A registered result in its source context

CELEST.WP5E.M3.01

Original ledger classification: Formal/Identity [E] (exact kernel/expansion/period/monodromy arithmetic: null-coordinate harmonicity for arbitrary kernels, residue V-ledger identity 1/(2r), kernel log coefficient 4 = |mu4| with first correction a0/eta^4 and exact m-grading, GLT tower p_{4k} = 4(-a0)^k with the n = 0 mod 4 selection rule, exceptional-locus log 4 log t0 + clock phase 4i, period response 1/4 integrating to monodromy i, a0-rigid node support and (2 pi i)^2 fluxes, all four controls -- sympy exact, Wolfram-mirrored) + [C] (the GLT dictionary eta_p log eta_p = the GH Kahler-potential kernel; Lindstrom-Rocek / Ivanov-Rocek) + [O] (the full nonlinear Kahler potential of the resolved A3 ALE; SEAM.EQUIV.TWISTOR.01 unchanged)

Original claim text

MILESTONE M3 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-22; CELEST.SEAM.01 WP5e back-reaction programme, milestone M3 -- 'the a0 uplift', SUCCESS on the preregistered v514 S8.3 criterion; exact sympy arithmetic, no floats, 23 checks, ~7 s): uplift the (l, m) = (4, +-4) GH multipole (= the a0 direction) to the Beltrami differential on PT/Z4 and compute the induced Kahler-potential correction (preregistered SUCCESS = a log-type correction whose coefficient is tied to the source charge 4 = |mu4|, matching the V-ledger; KILL = coefficient decoupled from the centre count). THE UPLIFT OBJECT (declared): the generalized-Legendre-transform kernel per centre, G(eta_p) = eta_p log eta_p (Lindstrom-Rocek / Ivanov-Rocek, typed [C] as a dictionary); its derivative kernel chi = sum_p log eta_p = log P4 is THE LOG OF THE v515 TWISTOR-FAMILY POLYNOMIAL. [E] KERNEL BRIDGE: the O(2) section is a NULL coordinate -- Laplacian(3d) G(eta_p) = 0 identically for an ARBITRARY kernel (the uplift is well-typed); RESIDUE IDENTITY d_x [transform(log eta_p)] = 1/r_p EXACTLY (residue 1/(2r)): the GH potential V = sum_p 1/r_p IS the x-derivative of the contour transform of chi; flux -4 pi per centre, total source charge 4 = |mu4| (the V-ledger matched). [E] FOUR COUPLED CENTRE-COUNT SCALES: (1) asymptotic kernel log chi = 4 log eta + a2/eta^2 + (a0 - a2^2/2)/eta^4 + ... -- LOG COEFFICIENT 4 = |mu4| = centre count (the CPS 'N log' analogue at kernel level); at the clock-fixed seam fibre zeta = 0 the a2 slot closes and the first correction is EXACTLY a0/eta^4 = the (4, +-4) multipole in kernel clothes; exact m-grading (zeta^k/eta^4 carries field charge m = 4 - k: m = +-4 pieces PURE a0, m = 0 piece -6 t0^4 phi0-free -- the U(1) -> Z4 breaking sits exclusively in the a0 pieces); NO log x power terms in the Laurent tower (the v514 honest zero -- no asymptotic METRIC log -- preserved); (2) the once-integrated GLT kernel sum_p eta_p log eta_p = 4 eta log eta + p_4/(12 eta^3) + p_8/(56 eta^7) + ... with p_n = 0 unless n = 0 mod 4 (selection rule at potential level) and p_{4k} = 4 (-a0)^k: EVERY correction carries the prefactor 4 = centre count; (3) the exceptional-locus log chi(eta = 0) = log a0 = 4 log t0 + i(4 phi0 + pi) -- t0-coefficient 4 = |mu4|, clock phase 4i; modulus response a0 d/da0 chi = a0/(eta^4 + a0) equals 1 AT the exceptional locus and falls off as a0/eta^4 at infinity (no constant, no log): the a0-log lives at the exceptional locus, exactly as v514 S6.3 demanded; (4) periods (v515 consistency): Pi_j(la) LINEAR in t0 (uniform first-order shift, lockstep + phase ratios (1, i, i^2) preserved exactly); d log Pi_j / d log a0 = 1/4 = 1/|mu4| uniformly; integrated over the a0 circle: monodromy e^{2 pi i/4} = i = ONE Coxeter clock step (the v493 monodromy REPRODUCED from the perturbative period shift; the finite monodromy is the cyclic centre permutation q_p -> q_{p+1}); node la-support t0-FREE (the Z8 seam bridge is a0-rigid); all 6 pair discriminants ~ t0^2 != 0 (K4 connectivity: the forced uniform flux vector persists); the BM kernel is modulus-free with int_{S^3} K = (2 pi i)^2 (the quantised fluxes cannot move). [E] NEGATIVE CONTROLS: the (4, 0) multipole is phi0-independent in V AND in the kernel (-6 t0^4): clock-invariant, breaks nothing; Z2/Eguchi-Hanson analogue: EVERY dial reads 2 = |Z2| ((2, +-2) amplitude (3/2) sin^2 th, kernel log 2, exceptional log 2 log t0, period response 1/2, monodromy -1 = the Z2 clock); wrong centre counts k = 3, 5: kernel log coefficient k, modulus slot O(6)/O(10) != O(8) -- the observable MOVES with the centre count; forbidden family Z^4 - Z: p3 = 3 != 0 (selection rule broken), e4 = 0 (no a0-log at the exceptional locus) -- fails BOTH uplift dials. [C] THE GLT DICTIONARY: 'eta_p log eta_p = THE GH Kahler-potential kernel' (Lindstrom-Rocek / Ivanov-Rocek). [O] the full nonlinear Kahler potential of the resolved A3 ALE (no closed form exists); the quantised BCOV coefficient stays with the delta-1 strand; SEAM.EQUIV.TWISTOR.01 untouched (preparation, not construction). VERDICT: SUCCESS -- the log-type correction delivered on four coupled dials (kernel log 4, exceptional-locus log 4 log t0, tower prefactor 4, period response 1/4 -> monodromy i), every one tied to the centre count (EH: 2, k-orbit: k); the KILL branch does NOT fire; NO marker moves anywhere.

Original classification detail

M3 [E]/[C]/[O] (SUCCESS on the preregistered criterion): M3 executed via v517 -- the (4, +-4) multipole uplifted to the GLT kernel chi = log P4 (the log of the v515 family polynomial; harmonic for any kernel, residue identity matches the V-ledger with flux -4 pi per centre); the log-type correction is delivered on FOUR COUPLED CENTRE-COUNT SCALES: asymptotic kernel log 4 = |mu4| (first seam-fibre correction exactly a0/eta^4, exact m-grading, no log x power terms -- the v514 honest zero preserved), GLT tower p_{4k} = 4(-a0)^k with the n = 0 mod 4 selection rule, exceptional-locus log chi(0) = log a0 = 4 log t0 + i(4 phi0 + pi), and period response d log Pi/d log a0 = 1/4 = 1/|mu4| integrating to the clock monodromy i (v493 reproduced) with a0-rigid node support and topological (2 pi i)^2 fluxes; controls: (4,0) clock-invariant, EH/Z2 reads 2 = |Z2| on every dial, k = 3/5 orbits move the coefficient with the centre count, the forbidden family fails both dials (p3 = 3, e4 = 0) -- the KILL (decoupling from the centre count) does NOT fire. FENCE [C]: the GLT dictionary; [O]: the full nonlinear Kahler potential of the resolved A3 ALE; no marker moves

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; CELEST.WP5E.M2.01; SEAM.EQUIV.TWISTOR.01; v493; v505; v509; v514; v515; v516