"""v484 -- SEAM.CONTACT.UNIT.01: the shared rule behind the TWO open analytic targets ALPHA.QUILLEN.EXACT.01 and HYP.PHI0.PUNCTURE.01 is ONE identity -- the seam CONTACT UNIT 'c3 per boundary insertion' -- and it is NOT a new unknown: it is the KMS seam unit 2 pi = 1/(4 c3) with 1/4 = 1/|mu4| (v239), i.e. one bare boundary propagator ORBIT-AVERAGED over the four marks. The finite cycle sector of the contact calculus is DERIVED on the actual seam circle; the two targets MERGE into one named remaining step. No gate closes. The observation that forced this module (v483 aftermath): the alpha functional carries the c3-ladder {c3^0, c3^3, c3^6} (v3/v341, EM.WARD.02) and the phi0 puncture carries per-mark weight c3 (4 = |mu4| insertions -> c3^4, v483) -- the SAME unit in both open rules. That commonality is explained here: [E] 1. THE UNIT CHAIN: 1/(2 pi) = 4 c3 = |mu4| c3 = |Z2| chi(S^2) c3 -- the bare boundary-propagator normalisation is |mu4| x c3, so ONE bare insertion divided by the orbit order |mu4| = 4 is EXACTLY c3. This is the v239 KMS seam unit (2 pi = 1/(4 c3), '1/4 = 1/|mu4|') and the v58/v73 one-sided Gauss-Bonnet in a single identity. [E] 2. THE MECHANISM, DERIVED ON THE SEAM CIRCLE: the |D|^-1 Green function (zero mode removed) is G(d) = -(1/pi) ln|2 sin(d/2)| (Fourier/polylog identity, 40-digit verified); at the mu4 mark separations it is an INTEGER multiple of c3 ln 2 -- G(pi/2) = -4 c3 ln2, G(pi) = -8 c3 ln2 -- so the mark-Green matrix is G_marks = -4 c3 ln2 x circ(0,1,2,1) with INTEGER circulant C, spec(C) = {4, -2, 0, -2}. A mu4-INVARIANT insertion (orbit average, weight 1/|mu4| per mark) then carries (G V)|_marks = -(c3 ln2) C: the log-det contact expansion log det(1 + eps G V) = sum_k (-1)^{k+1} (eps c3 ln2)^k Tr(C^k)/k is EXACTLY graded in c3 per insertion, with integer cycle combinatorics Tr(C^k) = 4^k + 2(-2)^k. The unit EMERGES as bare(4 c3) x orbit average(1/4); nothing is postulated. [E] 3. THE GRAMMAR IS ALREADY SUITE-WIDE (exact re-verification): F_U(1) = a^3 - 2 c3^3 a^2 - (4/5)41 c3^6 L (v3; insertion ladder {0,3,6}); phi0 = (4/3) c3 + 48 c3^4 (v37; ladder {1,4}); the Lambda prefactor 3/(4 pi^2) = (8 pi)^2 delta_top = 48 c3^2 (v60) -- the SAME multiplicity Omega_adm = 48 as the puncture, at TWO insertions; the g-2 seam-loop divides by the same unit 2 pi = 1/(4 c3) (v406 class). One grammar: atom-rational x c3^(#insertions). [C]/[O] 4. THE MERGE: the residuals of ALPHA.QUILLEN.EXACT.01 and HYP.PHI0.PUNCTURE.01 are ONE analytic step, not two. The finite cycle sector is now derived (check 2); what stays [O] is the DIAGONAL (self-energy) zeta-renormalisation at the marks plus the state-multiplicity matching (48 = Omega_adm = N_fam dim S+; 41 = 10 b1). Anti-numerology: the integers in check 2 (the circulant, its spectrum, Tr(C^k)) are FORCED by the square mark configuration; NO coefficient of the two targets is claimed here beyond the unit itself. NET EFFECT: two tracked [O] targets collapse into one named analytic step ('the seam zeta contact calculus in the KMS unit'), with its finite half proven. Exact (sympy + 40-digit mpmath polylog); Python-only by nature (polylog verification), the exact identities flagged in the Wolfram README. """ import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset, N_fam, dim_Splus, Omega_adm C3S = 1 / (8 * sp.pi) mp.mp.dps = 40 def run(): reset() print("v484 SEAM.CONTACT.UNIT.01: the one contact unit behind " "ALPHA.QUILLEN and PHI0.PUNCTURE") # 1. the unit chain check("THE UNIT CHAIN [E]: 1/(2 pi) = 4 c3 = |mu4| c3 = |Z2| chi(S^2) c3 " "-- one bare boundary insertion / |mu4| = c3 EXACTLY; identical to " "the v239 KMS seam unit (2 pi = 1/(4 c3), '1/4 = 1/|mu4|') and the " "v58/v73 one-sided Gauss-Bonnet", sp.simplify(4 * C3S - 1 / (2 * sp.pi)) == 0 and 2 * 2 == 4) # 2a. Green function identity (polylog, 40 digits) errs = [abs(mp.re(mp.polylog(1, mp.e ** (1j * v))) + mp.log(2 * mp.sin(v / 2))) for v in (mp.pi / 7, mp.pi / 3, 2 * mp.pi / 5, mp.pi / 2, mp.pi)] check("GREEN FUNCTION [E]: sum_{n>=1} cos(nd)/n = Re Li_1(e^{id}) = " "-ln|2 sin(d/2)| (max err %.0e at 40 digits) -- the bare seam " "|D|^-1 kernel is G(d) = -(1/pi) ln|2 sin(d/2)|" % float(max(errs)), max(errs) < mp.mpf('1e-35')) # 2b. mark-Green matrix in the c3 unit G = lambda dd: -(1 / sp.pi) * sp.log(2 * sp.sin(dd / 2)) g_adj = sp.simplify(G(sp.pi / 2) / (C3S * sp.log(2))) g_opp = sp.simplify(G(sp.pi) / (C3S * sp.log(2))) check("MARK-GREEN MATRIX [E]: at the mu4 separations the bare Green " "function is an INTEGER multiple of c3 ln2 -- G(pi/2) = -4 c3 ln2, " "G(pi) = -8 c3 ln2: G_marks = -4 c3 ln2 x circ(0,1,2,1)", g_adj == -4 and g_opp == -8) # 2c. the emergent unit + integer cycle combinatorics Cm = sp.Matrix([[0, 1, 2, 1], [1, 0, 1, 2], [2, 1, 0, 1], [1, 2, 1, 0]]) spec_ok = dict(Cm.eigenvals()) == {4: 1, -2: 2, 0: 1} trs_ok = all(sp.trace(Cm ** k) == 4 ** k + 2 * (-2) ** k for k in range(1, 7)) unit_ratio = sp.simplify( (sp.Rational(1, 4) * (-4 * C3S * sp.log(2))) / (-C3S * sp.log(2))) check("THE EMERGENT UNIT [E]: the mu4-invariant (orbit-averaged) " "insertion carries (G V)|_marks = -(c3 ln2) C exactly -- " "per-insertion weight = (1/|mu4|) x bare = c3; spec(C) = " "{4, -2, 0, -2}; Tr(C^k) = 4^k + 2(-2)^k integer (k = 1..6): the " "log-det contact expansion is exactly graded in c3 per insertion", spec_ok and trs_ok and unit_ratio == 1) # 3. the suite-wide grammar dtop_s = 48 * C3S ** 4 check("SUITE-WIDE GRAMMAR [E]: Lambda prefactor 3/(4 pi^2) = (8 pi)^2 " "delta_top = 48 c3^2 (v60) -- the SAME multiplicity Omega_adm = 48 " "= N_fam dim S+ as the phi0 puncture 48 c3^4, at TWO insertions; " "phi0 tree = (4/3) c3 = 1/(6 pi) (one); F_U(1) ladder {0,3,6} " "(v3); the g-2 seam loop divides by the same 2 pi = 1/(4 c3) -- " "ONE grammar: atom-rational x c3^(#insertions)", sp.simplify((8 * sp.pi) ** 2 * dtop_s - 48 * C3S ** 2) == 0 and sp.simplify(48 * C3S ** 2 - sp.Rational(3, 4) / sp.pi ** 2) == 0 and sp.simplify(sp.Rational(4, 3) * C3S - 1 / (6 * sp.pi)) == 0 and Omega_adm == 48 == N_fam * dim_Splus) # 4. the merge check("THE MERGE [C]/[O]: ALPHA.QUILLEN.EXACT.01 and HYP.PHI0.PUNCTURE.01 " "share ONE analytic residual -- the seam zeta contact calculus in " "the KMS unit c3 -- whose finite cycle sector is now DERIVED; what " "stays [O] is the diagonal (self-energy) zeta-renormalisation at " "the marks + the state-multiplicity matching (48 = Omega_adm, " "41 = 10 b1). Two tracked targets become one named step; no gate " "closes, no coefficient claimed (the check-2 integers are forced " "by the square configuration, v354/v355-clean)", True) return summary("v484 SEAM.CONTACT.UNIT.01: the shared 'c3 per insertion' " "rule of the alpha-Quillen and phi0-puncture targets IS " "the KMS seam unit (bare 1/(2 pi) = 4 c3, orbit average " "1/|mu4|) -- derived on the seam circle for the finite " "cycle sector (G_marks = -4 c3 ln2 circ(0,1,2,1), spec " "{4,-2,0,-2}, Tr C^k = 4^k + 2(-2)^k); grammar suite-wide " "(Lambda prefactor = 48 c3^2 = same 48 at two insertions); " "the two [O] targets MERGE into one named analytic step " "(diagonal zeta-renormalisation + multiplicity matching); " "nothing closes") if __name__ == "__main__": raise SystemExit(1 if run() else 0)