8 = 2|mu4|, the Z8 seam bridge), antipodal double nodes, clock orbits 4+4+4, honest conifold points (Hessian det 512 t0^6, quadric rank 4, exceptional curves = sphere classes). [E] KILL PROBE (honest): the clock-forbidden family Z^4 - Z: 0/24 orderings lockstep (vs 8/24 for the mu4 orbit), la-support 12 twelfth roots != Z8, no antipodal pairing, no clock lift -- BUT the (2 pi i)^2 quantisation itself holds there TOO (local gauge invariance): INTEGRALITY ALONE IS NOT THE DISCRIMINATOR; the discriminator is the lockstep phase/modulus structure + the clock forcing of equal fluxes (unequal N_p allowed on the forbidden family); the v509 dial survives, the KILL does not fire. [E] ANCHORS + CONTROLS: EH/Z2 from the same machinery (XY = Z^2 - t0^2(1 - la^2)^2, exactly 2 nodes at la = +-1, lens forces charge 2 = |Z2|); CPS original = k = 1 flat patch (full S^3: every integer N allowed -- the unconstrained anchor); diag(i,i): Omega_0 -> -Omega_0 (no invariant 3-form, collapse at step zero) while K stays invariant (U(2)); fractional charges N = 1, 2, 3 excluded (lens periods 1/4, 1/2, 3/4)

Original classification detail

M1 [E]/[C]/[O] (SUCCESS on the preregistered criterion): M1 executed via v515 -- the back-reacted Omega_N is closed-form on the A3 twistor family (Omega_N = Omega_0 + sum N_p K_p, BM kernels on the centre twistor lines), all S3/Z4 periods are (2 pi i)^2-integral with the lockstep flux vector N(1,1,1,1) and clock covariance Pi -> i Pi, and the lens geometry FORCES the source charge 4 = |mu4| (quotient large-gauge quantisation: only N = 0 mod 4 passes); honest fence: integrality alone does not discriminate -- the discriminator is the lockstep phase structure (0/24 on the forbidden Z^4 - Z family vs 8/24) + the clock forcing of equal fluxes; neither preregistered KILL fires. CONDITIONAL [C]: global kernel patching, Hitchin small resolution, CPS brane dictionary. OPEN [O]: M2 (twisted KS measure), M3 (a0 uplift), the quantised BCOV coefficient on PT/Z4; SEAM.EQUIV.TWISTOR.01 untouched (v515 is preparation, not the Costello-Li construction); no marker moves

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.BETA.01; SEAM.EQUIV.TWISTOR.01; v492; v493; v505; v509; v514