"""v517 -- CELEST.WP5E.M3.01: milestone M3 of the research contract CELEST.SEAM.01 (WP5e back-reaction programme) -- "the a0 uplift", executed (SUCCESS on the preregistered v514 S8.3 criterion). Question (preregistered VERBATIM in v514): 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. 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): for Gibbons-Hawking metrics the Kahler-potential data lives on twistor space as the generalized- Legendre-transform kernel per centre, G(eta_p) = eta_p log eta_p with eta_p(zeta) = eta(zeta) - q_p(zeta) the O(2) section distance (the Lindstrom-Rocek / Ivanov-Rocek construction, typed [C] as a dictionary); its derivative kernel is chi(zeta) = sum_p log eta_p = log prod_p eta_p = log P4(eta, zeta), THE LOG OF THE v515 TWISTOR-FAMILY POLYNOMIAL. Everything below is exact arithmetic on this kernel. References: Lindstrom-Rocek CMP 115 (1988) (generalized Legendre transform); Ivanov-Rocek CMP 182 (1996); Costello-Paquette-Sharma arXiv:2306.00940 ("N log" Kahler flux); Gibbons-Hawking Phys. Lett. 78B; v493 (clock family, monodromy), v514 (eps1 ledger, GH multipole ledger, M1-M3 preregistration), v515 (M1: the closed twistor family and the Omega_N period table). Exact sympy arithmetic throughout, no floats. [E] S0. REPLICATION: GH multipole ledger (selection rule m = 0 mod 4, (4, +-4) amplitude (35/16) sin^4 th carried by t0^4 e^{4 i phi0} = -a0, dipole/octupole 0, quadrupole m = 0 -- v514 S6.2) + the twistor family at general orbit phase phi0: e1 = e3 = 0, a2 = 4 z0 zb0 zeta^2, a0(zeta) = -(z0^2 - zb0^2 zeta^4)^2, seam value a0(0) = -z0^4; the phi0 = 0 slice is EXACTLY the v515 closed form XY = Z^4 + 4 t0^2 la^2 Z^2 - t0^4 (1 - la^4)^2. [E] S1. THE KERNEL BRIDGE: the O(2) section is a NULL coordinate -- Laplacian(3d) G(eta_p) = 0 identically for an ARBITRARY kernel G (the uplift is well-typed: every contour transform lands in the V-ledger's harmonic function space); RESIDUE IDENTITY d/dx [transform(log eta_p)] = 1/r_p EXACTLY (residue 1/(2r) at zeta_-): the GH potential V = sum_p 1/r_p IS the x-derivative of the contour transform of chi = log P4; source ledger: flux -4 pi per centre, total source charge 4 = |mu4| (the V-ledger the uplift must match). [E] S2. THE UPLIFT -- 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 (the term zeta^k/eta^4 carries field charge m = 4 - k: m = +-4 pieces PURE a0, m = 0 piece -6 t0^4 phi0-free); NO log x power terms in the Laurent tower -- the v514 honest zero (no asymptotic METRIC log) is 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. [E] S3. THE EXCEPTIONAL-LOCUS LOG (scale 3): chi(eta = 0) = log a0 = 4 log t0 + i(4 phi0 + pi) -- t0-coefficient 4 = |mu4| = centre count, clock phase 4i; the 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 (the EH log ledger uplifted to the A3 kernel). [E] S4. OMEGA_N CONSISTENCY (scale 4; v515): Pi_j(la) is LINEAR in t0 -- the a0 perturbation shifts every sphere period by the SAME relative amount (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] S5. 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 (the KILL "decoupled" does not fire); 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" is the generalized-Legendre-transform dictionary (Lindstrom-Rocek / Ivanov-Rocek), typed [C]. [O] O-FENCE: the full nonlinear Kahler potential of the resolved A3 ALE is not constructed here (no closed form exists); the quantised BCOV coefficient on PT/Z4 stays with the delta-1 strand; SEAM.EQUIV.TWISTOR.01 untouched (preparation, not construction); NO marker moves anywhere. VERDICT per the preregistered v514 S8.3 branches: SUCCESS -- the log-type correction is delivered on four coupled dials (kernel log 4, exceptional-locus log 4 log t0, tower prefactor p_{4k} = 4 (-a0)^k, period response 1/4 integrating to the clock monodromy i), every one tied to the centre count (EH: 2, k-orbit: k); the KILL branch (decoupling from the centre count) does NOT fire. Status: [E] exact kernel/expansion/period/monodromy arithmetic (sympy, no floats); [C] the GLT dictionary; [O] the full nonlinear Kahler potential of the resolved A3 ALE. Python; Wolfram-mirrored (kernel log series, m-grading, GLT tower selection rule, exceptional-locus log, period response 1/4 + monodromy i, negative controls), counted per GATE.WOLFRAM.02. Discovery provenance: experiments/tfpt-discovery/celestial_seam_wp5e_m2m3_a0_uplift_probe.py (2026-07-22, 23/23). """ from itertools import combinations import sympy as sp from tfpt_constants import check, summary, reset, N_fam I = sp.I PI = sp.pi MU4 = N_fam + 1 # 4 = |mu4|, the seam clock order # --------------------------------------------------------------------------- # shared symbols # --------------------------------------------------------------------------- eta, zeta, u = sp.symbols('eta zeta u') t0 = sp.symbols('t0', positive=True) z0, zb0 = sp.symbols('z0 zb0') # centre modulus + formal conjugate phi0 = sp.symbols('phi0', real=True) th, ph = sp.symbols('theta phi', real=True) Q4 = [sp.expand(I ** p * z0 - I ** (-p) * zb0 * zeta ** 2) for p in range(4)] P4 = sp.expand(sp.prod([eta - q for q in Q4])) Q4R = [q.subs({z0: t0, zb0: t0}) for q in Q4] # v515 real slice phi0 = 0 def fmt(xs): return "(" + ", ".join(str(x) for x in xs) + ")" def Pi_j(j, la): return sp.expand(2 * PI * I * (Q4R[(j + 1) % 4] - Q4R[j]) .subs(zeta, la)) # --------------------------------------------------------------------------- # S0 -- replication: GH multipole ledger + the family at general phi0 # --------------------------------------------------------------------------- def section0(): print(" -- S0: replication (multipole ledger + twistor family)") cosg = [sp.sin(th) * sp.cos(ph - phi0 - p * PI / 2) for p in range(4)] Vl = {l: sp.expand_trig(sp.expand( sum(sp.legendre(l, cg) for cg in cosg))) for l in (1, 2, 3, 4)} amp44 = sp.simplify(sp.integrate( Vl[4] * sp.cos(4 * (ph - phi0)), (ph, 0, 2 * PI)) / PI) sel = [sp.simplify(sp.integrate( Vl[4] * sp.cos(m * (ph - phi0)), (ph, 0, 2 * PI))) for m in (1, 2, 3)] check("S0.1 [GH MULTIPOLE LEDGER] dipole S_1 = %s and octupole " "S_3 = %s vanish; quadrupole S_2 = %s (m = 0 only); the first " "symmetry-breaking multipole is (4, +-4): cos 4(ph - phi0) " "amplitude = %s = (35/16) sin^4 th, carried by t0^4 e^{4 i " "phi0} = -a0; m = 1, 2, 3 Fourier integrals of S_4 all vanish " "(selection rule m = 0 mod 4) -- v514 S6.2 replicated" % (Vl[1], Vl[3], sp.simplify(Vl[2]), amp44), Vl[1] == 0 and Vl[3] == 0 and sp.simplify(Vl[2] - (3 * sp.sin(th) ** 2 - 2)) == 0 and sp.simplify(amp44 - sp.Rational(35, 16) * sp.sin(th) ** 4) == 0 and all(s == 0 for s in sel)) coeffs = {n: sp.expand(P4.coeff(eta, n)) for n in range(4)} a2 = sp.expand(4 * z0 * zb0 * zeta ** 2) a0z = sp.expand(-(z0 ** 2 - zb0 ** 2 * zeta ** 4) ** 2) check("S0.2 [FAMILY AT GENERAL phi0] prod_p (eta - q_p) = eta^4 + " "a2 eta^2 + a0(zeta) with e1 = %s, e3 = %s, a2 = 4 z0 zb0 " "zeta^2 (clock-invariant O(4) slot) and a0(zeta) = " "-(z0^2 - zb0^2 zeta^4)^2 in O(8); seam value a0(0) = %s = " "-z0^4 = -t0^4 e^{4 i phi0} (v514 S4.2 / v515 S2.2 at general " "orbit phase)" % (coeffs[3], coeffs[1], a0z.subs(zeta, 0)), coeffs[3] == 0 and coeffs[1] == 0 and sp.expand(coeffs[2] - a2) == 0 and sp.expand(coeffs[0] - a0z) == 0 and sp.expand(a0z.subs(zeta, 0) + z0 ** 4) == 0) fam_r = sp.expand(P4.subs({z0: t0, zb0: t0})) v515 = sp.expand(eta ** 4 + 4 * t0 ** 2 * zeta ** 2 * eta ** 2 - t0 ** 4 * (1 - zeta ** 4) ** 2) check("S0.3 [v515 SLICE] at z0 = zb0 = t0 (phi0 = 0) the family is " "EXACTLY the v515 closed form XY = Z^4 + 4 t0^2 la^2 Z^2 - " "t0^4 (1 - la^4)^2", sp.expand(fam_r - v515) == 0) return a0z # --------------------------------------------------------------------------- # S1 -- the kernel bridge: harmonicity + the residue V-ledger identity # --------------------------------------------------------------------------- def section1(): print(" -- S1: the kernel bridge (null coordinate + V-ledger match)") z, zb, x = sp.symbols('z zb x') zp, zbp, xp = sp.symbols('zp zbp xp') G = sp.Function('G') eta_p = (z - zp) + 2 * (x - xp) * zeta - (zb - zbp) * zeta ** 2 lap = sp.expand(4 * sp.diff(G(eta_p), z, zb) + sp.diff(G(eta_p), x, 2)) check("S1.1 [NULL COORDINATE] Laplacian(3d) G(eta_p) = G'' x " "[4 (d_z eta)(d_zb eta) + (d_x eta)^2] = G'' x [4(1)(-zeta^2) " "+ (2 zeta)^2] = %s IDENTICALLY for an ARBITRARY kernel G: " "the O(2) section is a null/harmonic coordinate -- every " "contour transform of every kernel lands in the V-ledger's " "harmonic function space (the uplift is well-typed)" % lap, lap == 0) dx, dz, dzb = sp.symbols('dx', real=True), *sp.symbols('dz dzb', positive=True) r = sp.sqrt(dx ** 2 + dz * dzb) eta_c = dz + 2 * dx * zeta - dzb * zeta ** 2 zeta_m = (dx - r) / dzb at_root = sp.simplify(eta_c.subs(zeta, zeta_m)) res = sp.simplify(1 / sp.diff(eta_c, zeta).subs(zeta, zeta_m)) check("S1.2 [RESIDUE V-LEDGER IDENTITY] eta_p(zeta_-) = %s at " "zeta_- = (dx - r)/dzb and Res_{zeta_-} 1/eta_p = " "1/eta_p'(zeta_-) = %s = 1/(2 r_p): with d_x log eta_p = " "2 zeta/eta_p the transform (1/2 pi i) oint dzeta/zeta of " "chi_p = log eta_p obeys d_x [transform] = 2 x 1/(2 r_p) = " "1/r_p EXACTLY -- the GH potential V = sum_p 1/r_p IS the " "x-derivative of the contour transform of chi = log P4 " "(contour = the GLT prescription enclosing zeta_-, typed " "[C])" % (at_root, res), at_root == 0 and sp.simplify(res - 1 / (2 * r)) == 0) rr = sp.symbols('r', positive=True) flux = sp.integrate(sp.integrate( sp.diff(1 / rr, rr) * rr ** 2 * sp.sin(th), (th, 0, PI)), (ph, 0, 2 * PI)) check("S1.3 [SOURCE LEDGER] flux of each 1/r_p through S^2 = %s = " "-4 pi per centre; total source charge of V = 4 = |mu4| = " "centre count (v514 S6.1 replicated): the V-ledger the " "uplift must match" % flux, flux == -4 * PI and 4 == MU4) # --------------------------------------------------------------------------- # S2 -- the uplift: log coefficient, correction tower, m-grading # --------------------------------------------------------------------------- def section2(a0z): print(" -- S2: the a0 uplift of the twistor-potential kernel") a0s, a2s = sp.symbols('a0s a2s') ser = sp.series(sp.log(1 + a2s * u ** 2 + a0s * u ** 4), u, 0, 7).removeO() c2 = ser.coeff(u, 2) c4 = ser.coeff(u, 4) odd = [ser.coeff(u, n) for n in (1, 3, 5)] check("S2.1 [ASYMPTOTIC EXPANSION] chi = log P4 = 4 log eta + " "log(1 + a2/eta^2 + a0/eta^4): LOG COEFFICIENT = 4 = |mu4| = " "centre count = total source charge (the CPS 'N log' " "analogue at kernel level, N -> 4); corrections c2 = %s, " "c4 = %s, all odd powers %s vanish; the tower is a PURE " "Laurent series (no log x power terms): the v514 honest " "zero -- no asymptotic METRIC log -- is preserved, the a0 " "correction is power-law at infinity" % (c2, c4, fmt(odd)), c2 == a2s and sp.expand(c4 - (a0s - a2s ** 2 / 2)) == 0 and all(o == 0 for o in odd) and not ser.has(sp.log(u))) c4z = sp.expand(c4.subs({a2s: 4 * z0 * zb0 * zeta ** 2, a0s: a0z})) c4_seam = c4z.subs(zeta, 0) c2_seam = sp.expand((4 * z0 * zb0 * zeta ** 2).subs(zeta, 0)) target = sp.expand(-z0 ** 4 - 6 * z0 ** 2 * zb0 ** 2 * zeta ** 4 - zb0 ** 4 * zeta ** 8) check("S2.2 [SEAM FIBRE: THE (4, +-4) UPLIFT] at the clock-fixed " "fibre zeta = 0 the a2 slot closes (c2 = %s) and the FIRST " "correction is c4(0)/eta^4 = %s/eta^4 = a0/eta^4 EXACTLY: " "the (l, m) = (4, +-4) multipole in kernel clothes, carried " "by a0 = -z0^4 = -t0^4 e^{4 i phi0}; full c4(zeta) = %s" % (c2_seam, c4_seam, sp.factor(c4z)), c2_seam == 0 and sp.expand(c4_seam + z0 ** 4) == 0 and sp.expand(c4z - target) == 0) alpha, w = sp.symbols('alpha w', real=True) z, zb, x = sp.symbols('z zb x') eta_f = z + 2 * x * zeta - zb * zeta ** 2 cov = sp.expand( eta_f.subs({z: sp.exp(I * alpha) * z, zb: sp.exp(-I * alpha) * zb}) - sp.exp(I * alpha) * eta_f.subs(zeta, sp.exp(-I * alpha) * zeta)) ok_grade = True grades = {} for k in (0, 4, 8): lhs = sp.expand((sp.exp(I * alpha) * w) ** (k - 1) * sp.exp(-4 * I * alpha) * sp.exp(I * alpha)) rhs = sp.expand(sp.exp(I * (k - 4) * alpha) * w ** (k - 1)) grades[k] = 4 - k ok_grade &= sp.simplify(lhs - rhs) == 0 m_pieces = {4: sp.expand(c4z.coeff(zeta, 0)), 0: sp.expand(c4z.coeff(zeta, 4)), -4: sp.expand(c4z.coeff(zeta, 8))} dphi = {m: sp.simplify(sp.diff( m_pieces[m].subs({z0: t0 * sp.exp(I * phi0), zb0: t0 * sp.exp(-I * phi0)}), phi0) * sp.exp(-I * m * phi0)) for m in (4, 0, -4)} check("S2.3 [EXACT m-GRADING] field rotation z -> e^{i alpha} z " "acts as eta -> e^{i alpha} eta(e^{-i alpha} zeta) " "(remainder %s), so the kernel term zeta^k/eta^4 carries " "field charge m = 4 - k (checked k = 0, 4, 8: m = %s); the " "c4 pieces: m = +-4 are -z0^4, -zb0^4 (PURE a0, " "d/dphi0 = 4i m-phase: %s) and m = 0 is -6 t0^4 " "(d/dphi0 = %s: the (4,0) direction is phi0-FREE): the " "symmetry breaking U(1) -> Z4 sits EXCLUSIVELY in the a0 " "pieces" % (cov, fmt(grades[k] for k in (0, 4, 8)), fmt([dphi[4], dphi[-4]]), dphi[0]), cov == 0 and ok_grade and m_pieces[4] == -z0 ** 4 and m_pieces[-4] == -zb0 ** 4 and sp.expand(m_pieces[0] + 6 * z0 ** 2 * zb0 ** 2) == 0 and dphi[4] == -4 * I * t0 ** 4 and dphi[-4] == 4 * I * t0 ** 4 and dphi[0] == 0) qs = [I ** p * z0 for p in range(4)] pn = {n: sp.expand(sum(q ** n for q in qs)) for n in range(1, 9)} ok_sel = all(pn[n] == 0 for n in range(1, 9) if n % 4 != 0) ok_val = (pn[4] == 4 * z0 ** 4 and pn[8] == 4 * z0 ** 8) tower = sp.expand(sum((1 / u - q) * sp.series( sp.log(1 - q * u), u, 0, 9).removeO() for q in qs)) tower = sp.expand(tower + sp.O(u ** 8)).removeO() coef3 = tower.coeff(u, 3) coef7 = tower.coeff(u, 7) lows = [sp.expand(tower.coeff(u, n)) for n in (0, 1, 2, 4, 5, 6)] check("S2.4 [GLT KERNEL TOWER + SELECTION RULE] holomorphic power " "sums at the seam fibre: p_n = sum_p (i^p z0)^n = 0 unless " "n = 0 mod 4 (all n <= 8 checked: the m = 0 mod 4 selection " "rule at potential level), p_4 = 4 z0^4 = -4 a0, p_8 = " "4 z0^8 = 4 a0^2; the once-integrated kernel sum_p eta_p " "log eta_p = 4 eta log eta + p_4/(12 eta^3) + p_8/(56 eta^7) " "+ ...: coefficients %s = (z0^4/3, z0^8/14) -- EVERY " "correction carries the prefactor 4 = centre count through " "p_{4k} = 4 (-a0)^k, and the first correction is " "-a0/(3 eta^3) (the a0 uplift of the GLT potential)" % fmt([coef3, coef7]), ok_sel and ok_val and sp.expand(coef3 - z0 ** 4 / 3) == 0 and sp.expand(coef7 - z0 ** 8 / 14) == 0 and all(v == 0 for v in lows)) # --------------------------------------------------------------------------- # S3 -- the exceptional-locus log # --------------------------------------------------------------------------- def section3(): print(" -- S3: the exceptional-locus log structure") a0sym = sp.symbols('a0', positive=True) # modulus of -a0 slice chi0 = sp.log(-t0 ** 4 * sp.exp(4 * I * phi0)) target = 4 * sp.log(t0) + I * (4 * phi0 + PI) exp_match = sp.simplify(sp.exp(target) + t0 ** 4 * sp.exp(4 * I * phi0)) dlog_t0 = sp.simplify(t0 * sp.diff(chi0, t0)) dlog_phi = sp.simplify(sp.diff(chi0, phi0)) check("S3.1 [THE LOG AT THE EXCEPTIONAL LOCUS] chi(eta = 0) = " "log P4(0, 0) = log a0: exp(4 log t0 + i(4 phi0 + pi)) = a0 " "(remainder %s), t0 d/dt0 chi(0) = %s = 4 = |mu4| = centre " "count and d/dphi0 chi(0) = %s = 4i = the clock charge: the " "first nontrivial log structure generated by the a0 " "direction sits at the exceptional locus with coefficient " "EXACTLY the centre count -- 'log-type correction tied to " "the source charge 4' (the preregistered SUCCESS wording)" % (exp_match, dlog_t0, dlog_phi), exp_match == 0 and dlog_t0 == MU4 and dlog_phi == 4 * I) resp = sp.simplify(a0sym * sp.diff( sp.log(eta ** 4 + a0sym), a0sym)) at0 = resp.subs(eta, 0) ser_inf = sp.series(resp.subs(eta, 1 / u), u, 0, 5).removeO() lead = sp.expand(ser_inf.coeff(u, 4)) check("S3.2 [WHERE THE LOG LIVES] the modulus response a0 d/da0 " "chi = a0/(eta^4 + a0): equals %s at the exceptional locus " "eta = 0 and falls off as %s u^4 = a0/eta^4 at infinity " "(leading term of the series, no constant or log term): the " "a0-log lives AT the exceptional locus, the asymptotic " "region sees only the power-law (4, +-4) tail -- v514 S6.3 " "(EH log ledger: log at u -> 0, zero at infinity) uplifted " "to the A3 kernel" % (at0, lead), at0 == 1 and lead == a0sym and ser_inf.coeff(u, 0) == 0 and not ser_inf.has(sp.log(u))) # --------------------------------------------------------------------------- # S4 -- Omega_N / period consistency (v515) # --------------------------------------------------------------------------- def section4(): print(" -- S4: Omega_N consistency -- period shift, monodromy, " "rigidity") la = sp.symbols('la') lin = [sp.expand(t0 * sp.diff(Pi_j(j, la), t0) - Pi_j(j, la)) for j in range(3)] ratios = [sp.simplify(Pi_j(j, 0) / Pi_j(0, 0)) for j in range(3)] check("S4.1 [UNIFORM FIRST-ORDER SHIFT] Pi_j(la) is LINEAR in t0 " "for all j and all la (Euler remainders %s): the a0 " "perturbation shifts every sphere period by the SAME " "relative amount delta Pi/Pi = delta t0/t0 -- lockstep and " "the phase ratios %s = (1, i, i^2) are preserved EXACTLY at " "first order (v515 S3.2 structure undisturbed)" % (fmt(lin), fmt(ratios)), all(v == 0 for v in lin) and ratios == [1, I, sp.expand(I ** 2)]) Pi_h = 2 * PI * I * (I - 1) * z0 # Pi_0 at la = 0, holomorphic a0_h = -z0 ** 4 dlog = sp.simplify((sp.diff(Pi_h, z0) * z0 / Pi_h) / (sp.diff(a0_h, z0) * z0 / a0_h)) mono = sp.simplify(sp.exp(2 * PI * I * dlog)) check("S4.2 [PERIOD RESPONSE = 1/|mu4| => CLOCK MONODROMY] " "holomorphic first-order response d log Pi_j / d log a0 = " "%s = 1/4 = 1/|mu4| (Pi ~ z0, a0 = -z0^4); integrating the " "first-order shift around the a0 circle: e^{2 pi i x 1/4} = " "%s = i = ONE Coxeter clock step -- the v493 monodromy " "reproduced from the perturbative period shift" % (dlog, mono), dlog == sp.Rational(1, MU4) and mono == I) shifted = [sp.expand(Q4[p].subs({z0: I * z0, zb0: zb0 / I}, simultaneous=True) - Q4[(p + 1) % 4]) for p in range(4)] a0_shift = sp.expand((-z0 ** 4).subs(z0, I * z0) - (-z0 ** 4)) per_step = [sp.expand(Pi_j((j + 1) % 4, 0) - I * Pi_j(j, 0)) for j in range(3)] check("S4.3 [FINITE MONODROMY = CENTRE PERMUTATION] the full " "circle a0 -> e^{2 pi i} a0 is z0 -> i z0, zb0 -> -i zb0 " "(a0 = -z0^4 returns exactly: remainder %s): the O(2) " "sections permute q_p -> q_{p+1} exactly (remainders %s) " "and Pi_{j+1}(0) = i Pi_j(0) (%s): the integrated shift is " "the cyclic centre permutation with period phase i " "(v515 S3.3 / v514 S4.3 replicated)" % (a0_shift, fmt(shifted), fmt(per_step)), a0_shift == 0 and all(v == 0 for v in shifted) and all(v == 0 for v in per_step)) node_free = [] for j in range(4): d = sp.expand((Q4R[(j + 1) % 4] - Q4R[j]) / t0) node_free.append(t0 not in d.free_symbols) lam_c = sp.exp(3 * I * PI / 4) at_node = sp.expand(sp.expand_complex( (Q4R[1] - Q4R[0]).subs(zeta, lam_c))) discs = {} for p, r_ in combinations(range(4), 2): discs[(p, r_)] = sp.factor(sp.discriminant( sp.expand(Q4R[p] - Q4R[r_]), zeta)) ok_disc = all(sp.simplify(d) != 0 for d in discs.values()) ok_t2 = all(sp.degree(sp.expand(d), t0) == 2 for d in discs.values()) check("S4.4 [a0-RIGID FLUX STRUCTURE] (q_{j+1} - q_j)/t0 is " "t0-FREE for all adjacent pairs (%s): the node la-support " "(the 8 eighth roots of unity, e.g. (q_1 - q_0)(e^{3 i " "pi/4}) = %s) does NOT move under the a0 deformation -- the " "Z8 seam bridge is a0-rigid; all 6 pair discriminants are " "nonzero and ~ t0^2 (%s): the K4 connectivity that FORCES " "the uniform flux vector N(1,1,1,1) persists for every " "t0 > 0" % (fmt(node_free), at_node, fmt(discs[k] for k in sorted(discs))), all(node_free) and at_node == 0 and ok_disc and ok_t2) v1s, v2s, w1s, w2s = sp.symbols('v1 v2 w1 w2') r4 = (v1s * w1s + v2s * w2s) ** 2 Kf = {(0, 1, 3): w1s / r4, (0, 1, 2): -w2s / r4} kernel_free = all(t0 not in sp.sympify(v).free_symbols and z0 not in sp.sympify(v).free_symbols for v in Kf.values()) ths, phs, pss = sp.symbols('th2 ph2 ps2', real=True) v1 = sp.cos(ths) * sp.exp(I * phs) v2 = sp.sin(ths) * sp.exp(I * pss) coords = (ths, phs, pss) dv1 = [sp.diff(v1, c) for c in coords] dv2 = [sp.diff(v2, c) for c in coords] etaR = [sp.expand(sp.conjugate(v1) * sp.diff(sp.conjugate(v2), c) - sp.conjugate(v2) * sp.diff(sp.conjugate(v1), c)) for c in coords] dens = sp.simplify(sp.expand(sp.Matrix([dv1, dv2, etaR]).det())) per = sp.integrate(dens, (ths, 0, PI / 2)) * (2 * PI) * (2 * PI) check("S4.5 [QUANTISED FLUX CANNOT MOVE] the CPS/BM kernel K " "contains NO modulus (free of t0, z0: %s) and int_{S^3} K = " "%s = (2 pi i)^2 (v515 S4.1 replicated): the (2 pi i)^2 x " "N(1,1,1) flux vector is topological -- the first-order a0 " "shift moves the seam-fibre PHASE vector 2 pi i t0 (i-1)" "(1, i, i^2) only through t0, never the integer charges: " "the Omega_N period structure is PRESERVED" % (kernel_free, per), kernel_free and sp.simplify(per - (2 * PI * I) ** 2) == 0) # --------------------------------------------------------------------------- # S5 -- negative controls # --------------------------------------------------------------------------- def section5(): print(" -- S5: negative controls") cosg = [sp.sin(th) * sp.cos(ph - phi0 - p * PI / 2) for p in range(4)] V4 = sp.expand_trig(sp.expand(sum(sp.legendre(4, cg) for cg in cosg))) A40 = sp.simplify(sp.integrate(V4, (ph, 0, 2 * PI)) / (2 * PI)) target40 = sp.Rational(1, 16) * (105 * sp.sin(th) ** 4 - 120 * sp.sin(th) ** 2 + 24) check("S5.1 [NC-a: THE (4,0) MULTIPOLE] the m = 0 component of " "S_4 is %s: phi0-INDEPENDENT (d/dphi0 = %s) -- the (4,0) " "direction is clock- and U(1)-invariant and does NOT break " "the symmetry, exactly as preregistered (the kernel-side " "m = 0 piece -6 t0^4 of S2.3 is its uplift)" % (A40, sp.diff(A40, phi0)), sp.simplify(A40 - target40) == 0 and sp.diff(A40, phi0) == 0) cosg2 = [sp.sin(th) * sp.cos(ph - phi0 - p * PI) for p in range(2)] V1 = sp.expand_trig(sp.expand(sum(sp.legendre(1, cg) for cg in cosg2))) V2 = sp.expand_trig(sp.expand(sum(sp.legendre(2, cg) for cg in cosg2))) amp22 = sp.simplify(sp.integrate( V2 * sp.cos(2 * (ph - phi0)), (ph, 0, 2 * PI)) / PI) qeh = [z0, -z0] PEH = sp.expand(sp.prod([eta - q for q in qeh])) serEH = sp.series(sp.log(1 - z0 ** 2 * u ** 2), u, 0, 4).removeO() chiEH0 = sp.log(-t0 ** 2 * sp.exp(2 * I * phi0)) dlogEH_t0 = sp.simplify(t0 * sp.diff(chiEH0, t0)) PiEH = 4 * PI * I * z0 a0EH = -z0 ** 2 dlogEH = sp.simplify((sp.diff(PiEH, z0) * z0 / PiEH) / (sp.diff(a0EH, z0) * z0 / a0EH)) monoEH = sp.simplify(sp.exp(2 * PI * I * dlogEH)) check("S5.2 [NC-b: Z2/EGUCHI-HANSON ANALOGUE] two centres +-z0: " "dipole = %s; the first symmetry-breaking multipole is " "(2, +-2) with amplitude %s = (3/2) sin^2 th carried by " "t0^2 e^{2 i phi0} = -a0^EH; kernel log(eta^2 + a0^EH): log " "coefficient 2 = |Z2|, first correction %s u^2 = a0^EH/" "eta^2; exceptional log t0 d/dt0 chi(0) = %s = 2 = centre " "count; period response d log Pi/d log a0 = %s = 1/2 = " "1/|Z2|, monodromy e^{2 pi i/2} = %s = -1 = the Z2 clock: " "EVERY dial reads 2 -- the coefficient is COUPLED to the " "centre count (2 vs 4), not universal" % (V1, amp22, serEH.coeff(u, 2), dlogEH_t0, dlogEH, monoEH), V1 == 0 and sp.simplify(amp22 - sp.Rational(3, 2) * sp.sin(th) ** 2) == 0 and sp.expand(PEH - (eta ** 2 - z0 ** 2)) == 0 and serEH.coeff(u, 2) == -z0 ** 2 and dlogEH_t0 == 2 and dlogEH == sp.Rational(1, 2) and monoEH == -1) rows = {} for k in (3, 5): w = sp.exp(2 * PI * I / k) qs = [w ** p * z0 for p in range(k)] Pk = sp.expand(sp.prod([eta - q for q in qs])) cf = {n: sp.simplify(sp.expand_complex( sp.expand(Pk.coeff(eta, n)))) for n in range(k + 1)} mids_zero = all(cf[n] == 0 for n in range(1, k)) logcoef = k if (cf[k] == 1 and mids_zero) else None const_ok = sp.simplify(sp.expand_complex( cf[0] + z0 ** k)) == 0 slot = 2 * k rows[k] = (logcoef, const_ok, slot) check("S5.3 [NC-c: WRONG CENTRE COUNT] k-orbits: k = 3 gives " "kernel log coefficient %d and modulus slot O(%d), k = 5 " "gives %d and O(%d) -- the log coefficient MOVES with the " "centre count (3, 4, 5 all distinct: the observable " "discriminates, the KILL 'decoupled from the centre count' " "does not fire); only k = 4 = |mu4| hits the O(8) a0 slot " "of the seam weight ledger (v505 S5.3); constant terms " "prod(eta - q_p)|_0 = -z0^k verified: %s, %s (the " "k-th-power moduli)" % (rows[3][0], rows[3][2], rows[5][0], rows[5][2], rows[3][1], rows[5][1]), rows[3][0] == 3 and rows[5][0] == 5 and rows[3][2] == 6 and rows[5][2] == 10 and rows[3][1] and rows[5][1]) OM = sp.Rational(-1, 2) + sp.sqrt(3) * I / 2 ZF = [sp.Integer(0), sp.Integer(1), OM, sp.expand(OM ** 2)] pnF = {n: sp.simplify(sp.expand_complex( sum(z ** n for z in ZF))) for n in (1, 2, 3)} e4F = sp.simplify(sp.expand_complex(sp.prod(ZF))) serF = sp.series(sp.log(1 - u ** 3), u, 0, 4).removeO() check("S5.4 [NC-d: CLOCK-FORBIDDEN FAMILY] centres {0, 1, w, w^2} " "(Z^4 - Z, v515 S5): power sums (p1, p2, p3) = %s -- p3 = 3 " "!= 0 breaks the m = 0 mod 4 potential selection rule (first " "kernel correction %s u^3 sits in the forbidden m = 3 " "slot); e4 = prod z_p = %s = 0: NO a0-log at the exceptional " "locus (a centre sits at eta = 0, the modulus direction is " "absent) -- the forbidden family fails BOTH uplift dials" % (fmt([pnF[1], pnF[2], pnF[3]]), serF.coeff(u, 3), e4F), pnF[1] == 0 and pnF[2] == 0 and pnF[3] == 3 and e4F == 0 and serF.coeff(u, 3) == -1) # --------------------------------------------------------------------------- # S6 -- verdict # --------------------------------------------------------------------------- def section6(): print(" -- S6: verdict per the preregistered v514 S8.3 criterion") check("S6.1 [WHAT IS EXACT] (a) the kernel chi = log P4 (log of " "the v515 family polynomial) is harmonic-typed (null " "coordinate) and reproduces the V-ledger through the exact " "residue identity 1/(2r); (b) asymptotic log coefficient " "4 = |mu4|, seam-fibre first correction a0/eta^4 with exact " "m-grading (m = +-4 pure a0, m = 0 phi0-free), GLT tower " "p_{4k} = 4(-a0)^k, no asymptotic log corrections; (c) " "exceptional-locus log: chi(0) = log a0, t0-coefficient 4, " "clock phase 4i, modulus response 1 at the locus vs " "power-law at infinity; (d) periods: uniform first-order " "shift (lockstep preserved), d log Pi/d log a0 = 1/4, " "integrated monodromy = i = one clock step, node support " "and (2 pi i)^2 fluxes a0-rigid; (e) all controls", True) check("S6.2 [VERDICT: SUCCESS -- with the honest fence] " "preregistered SUCCESS 'log-type correction whose " "coefficient is tied to the source charge 4 = |mu4| " "(matching the V-ledger)': DELIVERED on four coupled dials " "(kernel log 4, exceptional-locus log 4 log t0, tower " "prefactor 4, period response 1/4 -> monodromy i), each " "moving to 2 for EH/Z2 and to k for k-orbits: the KILL " "(decoupling from the centre count) does NOT fire. FENCE: " "'eta log eta = THE GH Kahler-potential kernel' is the " "generalized-Legendre-transform dictionary, typed [C]; the " "full nonlinear Kahler potential of the resolved A3 ALE is " "not constructed here (no closed form exists); NO marker " "moves anywhere", True) # --------------------------------------------------------------------------- def run(): reset() print("v517 CELEST.WP5E.M3.01: the a0 uplift -- twistor-potential " "kernel, exceptional-locus log, period response (M3 of " "CELEST.SEAM.01, SUCCESS)") a0z = section0() section1() section2(a0z) section3() section4() section5() section6() return summary("v517 CELEST.WP5E.M3.01: the a0 uplift -- M3 " "executed (SUCCESS on the preregistered v514 S8.3 " "criterion): the (4, +-4) multipole uplifted to " "the GLT kernel chi = log P4 (the log of the v515 " "family polynomial; null coordinate => harmonic " "for any kernel, residue identity d_x transform = " "1/r_p matches the V-ledger, flux -4 pi per " "centre); FOUR COUPLED CENTRE-COUNT SCALES: " "asymptotic kernel log = 4 = |mu4| with 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); period response d log Pi/" "d log a0 = 1/4 = 1/|mu4| uniform, integrating to " "the clock monodromy i (v493 reproduced), node " "support and (2 pi i)^2 fluxes a0-rigid; controls: " "(4,0) clock-invariant, EH/Z2 reads 2 = |Z2| on " "every dial, k = 3/5 orbits move the coefficient " "with the centre count, forbidden family fails " "both dials (p3 = 3, e4 = 0); FENCE: the GLT " "dictionary is [C], the full nonlinear Kahler " "potential of the resolved A3 ALE stays [O]; no " "marker moves") if __name__ == "__main__": raise SystemExit(1 if run() else 0)