A registered result in its source context

CELEST.WP2.01

Original ledger classification: Formal/Identity [E] (exact deformation geometry/period/algebra arithmetic, sympy, negative controls) + [C] (three named identifications, verdict B; v216 precision typed) + [O] (algebra/contract programme; SEAM.EQUIV.01 and v216 unchanged)

Original claim text

WP2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-20; CELEST.SEAM.01 work package 2, verdict B; companion of CELEST.WP1.01/v492): the clock-invariant deformation XY = Z^4 + a0 of the A3 seam orbifold -- geometry, period/monodromy household, and the Bittleston-Homans-Sharma deformed-algebra pattern (arXiv:2305.09451, JHEP 09 (2023) 008; the Z2/Eguchi-Hanson case, fibre XY - Z^2 = -c^2) transferred to k = 4. Sympy exact throughout (47 checks). [E] GEOMETRY: P(iZ) = P(Z) forces a3 = a2 = a1 = 0 SHARPLY and two-sided (prod_k (Z - i^k w) = Z^4 - w^4: mu4-stable root set <=> clock-invariant quartic); XY = Z^4 + a0 smooth iff a0 != 0; disc = 256 a0^3; branch points Z^4 = -a0 = ONE free mu4 orbit in every fibre; cross-ratio 2 / j = 1728 for ALL a0 (binary-quartic invariants I = 12 a0, J = 0 IDENTICALLY) => a0 is a PURE SCALE, the tau = i pillowcase (v168/v214) is FROZEN -- no shape modulus survives clock-invariance. [E] PERIODS / MONODROMY: Pi = t(i-1)*(1, i, i^2) -- all 3 vanishing spheres carry the SAME holomorphic volume sqrt(2) t (lockstep, no relative modulus); the clock acts on H2 as a COXETER ELEMENT of W(A3) (char x^3+x^2+x+1, eigenvalues {i,-1,-i} = the A3 exponents 1,2,3; order = Coxeter number h(A3) = 4 = |mu4| = N_fam + 1); NO invariant cycle (det(A-1) = -4); the surviving deformation direction is the weight-1 chi_1-Fourier diagonal (1, i, i^2), NOT the naive invariant diagonal; the clock IS the Picard-Lefschetz monodromy of the family (a0 -> e^{2 pi i} a0 = the 4-cycle); versal weights (a3,a2,a1,a0) = (1,2,3,0) = the REGULAR mu4 representation, invariant line = a0 alone. [E] DEFORMED ALGEBRA (BHS Z2 -> Z4): mode dictionary {w[m,n]: m = n mod 4} <-> {X^p Y^q Z^r: r <= 3}; flat deformation (counting level); bracket {f,g} = -4*Nambu(XY - Z^4 - a0) ANCHORED at the a0 = 0 orbifold w-sector ({X,Y} = 16 Z^3, {Y,Z} = -4Y, {Z,X} = -4X), Jacobi + Casimir exact, uniform clock-degree -1; S-algebra corrections exactly LINEAR in a0 at the Z4 wrap r+r' >= 4, conserving the mu4 grade -- NO sector leak, so the v492 glue-equivariant SDYM(E8) sector deforms consistently; a0 <-> the BHS c^2 slot (weight 0, the order-c^2 splitting-function correction slot); matrix factorisations (McKay modules) deform along every 1+3 and 2+2 orbit splitting; the clock fixes exactly the antipodal 2+2 sector = the EH/Z2 middle node (4 = 2 x 2). [E] NEGATIVE CONTROLS (doubly sharp): a1 Z => j = 0 (hexagonal CM point, maximally non-square); pure a2 Z^2 stays SINGULAR (A1 node); a2 != 0 => j = 1556068/81 != 1728 (exact instance, cross-ratio and binary-quartic routes agree); grading leak explicit (Z^2 * Z^2 picks up grades {0,1,2} in the variant ring). [C] THREE NAMED IDENTIFICATIONS (why verdict B, not A): (1) the fibrewise -4*Nambu bracket is THE k = 4 CCA bracket (BHS prove k = 2; anchored here at a0 = 0, twistorial derivation not redone); (2) period = root difference (standard A-series residue normalisation); (3) the seam-scale reading of a0 presupposes the WP1 clock-deck bridge (clock^2 = deck, Z8, 8 = 2|mu4|). [C] v216 PRECISION (typed, NO marker move): GIVEN the order-4 clock, the clock-invariant deformation freezes the (2,2,2,2) modulus to the square AUTOMATICALLY (only the scale a0 stays free) -- a RELOCATION of the same order-4 carrier input to the celestial side (the clock IS the input), not a new derivation; v216 keeps its [O]-residual. [O] OPEN: full deformed twisted-sector OPEs; the W(mu)-type scaling-limit identification for k = 4 (BHS announce the ADE case); the spacetime splitting-function check; WP3 (GS coefficient alignment), WP4 (character vs S-algebra grading), WP5 (constructive holography); SEAM.EQUIV.01 (continuum (E8)_1 net on the seam) stays [O], untouched. No gate closes, no marker moves.

Original classification detail

WP2 [E]/[C]/[O]: clock-invariance selects XY = Z^4 + a0 sharply; a0 = pure seam scale, tau = i pillowcase frozen (j = 1728 for all a0); clock on H2 = Coxeter element of W(A3) = Picard-Lefschetz monodromy; surviving direction = chi_1-Fourier diagonal, spheres in lockstep; BHS Z2 -> Z4: -4*Nambu bracket anchored at the orbifold, corrections linear in a0, grade-conserving (no sector leak), a0 <-> c^2 slot; negative controls j = 0 / j != 1728 / grading leak; verdict B; SEAM.EQUIV.01 and v216 [O] untouched

Dependencies as recorded

CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v92; v125; v128; v168; v179; v180; v214; v216; v492