A registered result in its source context

CELEST.WP1.01

Original ledger classification: Formal/Identity [E] (exact Z4-orbifold/character/spin arithmetic, sympy, negative controls) + [C] (the named clock^2 = deck identification, verdict B) + [O] (celestial programme; SEAM.EQUIV.01 unchanged)

Original claim text

WP1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-20; CELEST.SEAM.01 work package 1, verdict B): the E8 mu4-glue grading (v128) IS the flat Z4 monodromy of the glue-equivariant celestial chiral algebra on the A3 ALE space C^2/Z4 -- the Bittleston-Homans-Sharma CCA construction (arXiv:2305.09451, JHEP 09 (2023) 008; the Z2/Eguchi-Hanson case) refined to Z4 with a Kronheimer-Nakajima flat connection at infinity. Sympy exact throughout. [E] INNER GRADING / FLAT MONODROMY: h = (2,2,2,2,2;0^4) gives <alpha,h> = glue class mod 4 for ALL 240 roots => glue = Ad(exp(2 pi i h/4)), an order-4 torus element = a flat Z4 monodromy; the A3-side detector h' = (0^5;1,1,1,-3) reads the SAME class -- the glue charge is the diagonal (1,1) of the discriminant group Z4 x Z4 (the v92/v125 Lagrangian glue); Z2 halfway: 120 = SO(16), 128 = spinor (v125 sub-Q {0,2}). [E] SPATIAL SIDE: C^2/Z4 under (z1,z2) -> (i z1, i^-1 z2) IS the A3 singularity XY = Z^4 (invariant ring exact; Hilbert series (1+t^4)/((1-t^2)(1-t^4)); Molien table verified; conjugation symmetry N[d][1] = N[d][3]). [E] EQUIVARIANT SECTOR CLOSES: the glue-equivariant SDYM(E8) modes (j + m = 0 mod 4) form a Lie subalgebra with D[d] = 60(d+1) even / 64(d+1) odd -- uniform per parity ONLY because dim g_j = (60,64,60,64) (the v128 dims are the consistency condition); zero modes (d=0) = g0 = carrier D5+A3+Cartan = 60; asymptotic density 62/248 = 1/4 = 1/|Z4|; EH/Z2 halfway towers 120/128. [E] CHARACTER: Theta_E8 = sum of the 4 glue-coset thetas (shell 1 = (52,64,60,64) from the LATTICE, independent of the root construction); the 4 sector characters Theta_Cj/eta^8 sum EXACTLY to chi_(E8)_1 = 1+248q+4124q^2+34752q^3 (v377 series); sector weights h_D5 = (0,5/8,1/2,5/8), h_A3 = (0,3/8,1/2,3/8) = the v92/v125 discriminant form (5x^2+3y^2)/8 mod 1; glue diagonal h = (0,1,1,1) INTEGER = v125 isotropy = the extension to (E8)_1 is local. [E] CRITICAL CORRECTION (why verdict B): the A3 deck diag(i,i^-1) induces z -> -z on the celestial sphere (projective order 2, the sheet flip) -- NOT the order-4 clock z -> iz; any SU(2) lift of the clock has order 8 (spin group Z8, would quotient to A7); exact bridge (spin clock)^2 = deck, |Z8| = 8 = 2|mu4| (the c3 = 1/(8 pi) winding integer); the order-4 clock IS realised as the U(2) phase diag(i,1) (order 4, fixed {0,inf}, free mu4 orbit -- all v180 properties; normalises the deck, det = i rotates the holomorphic symplectic form by the mu4 generator -- Kaehler, not triholomorphic). [E] GRAVITY SIDE: clock-invariance selects from the versal A3 family XY = Z^4 + a2 Z^2 + a1 Z + a0 EXACTLY the 1-parameter family XY = Z^4 + a0; its 4 branch points are ONE mu4 orbit with cross-ratio 2 -- the seam marks (v179 clock orbit; v168/v214 pillowcase j = 1728). [E] NEGATIVE CONTROLS: false spatial action diag(i,i) killed 3 ways (det = -1 not in SL(2,C); Poisson closure fails 14/14; grades unsymmetric); false glue (60/64 swap) breaks additivity on 3112/6720 root pairs; rigidity: exactly {id, j -> -j} = Aut(Z4) of 24 relabelings are additive. [C] the clock-deck identification is a NAMED extra identification (clock^2 = deck on the sphere, sheet-Z2 spin bookkeeping) -- verdict B, not A. [O] structural arithmetic ONLY: no celestial duality is constructed; CELEST.SEAM.01 (WP2-WP5) stays a research contract; SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.

Original classification detail

WP1 [E]/[C]/[O]: E8 mu4-glue = inner order-4 torus element = flat Z4 monodromy on the A3 ALE; equivariant SDYM(E8) sector closes (60/64 towers, zero modes = carrier, density 1/4); (E8)_1 character = sum of the 4 glue sectors, weights = discriminant form; clock = U(2) diag(i,1), clock^2 = deck (Z8, 8 = 2|mu4|); clock-invariant deformation XY = Z^4 + a0, branch points = one mu4 orbit, cross-ratio 2; negative controls + Aut(Z4) rigidity; verdict B; SEAM.EQUIV.01 stays [O]

Dependencies as recorded

CELEST.SEAM.01; SEAM.EQUIV.01; E8.GLU.01; v92; v125; v128; v168; v179; v180; v214; v377