"""v492 -- CELEST.WP1.01: the Z4-orbifold structure of the celestial chiral algebra on C^2/Z4 (the A3 ALE space) vs the E8 mu4-glue grading -- WP1 of the research contract CELEST.SEAM.01 (tfpt_research_contracts). Sympy exact. Question: does the mu4 clock on the seam celestial sphere select the SAME Z4 structure as the E8 mu4-glue grading (v125/v128), when both are read through the celestial-chiral-algebra (CCA) lens of Bittleston-Homans-Sharma (arXiv:2305.09451, JHEP 09 (2023) 008; the Eguchi-Hanson Z2 case) with a flat connection at infinity (Kronheimer-Nakajima / heterotic-on-ALE standard)? Here k = 4: the spatial deck is (z1,z2) -> (i z1, i^-1 z2) (the unique Z4 in SU(2) up to conjugacy, quotient = A3), and rho = the E8 mu4 glue (v128 grading, exact on all 240 roots). [E] S1. INNER GRADING / FLAT MONODROMY. v128 replication (240 = 52+64+60+64, additive on all 6720 root pairs, dims with Cartan (60,64,60,64), g0 = D5+A3 carrier = 60) PLUS the new statement: the grading is INNER -- h = (2,2,2,2,2;0^4) gives = class mod 4 for all 240 roots, so the glue IS Ad(exp(2 pi i h/4)), an order-4 torus element = a flat Z4 monodromy in the Kronheimer-Nakajima sense; 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: g0+g2 = 120 = SO(16), g1+g3 = 128 = spinor (v125 sub-Q-system {0,2}). [E] S2. C^2/Z4 IS THE A3 SINGULARITY. Molien table of C[z1,z2] under the deck (grade j = (p-q) mod 4) reproduced exactly by the character average; invariant ring C[X,Y,Z]/(XY - Z^4) (exact relation); Hilbert series (1+t^4)/((1-t^2)(1-t^4)) = grade-0 column; conjugation symmetry N[d][1] = N[d][3] (the spatial 64/64 pairing slot). [E] S3. GLUE-EQUIVARIANT SDYM(E8) SECTOR CLOSES. The rho-equivariant modes g_j (x) C[z1,z2]_m with j + m = 0 mod 4 form a Lie subalgebra (needs EXACTLY the v128 additivity); graded dims D[d] = 60(d+1) even / 64(d+1) odd -- uniform per parity BECAUSE dim g0 = dim g2 = 60, dim g1 = dim g3 = 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| (orbifold index); halfway EH/Z2 sector = 120/128 towers (the BHS Z2 case refined by mu4 exactly as 4 = 2 x 2, v125). [E] S4. (E8)_1 CHARACTER = SUM OF THE 4 GLUE SECTORS. Theta_E8 splits into the 4 glue-coset thetas; shell 1 = (52,64,60,64) re-derived from the LATTICE (independent of the root construction); sector characters Theta_Cj/eta^8 sum exactly to chi_(E8)_1 = 1 + 248q + 4124q^2 + 34752q^3 (the v377 series); level-1 currents per sector = (60,64,60,64); 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] S5. THE CRITICAL CORRECTION (clock != deck; clock^2 = deck). 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); the exact bridge is (spin clock)^2 = deck generator, |Z8| = 8 = 2|mu4| (the c3 = 1/(8 pi) seam winding integer). The order-4 clock IS realised as the U(2) phase diag(i,1): projective order 4, fixed points {0,inf}, free orbit {1,i,-1,-i} = mu4 (all v180 clock properties), normalising the deck (preserves the glue flat connection), det = i rotating the holomorphic symplectic form by the mu4 generator (a Kaehler, NOT triholomorphic isometry). 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, whose 4 branch points are ONE mu4 orbit with cross-ratio 2 -- the seam marks (v179 clock orbit; v168/v214 cross-ratio 2 => j = 1728 pillowcase). [E] S6. NEGATIVE CONTROLS. False spatial action diag(i,i): det = -1 not in SL(2,C) (no CY/ALE resolution), Poisson bracket leaves the invariant sector in 14/14 nonzero cases, monomial grades unsymmetric -- killed three ways. False glue (60- and 64-blocks swapped): additivity breaks on 3112/6720 root pairs. Rigidity: of all 24 relabelings exactly {identity, j -> -j} = Aut(Z4) are additive. [C] THE CLOCK-DECK IDENTIFICATION IS A NAMED EXTRA IDENTIFICATION (verdict B, not A): the match holds under ONE precisely named identification -- clock^2 = deck on the sphere (sheet-Z2 spin bookkeeping via Z8) -- which is consistent with v109/v233 sheet structure but is NOT itself forced by the arithmetic here. [O] FENCE: this module is STRUCTURAL ARITHMETIC on finite data (roots, cosets, monomials, Moebius/spin matrices). It does NOT construct a celestial holography / twistorial bulk for TFPT; the celestial path stays the research contract CELEST.SEAM.01 (WP2-WP5); SEAM.EQUIV.01 (the continuum (E8)_1 net on the seam) stays [O] and is untouched. Status: [E] exact arithmetic (sympy, no floats); [C] the named clock^2 = deck identification; [O] the celestial-dual construction (contract). Python (sympy); Wolfram-mirrorable (exact), deferred to the next verified engine run.""" from fractions import Fraction as F from itertools import combinations, permutations, product import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8 HALF = F(1, 2) DMAX = 9 # polynomial degree cutoff for the mode tables # --------------------------------------------------------------------------- # E8 roots in D5 (+) A3 + glue coordinates (the v128 construction) # --------------------------------------------------------------------------- def build_roots(): d5_roots, d5_v = [], [] for i, j in combinations(range(5), 2): for si in (1, -1): for sj in (1, -1): v = [F(0)] * 5 v[i], v[j] = F(si), F(sj) d5_roots.append(tuple(v)) for i in range(5): for s in (1, -1): v = [F(0)] * 5 v[i] = F(s) d5_v.append(tuple(v)) d5_s, d5_c = [], [] for signs in product((1, -1), repeat=5): v = tuple(HALF * s for s in signs) (d5_s if signs.count(-1) % 2 == 0 else d5_c).append(v) a3_roots = [] for i in range(4): for j in range(4): if i != j: v = [F(0)] * 4 v[i], v[j] = F(1), F(-1) a3_roots.append(tuple(v)) def wclass(k): out = [] for sub in combinations(range(4), k): v = [F(-k, 4)] * 4 for i in sub: v[i] += 1 out.append(tuple(v)) return out z5, z4 = tuple([F(0)] * 5), tuple([F(0)] * 4) roots = {} for r in d5_roots: roots[(r, z4)] = 0 for r in a3_roots: roots[(z5, r)] = 0 for d in d5_s: for w in wclass(1): roots[(d, w)] = 1 for d in d5_v: for w in wclass(2): roots[(d, w)] = 2 for d in d5_c: for w in wclass(3): roots[(d, w)] = 3 return roots def additivity(roots, classmap): """(# root pairs whose sum is a root, # violating grade additivity).""" items = list(roots.keys()) pairs = viol = 0 for i in range(len(items)): a1, b1 = items[i] for j in range(i + 1, len(items)): a2, b2 = items[j] s = (tuple(x + y for x, y in zip(a1, a2)), tuple(x + y for x, y in zip(b1, b2))) if s in roots: pairs += 1 if classmap[s] != (classmap[items[i]] + classmap[items[j]]) % 4: viol += 1 return pairs, viol def monomial_table(action, dmax=DMAX, modulus=4): """N[d][j] = # monomials z1^p z2^q of degree d with grade j (mod modulus).""" N = [] for d in range(dmax + 1): row = [0] * modulus for p in range(d + 1): row[action(p, d - p) % modulus] += 1 N.append(row) return N def section1(): print(" -- S1: E8 mu4-glue grading (v128 replication + inner-torus test)") roots = build_roots() counts = [sum(1 for c in roots.values() if c == k) for k in range(4)] norm_ok = all(sum(x * x for x in a) + sum(x * x for x in b) == 2 for (a, b) in roots) check("S1 [E]: 240 roots, all norm 2, glue split 240 = 52+64+60+64: %s" % counts, len(roots) == 240 and norm_ok and counts == [52, 64, 60, 64]) dims = [counts[0] + rankE8] + counts[1:] # Cartan (rank 8) sits in grade 0 check("S1 [E]: graded dims with Cartan dim g_j = %s = (60,64,60,64); " "g0 = D5+A3 carrier (45+15=60)" % dims, dims == [60, 64, 60, 64] and 45 + 15 == 60) pairs, viol = additivity(roots, roots) check("S1 [E]: grading additive on ALL %d root pairs (violations: %d)" % (pairs, viol), pairs == 6720 and viol == 0) h_d5 = tuple(F(2) for _ in range(5)) ok_h = all((sum(x * y for x, y in zip(a, h_d5))) % 4 == c and (sum(x * y for x, y in zip(a, h_d5))).denominator == 1 for (a, b), c in roots.items()) check("S1 [E]: INNER grading -- h = (2,2,2,2,2;0^4) gives = class mod 4 " "for all 240 roots => glue = Ad(exp(2 pi i h/4)), an order-4 torus element " "(flat Z4 connection / Kronheimer-Nakajima monodromy)", ok_h) h_a3 = (F(1), F(1), F(1), F(-3)) ok_h2 = all((sum(x * y for x, y in zip(b, h_a3))) % 4 == c and (sum(x * y for x, y in zip(b, h_a3))).denominator == 1 for (a, b), c in roots.items()) check("S1 [E]: the SAME class read off the A3 side by h' = (0^5;1,1,1,-3) -- the " "glue charge is the DIAGONAL (1,1) of the discriminant group Z4 x Z4 of " "D5 (+) A3 (the v92/v125 Lagrangian glue <(1,1)>)", ok_h2) even, odd = dims[0] + dims[2], dims[1] + dims[3] check("S1 [E]: Z2 halfway refinement g0+g2 = %d = dim SO(16), g1+g3 = %d = spinor " "128 (the v125 sub-Q-system {0,2} = SO(16)_1 step)" % (even, odd), even == 120 and odd == 128) return roots, dims def section2(): print(" -- S2: Molien/Hilbert data of C[z1,z2] under the A3 deck Z4") N = monomial_table(lambda p, q: p - q) t = sp.symbols('t') i4 = sp.I molien = [] for j in range(4): Fj = sp.Rational(1, 4) * sum( i4 ** ((-j * k) % 4) / ((1 - i4 ** k * t) * (1 - i4 ** (-k) * t)) for k in range(4)) ser = sp.series(sp.simplify(Fj), t, 0, DMAX + 1).removeO() molien.append([int(sp.expand(ser).coeff(t, d)) for d in range(DMAX + 1)]) ok_molien = all(molien[j][d] == N[d][j] for j in range(4) for d in range(DMAX + 1)) check("S2 [E]: Molien series (character average) reproduces the monomial table " "exactly for all grades j = 0..3, d <= %d" % DMAX, ok_molien) z1, z2 = sp.symbols('z1 z2') X, Y, Z = z1 ** 4, z2 ** 4, z1 * z2 check("S2 [E]: invariant ring generated by X=z1^4, Y=z2^4, Z=z1 z2 with the A3 " "relation XY = Z^4 (exact): C^2/Z4 IS the A3 ALE singularity", sp.simplify(X * Y - Z ** 4) == 0) hilb = sp.series((1 + t ** 4) / ((1 - t ** 2) * (1 - t ** 4)), t, 0, DMAX + 1).removeO() ok_hilb = all(int(sp.expand(hilb).coeff(t, d)) == N[d][0] for d in range(DMAX + 1)) check("S2 [E]: Hilbert series of C[X,Y,Z]/(XY - Z^4) = (1+t^4)/((1-t^2)(1-t^4)) " "matches the grade-0 Molien column", ok_hilb) ok_conj = all(N[d][1] == N[d][3] for d in range(DMAX + 1)) check("S2 [E]: conjugation symmetry N[d][1] = N[d][3] (z1 <-> z2 swap = Weyl of " "SU(2)) -- the spatial side offers the 64/64 pairing slot", ok_conj) return N def equivariant_dims(dims, N, modulus=4): """D[d] = sum_j dim(g_j) * N[d][(-j) mod modulus] (j + m = 0 mod modulus).""" return [sum(dims[j % len(dims)] * N[d][(-j) % modulus] for j in range(len(dims))) for d in range(len(N))] def section3(dims, N): print(" -- S3: glue-equivariant CCA sector of SDYM(E8) on C^2/Z4") D = equivariant_dims(dims, N) ok_shape = all(D[d] == (60 if d % 2 == 0 else 64) * (d + 1) for d in range(DMAX + 1)) check("S3 [E]: D[d] = 60(d+1) for even d, 64(d+1) for odd d -- uniform per parity " "BECAUSE dim g0 = dim g2 = 60 and dim g1 = dim g3 = 64 (the v128 dims are " "exactly what uniformity requires); D[0..%d] = %s" % (DMAX, D), ok_shape) check("S3 [E]: zero modes (d=0) = g0 alone: dim %d = D5 (+) A3 carrier + Cartan " "-- the unbroken algebra on the orbifold IS the carrier" % D[0], D[0] == 60) check("S3 [E]: asymptotic density D[d]/(248(d+1)) -> (60+64)/2/248 = 1/4 = 1/|Z4| " "(orbifold index density)", sp.Rational(62, 248) == sp.Rational(1, 4)) # closure: root-grade additivity (S1, 0 violations) + monomial grade (p-q) # additive under multiplication by construction => equivariant subalgebra check("S3 [E]: closure -- [g_j (x) m, g_k (x) m'] in g_{j+k} (x) mm' with " "(j+k) + (grade m + grade m') = 0 mod 4: the equivariant sector is a Lie " "subalgebra (needs EXACTLY the v128 additivity, 0 violations)", True) N2 = monomial_table(lambda p, q: p + q, modulus=2) D2 = [sum((dims[0] + dims[2] if j == 0 else dims[1] + dims[3]) * N2[d][j] for j in range(2)) for d in range(DMAX + 1)] ok_eh = all(D2[d] == (120 if d % 2 == 0 else 128) * (d + 1) for d in range(DMAX + 1)) check("S3 [E]: halfway EH/Z2 sector (deck^2 = -1, the BHS Z2 case): dims 120(d+1) " "/ 128(d+1) = SO(16) adjoint / spinor -- the A3 tower refines the EH tower " "exactly as mu4 refines the sheet Z2 (4 = 2 x 2, v125)", ok_eh) return D def d5_coset_norms(maxnorm): """norm -> count for the 4 cosets of D5*/D5: 0=D5, v, s, c.""" out = {k: {} for k in ('0', 'v', 's', 'c')} rng_int = range(-2, 3) for v in product(rng_int, repeat=5): n = F(sum(x * x for x in v)) if n > maxnorm: continue key = '0' if sum(v) % 2 == 0 else 'v' out[key][n] = out[key].get(n, 0) + 1 rng_half = [F(k, 2) for k in (-3, -1, 1, 3)] for v in product(rng_half, repeat=5): n = sum(x * x for x in v) if n > maxnorm: continue key = 's' if (sum(v) - F(5, 2)) % 2 == 0 else 'c' out[key][n] = out[key].get(n, 0) + 1 return out def a3_coset_norms(maxnorm): """norm -> count for the 4 cosets of A3*/A3: w0=A3, w1, w2, w3.""" out = {k: {} for k in range(4)} for k in range(4): shift = F(-k, 4) rng = [m + shift for m in range(-3, 4)] for v in product(rng, repeat=4): if sum(v) != 0: continue n = sum(x * x for x in v) if n > maxnorm: continue out[k][n] = out[k].get(n, 0) + 1 return out def series_mul(a, b, nmax): out = [0] * (nmax + 1) for i, ai in enumerate(a): if i > nmax: break for j, bj in enumerate(b): if i + j > nmax: break out[i + j] += ai * bj return out def section4(): print(" -- S4: (E8)_1 character = sum of the 4 mu4 glue sectors (exact)") NMAX = 3 # character levels q^0..q^3 MAXNORM = 2 * NMAX d5 = d5_coset_norms(MAXNORM) a3 = a3_coset_norms(MAXNORM) pairing = {0: ('0', 0), 1: ('s', 1), 2: ('v', 2), 3: ('c', 3)} theta = {} for j, (dk, ak) in pairing.items(): coeffs = [0] * (NMAX + 1) for n5, c5 in d5[dk].items(): for n3, c3l in a3[ak].items(): n = n5 + n3 if n <= MAXNORM and n % 2 == 0 and F(n, 2).denominator == 1: coeffs[int(n) // 2] += c5 * c3l theta[j] = coeffs shell1 = [theta[j][1] for j in range(4)] check("S4 [E]: shell 1 (norm 2): %s = (52,64,60,64) -- the v128 split re-derived " "from the LATTICE, independent of the root construction" % shell1, shell1 == [52, 64, 60, 64]) shell2 = [theta[j][2] for j in range(4)] check("S4 [E]: shell 2 (norm 4): %s sums to 2160 = |E8 shell 2| -- the Z4 glue " "grading persists shell by shell (Theta_E8 = sum Theta_Cj)" % shell2, sum(shell2) == 2160) q = sp.symbols('q') P8 = sp.Integer(1) for n in range(1, NMAX + 1): P8 *= sum(sp.binomial(7 + k, 7) * q ** (n * k) for k in range(0, NMAX // n + 1)) P8 = sp.expand(P8) p8 = [int(P8.coeff(q, k)) for k in range(NMAX + 1)] sigma3 = lambda n: sum(d ** 3 for d in range(1, n + 1) if n % d == 0) e4 = [1] + [240 * sigma3(n) for n in range(1, NMAX + 1)] chi_e8 = series_mul(e4, p8, NMAX) sector = {j: series_mul(theta[j], p8, NMAX) for j in range(4)} total = [sum(sector[j][m] for j in range(4)) for m in range(NMAX + 1)] check("S4 [E]: sum of the 4 sector characters Theta_Cj/eta^8 = chi_(E8)_1 = %s " "(1, 248, 4124, 34752 -- the v377 series)" % total, total == chi_e8 and chi_e8 == [1, 248, 4124, 34752]) lvl1 = [sector[j][1] for j in range(4)] check("S4 [E]: level-1 currents per sector: %s = (60,64,60,64) -- the v128 dims " "WITH Cartan appear as the 4 branching blocks of the 248" % lvl1, lvl1 == [60, 64, 60, 64]) hmin = [min(n for n, c in {0: d5['0'], 1: d5['s'], 2: d5['v'], 3: d5['c']}[j].items() if c > 0 and n > 0) for j in range(4)] hd5 = [F(0)] + [F(h, 2) for h in hmin[1:]] ha3 = [F(0)] + [F(min(n for n in a3[k] if n > 0), 2) for k in (1, 2, 3)] qform = lambda x, y: F(5 * x * x + 3 * y * y, 8) ok_h = all(hd5[j] % 1 == F(5 * j * j, 8) % 1 and ha3[j] % 1 == F(3 * j * j, 8) % 1 for j in range(4)) check("S4 [E]: sector weights h(D5) = (0,5/8,1/2,5/8), h(A3) = (0,3/8,1/2,3/8): " "equal mod 1 to 5x^2/8 and 3y^2/8 -- the v92/v125 DISCRIMINANT FORM " "(5x^2+3y^2)/8 IS the pair of conformal weights of the lattice-CFT sectors", ok_h) ok_int = all(qform(j, j) % 1 == 0 and (hd5[j] + ha3[j]) % 1 == 0 for j in range(4)) check("S4 [E]: glue-diagonal weights h_j = h_D5 + h_A3 INTEGER for all j " "(h = 0,1,1,1): v125 isotropy q(k(1,1)) = k^2 in Z <=> all four sectors " "are CURRENTS <=> the extension to (E8)_1 is local", ok_int) def moebius_of_diag(a, d): """induced factor on z = z1/z2 for (z1,z2) -> (a z1, d z2).""" return sp.simplify(a / d) def proj_order(factor, nmax=16): f = sp.Integer(1) for n in range(1, nmax + 1): f = sp.simplify(f * factor) if f == 1: return n return None def section5(): print(" -- S5: which Z4 rotates the celestial sphere? (exact spin arithmetic)") I = sp.I zeta8 = sp.exp(sp.I * sp.pi / 4) fac_deck = moebius_of_diag(I, 1 / I) check("S5 [E]: deck diag(i,i^-1) induces z -> (%s) z on the celestial sphere: " "PROJECTIVE ORDER %s -- the A3 deck acts as the sheet flip z -> -z, NOT as " "the order-4 clock z -> iz" % (fac_deck, proj_order(fac_deck)), fac_deck == -1 and proj_order(fac_deck) == 2) zeta = sp.symbols('zeta') sols = sp.solve(sp.Eq(zeta ** 2, I), zeta) orders = [proj_order(sp.simplify(s ** 2)) for s in sols] su2_orders = [] for s in sols: g = sp.Matrix([[s, 0], [0, 1 / s]]) p, m = sp.eye(2), g k = 1 while not m.equals(p): m = sp.simplify(m * g) k += 1 if k > 20: break su2_orders.append(k) check("S5 [E]: SPIN OBSTRUCTION -- any SU(2) lift diag(zeta,1/zeta) of the clock " "z -> iz needs zeta^2 = i (primitive 8th root): SU(2) orders %s = 8 -- the " "clock's spin group is Z8 (would quotient to A7), NEVER the A3 deck Z4" % su2_orders, all(o == 8 for o in su2_orders) and all(o == 4 for o in orders)) g8 = sp.Matrix([[zeta8, 0], [0, 1 / zeta8]]) deck = sp.Matrix([[I, 0], [0, 1 / I]]) check("S5 [E]: SPIN BRIDGE -- (spin clock)^2 = diag(zeta8,zeta8^-1)^2 = " "diag(i,i^-1) = the A3 deck generator EXACTLY; the deck Z4 is the unique " "index-2 subgroup of the clock's Z8; |Z8| = 8 = 2|mu4| (the c3 = 1/(8 pi) " "seam winding integer)", sp.simplify(g8 ** 2 - deck) == sp.zeros(2, 2)) fac_clock = moebius_of_diag(I, 1) z = sp.symbols('z') fixed = sp.solve(sp.Eq(I * z, z), z) orbit = {sp.simplify(I ** k) for k in range(4)} check("S5 [E]: U(2) CLOCK -- diag(i,1) induces z -> iz with projective order %s, " "fixed points {0,inf}, free orbit {1,i,-1,-i} = mu4: ALL v180 clock " "properties hold for this (non-deck) isometry" % proj_order(fac_clock), fac_clock == I and proj_order(fac_clock) == 4 and fixed == [0] and orbit == {1, I, -1, -I}) clock = sp.Matrix([[I, 0], [0, 1]]) comm = sp.simplify(clock * deck - deck * clock) det_clock = sp.simplify(clock.det()) check("S5 [E]: the clock NORMALISES the deck (commutator = 0, preserves the glue " "flat connection); det = i of order 4: it rotates the holomorphic " "symplectic form omega = dz1^dz2 by exactly the mu4 generator (a Kaehler, " "NOT triholomorphic isometry -> rotates the twistor sphere by pi/2)", comm == sp.zeros(2, 2) and det_clock == I and proj_order(det_clock) == 4) check("S5 [E]: clock vs deck on the sphere -- clock^2 induces z -> (%s) z = the " "deck's sphere image: projectively the deck is the SQUARE of the clock " "(the sheet pair Z2 inside the clock Z4)" % sp.simplify(fac_clock ** 2), sp.simplify(fac_clock ** 2 - fac_deck) == 0) Z, a3c, a2c, a1c, a0c = sp.symbols('Z a3 a2 a1 a0') P = Z ** 4 + a3c * Z ** 3 + a2c * Z ** 2 + a1c * Z + a0c diff = sp.expand(P.subs(Z, sp.I * Z) - P) sol = sp.solve([sp.Poly(diff, Z).coeff_monomial(Z ** k) for k in range(4)], [a3c, a2c, a1c], dict=True) ok_def = sol == [{a3c: 0, a2c: 0, a1c: 0}] check("S5 [E]: CLOCK-INVARIANT DEFORMATION -- P(iZ) = P(Z) forces a3 = a2 = a1 = " "0: of the 3-parameter versal A3 family XY = Z^4 + a2 Z^2 + a1 Z + a0 " "(3 = rank A3 = N_fam resolution spheres), the clock fixes EXACTLY the " "1-parameter family XY = Z^4 + a0", ok_def and N_fam == 3) c = sp.symbols('c', positive=True) roots4 = sp.solve(sp.Eq(Z ** 4, c ** 4), Z) ratios = {sp.simplify(r / c) for r in roots4} z1r, z2r, z3r, z4r = c, sp.I * c, -c, -sp.I * c lam = sp.simplify((z1r - z3r) * (z2r - z4r) / ((z1r - z4r) * (z2r - z3r))) check("S5 [E]: the 4 branch points of XY = Z^4 - c^4 are ONE mu4 orbit " "{c, ic, -c, -ic} with CROSS-RATIO %s = 2: the seam marks (v179 clock " "orbit; v168/v214 cross-ratio 2 => j = 1728 pillowcase) appear as the " "celestial deformation branch points" % lam, ratios == {1, sp.I, -1, -sp.I} and lam == 2) def section6(roots): print(" -- S6: negative controls (the test must kill the fakes)") I = sp.I det_false = sp.simplify(sp.Matrix([[I, 0], [0, I]]).det()) check("S6 [E]: NC1a -- diag(i,i) has det = %s != 1: NOT in SL(2,C) => quotient " "not Gorenstein/CY, NO ALE hyperkaehler resolution, no SD-gravity " "background" % det_false, det_false == -1) z1, z2 = sp.symbols('z1 z2') pb = lambda f, g: (sp.diff(f, z1) * sp.diff(g, z2) - sp.diff(f, z2) * sp.diff(g, z1)) grade_false = lambda p_, q_: (p_ + q_) % 4 viol = tested = 0 invmons = [(p_, q_) for p_ in range(5) for q_ in range(5) if grade_false(p_, q_) == 0 and p_ + q_ > 0] for (pa, qa), (pb_, qb_) in combinations(invmons, 2): b = sp.expand(pb(z1 ** pa * z2 ** qa, z1 ** pb_ * z2 ** qb_)) if b == 0: continue tested += 1 if grade_false(pa + pb_ - 1, qa + qb_ - 1) != 0: viol += 1 check("S6 [E]: NC1b -- Poisson bracket of diag(i,i)-invariant monomials leaves " "the invariant sector in %d/%d nonzero cases (bracket shifts the grade by " "-2 mod 4): the false w-algebra sector does NOT close -- KILLED" % (viol, tested), viol == tested and tested > 0) Nf = monomial_table(lambda p_, q_: p_ + q_) asym = [(d, Nf[d][1], Nf[d][3]) for d in range(DMAX + 1) if Nf[d][1] != Nf[d][3]] check("S6 [E]: NC1c -- under diag(i,i) the monomial grades have N[d][1] != " "N[d][3] at %d of %d levels: the conjugate 64/64 pairing of g1/g3 has NO " "spatial partner slots -- KILLED" % (len(asym), DMAX + 1), len(asym) > 0) swap = {0: 0, 1: 2, 2: 1, 3: 3} fake = {r: swap[c] for r, c in roots.items()} pairs, viol2 = additivity(roots, fake) check("S6 [E]: NC2 -- swapping the glue classes of the 64-block and the 60-block " "breaks grading additivity on %d of %d root pairs (>0) and the conjugate " "pairing (dim fake-g1 = 60 != 64 = dim fake-g3): the false glue does NOT " "define a graded Lie algebra -- KILLED" % (viol2, pairs), viol2 > 0) good = [] for perm in permutations(range(4)): relab = {r: perm[c] for r, c in roots.items()} _, v = additivity(roots, relab) if v == 0: good.append(perm) check("S6 [E]: NC3 RIGIDITY -- of all 24 relabelings of the glue classes exactly " "%d are additive: %s = identity and negation j -> -j = Aut(Z4): the mu4 " "glue grading is rigid up to the group automorphism" % (len(good), good), sorted(good) == [(0, 1, 2, 3), (0, 3, 2, 1)]) def run(): reset() print("v492 CELEST.WP1.01: Z4-orbifold structure of the celestial chiral algebra " "on C^2/Z4 (A3 ALE) vs the E8 mu4-glue grading (WP1 of CELEST.SEAM.01)") roots, dims = section1() N = section2() section3(dims, N) section4() section5() section6(roots) # fences (recorded, not computed): verdict B + the [O] contract scope check("FENCE [C]: the clock-deck identification is a NAMED extra identification " "(verdict B, not A): the match holds under clock^2 = deck on the sphere " "(sheet-Z2 spin bookkeeping, Z8 with 8 = 2|mu4| = %d); consistent with the " "v109/v233 sheet structure, not forced by the arithmetic here" % (2 * 4), 2 * 4 == 8 == g_car + N_fam) check("FENCE [O]: structural arithmetic ONLY -- no celestial duality is " "constructed; the celestial path stays the research contract " "CELEST.SEAM.01 (WP2-WP5); SEAM.EQUIV.01 stays [O], untouched", True) return summary("v492 CELEST.WP1.01: E8 mu4-glue = inner order-4 torus automorphism = flat " "Z4 monodromy on the A3 ALE C^2/Z4; glue-equivariant SDYM(E8) sector closes " "with dims 60/64 alternating, zero modes = carrier; (E8)_1 character = sum of " "the 4 glue sectors with h = discriminant form; the mu4 clock is the U(2) " "phase diag(i,1) with clock^2 = deck (spin bridge Z8, 8 = 2|mu4|); " "clock-invariant A3 deformation = 1-parameter, branch points = one mu4 orbit, " "cross-ratio 2; negative controls kill false action/false glue; rigidity = " "Aut(Z4). Verdict B; SEAM.EQUIV.01 untouched") if __name__ == "__main__": raise SystemExit(1 if run() else 0)