#!/usr/bin/env python3 """v790 -- HECKE.SNF_THETA.01: the SNF fingerprint of the correction theta against the ramified E8 inclusion, plus the exact mu4 -> C2 packet-channel split (the claim also covers HECKE.MU4_PACKET.01), ONE module from two probes (18/18 + 17/17 checks, ~3 s; discovery probes hecke_snf_theta_probe.py SNF-FINGERPRINT and hecke_mu4_packet_probe.py MU4-PARTIAL, both 2026-08-05). THE FINGERPRINT (part 1): the representation identity 256 R(n) = #{(u,v) in (2Z+1)^8 : sum u_i^2 + 2 sum v_j^2 = 4n} for the correction theta H(q) = q^3 psi(q^2)^4 psi(q^4)^4 is exact by direct lattice enumeration (n <= 40; two series routes agree to O(q^48)), and the SNF match divisors(Gram_M) == divisors((1+i)L c L) == (1,1,1,1,2,2,2,2) is exact (v689 replication, |det| = 16 = N(1+i)^4 on both sides) -- but the isometry census is EMPTY at every admissible level: AMBIENT kissing 240 vs 8 with det*min^-8 mismatch on BOTH pair members (1/256 vs 16 -- proof-grade: no R-linear isometry at ANY scale lambda can carry L -> lambda M* or (1+i)L -> lambda M); QUOTIENT census 0/20160 over ALL of GL(4,F2) (the ramified form b_V is alternating, the disc form b_D of M is the identity form -- b_V(x,x) = 0 for all x vs b_D(e_i,e_i) = 1; |Aut(b_V)| = 720 = |Sp(4,F2)|, |Aut(b_D)| = 48; the sigma/mu4-equivariance census is vacuous); and the naive 4+4 coordinate split is inadmissible BY THEOREM (v638 S1.2 replicated independently: 0 of 70 coordinate 4-subsets are (pi_J, pi_sigma)-invariant). The wrong-signature control diag(1,1,1,2,2,2,2,2) fires (divisors (1^3, 2^5), disc order 32 != 16). The SNF match is therefore typed as a FINGERPRINT, not an isometry. THE PACKET SPLIT (part 2): r-hat = (240, -8, -16, -8) and j-hat = (248, 0, -8, 0) exact (Gaussian-integer DFT on the glue group mu4 = E8/(D5 (+) A3); corpus census r = (52, 64, 60, 64), signed 52 - 60 = -8 = -rank(E8); N1 = N3 for EVERY shell proof-grade via x -> -x); the NEW exact identity: csig = (1/15) E4(q^2) - (6/5) E4(q^4) + (32/15) E4(q^8) - 8 f8 to O(q^64), i.e. the cusp part of the QUARTIC mu4 character is EXACTLY -8 f8 with -8 = 52 - 60 = -rank(E8) (ties the v537 lift Sh(g) = -8 f8); the all-shell quartic vanishing is FALSE -- the Cartan-closed quartic series has the exact closed form prod (1 + q^{2n})^{-4} and fails from grade 2 (the grade-1 vanishing IS the corpus j-hat anchor); the C2 (quadratic) channel is PURE Eisenstein, theta4(q)^8 = (16 E4(q^2) - E4(q))/15 termwise -- it carries NO cusp form; csig(p) = -8 a_p(f8) at all odd primes p <= 47 with the corpus anchors a_p = (-4, -2, 24, -44, 22) at p = (3, 5, 7, 11, 13) (v536 Eichler reading a_p = -c(p)/8). Both controls fire (scrambled sector assignment, wrong Cartan completion). No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. PROVENANCE: discovery probes hecke_snf_theta_probe.py (2026-08-05, 18/18, 2.2 s, SNF-FINGERPRINT) + hecke_mu4_packet_probe.py (2026-08-05, 17/17, 0.2 s, MU4-PARTIAL); no spec corrections are disclosed in either probe docstring; both re-run identically at promotion. Promoted verbatim, part 1's top-level executable statements moved unchanged into main() and part 2 wrapped in a function scope; each part's banner print of its own docstring is served by a local __doc__ binding that holds the original probe docstring verbatim; part 2's check() bookkeeping switches global -> nonlocal and additionally records the (name, ok) pairs (print/bookkeeping only, no gate, no number touched); numbers unchanged; run() encodes both patterns (v757 precedent). Original hecke_snf_theta_probe.py docstring (verbatim): hecke_snf_theta_probe -- HECKE.SNF_THETA.01 (positive-protocol round, strand C, probe 1 of 2). THE CLAIM UNDER TEST [C neu -> decidable]: the correction theta H(q) = q^3 psi(q^2)^4 psi(q^4)^4 (psi = Ramanujan's theta, psi(q) = sum_{k>=0} q^{k(k+1)/2}) has the quadratic form Q(u, v) = u_1^2 + ... + u_4^2 + 2(v_1^2 + ... + v_4^2) on ODD vectors: 256 R(n) = #{(u,v) in (2Z+1)^8 : Q(u,v) = 4n}, where R(n) is the q^n coefficient of H. Its Gram matrix diag(1,1,1,1,2,2,2,2) has Smith signature (1,1,1,1,2,2,2,2) -- EXACTLY the elementary divisors of the ramified inclusion (1+i)L c L of the unimodular hermitian E8 lattice (v689 machinery, read-only replication). THE DECIDER: does an isometry exist between the correction-theta lattice pair (M = Z^4 (+) sqrt2 Z^4 with the odd-coset structure, i.e. M c M* with M*/M = F2^4) and the ramified inclusion pair ((1+i)L c L with L/(1+i)L = F2^4), under ADMISSIBLE basis changes only? ADMISSIBLE CLASS (predeclared, frozen): maps definable from the Z[i]/transversal structure -- (i) ambient: R-linear isometries T (a global scale factor lambda > 0 allowed) with T(L) = lambda M* and T((1+i)L) = lambda M, composed with U(L,h)-automorphisms and the ramified twist x -> (1+i)x; (ii) quotient: F2-linear maps V = L/(1+i)L -> D = M*/M intertwining the induced finite bilinear structures (the symplectic / Fourier / Weil-representation class); INADMISSIBLE (the brake): any map whose definition REQUIRES a J/sigma-equivariant 4+4 coordinate split of the 8 real coordinates. The corpus has PROVEN no such split exists: v638_code_semantics.py S1.2 ("Z6-invariant 4-subsets: 0 => NO equivariant block-systematic 4+4 split exists"); replicated independently in S3c below. The naive coordinatewise map (first 4 coords -> u-block, last 4 -> v-block) is therefore inadmissible BY THEOREM, and the decider must census the transversal-coded maps only. SLICES: S1 the representation-number identity: H(q) two routes (psi-series and eta-quotient eta(4)^4 eta(8)^8 / eta(2)^4), then 256 R(n) = #{odd (u,v) : Q = 4n} by DIRECT lattice enumeration (meet-in-the-middle over the two 4-blocks) for n <= NCAP = 40. S2 Smith side: SNF(Gram_M) = (1,1,1,1,2,2,2,2); v689 replication (C* placement, L = Construction A, A = coords of (1+J)b_i), SNF(A) = (1,1,1,1,2,2,2,2); EXACT elementary-divisor match. Census info: per-class theta of L (shells 1..3) vs the odd-coset theta of M (normalized) -- the two 16-element quotient structures side by side. S3 THE DECIDER, three levels: (a) ambient isometry census: scale-invariant obstructions (kissing number, det x min^{-8}) for L vs M* and (1+i)L vs M -> census count at level (i); (b) quotient census: b_V = hbar (v752 convention, alternating -- replicated from the concrete lattice) vs b_D = identity form on F2^4 (disc form of M, NON-alternating); FULL GL(4,F2) census (20160 invertible maps): # intertwiners, |Aut(b_V)|, |Aut(b_D)|; sigma/mu4 equivariance of any intertwiner found; (c) the brake demonstration: all 70 coordinate 4-subsets tested for (pi_J, pi_sigma)-invariance (expect 0 -> the naive coordinatewise map FAILS the admissibility gate). S4 controls: wrong-signature lattice diag(1,1,1,2,2,2,2,2) must FAIL the SNF match (divisors (1^3,2^5), disc order 32 != 16); the naive map must fail the gate (S3c). Verdict adjudication. VERDICTS (frozen): SNF-ISOMETRIC (admissible isometry exists), SNF-FINGERPRINT (SNF match exact but NO admissible isometry -- the match is typed as a fingerprint), SNF-FALSE (theta identity or SNF match fails). FENCES: [C neu] readings typed; v775 ROOTCLASS-MIXED cited -- the census is lattice/character-level, NO root-level matter reading and NO physics claim is attached. Firewall: experiments/ discovery sandbox; writes nothing; no verification/, ledger, paper or website surface touched. Exact integer/Fraction arithmetic; sympy only for SNF; numpy only for integer enumeration. Deterministic (no RNG). Sources (read-only): v689_gaussian_code_bridge.py (C*, L, J, A, SNF), v752_projective_hamming_incidence.py (hbar convention), v638_code_semantics.py (no-4+4-split theorem), v775 (ROOTCLASS-MIXED). Original hecke_mu4_packet_probe.py docstring (verbatim): hecke_mu4_packet_probe -- HECKE.MU4_PACKET.01 (positive-protocol round, strand C, probe 2 of 2): the mu4 -> C2 collapse [E-candidate]. CORPUS ANCHORS (located): first-shell glue sector counts r = (52, 64, 60, 64) (part-11 probe e8_glue_chi4_signed_theta_probe.py, v492/v505 replications); weight-1 current sectors j = (60, 64, 60, 64) = r + (8, 0, 0, 0) (Cartan; the beta-equivariant route gives the graded character (248, 0, -8, 0)); signed root census 52 - 60 = -8 = -rank(E8); f8 = q eta(2t)^4 eta(4t)^4 (weight-4 newform, level 8); v537: Sh_{t=2}(g) = -8 f8; v536 Eichler reading a_p = -c(p)/8. DFT CONVENTION (frozen): characters chi_t(c) = i^{t c} on the glue group mu4 = Z4 = E8/(D5 (+) A3); ordering t = (0, 1, 2, 3) = (trivial, quartic i, quadratic -1, quartic i^3). r-hat = (240, -8, -16, -8); j-hat = (248, 0, -8, 0). CARTAN CLOSURE (frozen, all shells): the level-1 current-algebra grading -- multiply each sector theta by the 8-fold oscillator factor 1/phi(q)^8, phi(q) = prod (1 - q^n) (lattice-VOA grading: e^lambda (x) oscillators; oscillators are glue-neutral). Grade 1 reproduces j = (60, 64, 60, 64) with the 8 Cartan modes in the trivial sector. FROZEN CANDIDATE IDENTITIES (declared BEFORE computing): K1 quadratic channel (unclosed) IS pure Eisenstein: Th0 - Th1 + Th2 - Th3 = theta4(q)^8 = (16 E4(q^2) - E4(q))/15 termwise to O(q^NSER); in particular its f8-component is 0. K2 THE -8 f8 NORMALIZATION (the Section-12 claim, sharpened): the cusp content of the QUARTIC character series csig = Th0 - Th2 is EXACTLY -8 f8, with -8 = the shell-1 quartic DFT = signed census 52 - 60 = -rank(E8): csig + 8 f8 lies in the pure sigma3-Eisenstein span {E4(q^d) : d = 1,2,4,8,16} + Eis(chi4,chi4), exactly to O(q^NSER). Consequence tested separately: csig(n) = -8 f8(n) for ALL odd n. K3 prime anchors: a_p(f8) = (-4, -2, 24, -44, 22) at p = (3, 5, 7, 11, 13) (v536/v537 values) and csig(p) = -8 a_p(f8) for all odd primes p <= 47. K4 all-shell quartic vanishing (the user's collapse claim): the Cartan-closed quartic series csig(q)/phi(q)^8 vanishes in EVERY grade n >= 1 (not just grade 1). Exact eta form of the closed series: csig/phi^8 = prod (1 + q^{2n})^{-4} (machine-verified), so the claim is decidable termwise. TASKS: S1 glue machinery replication (part-11 D5 (+) A3 basis, linear classifier): census (52, 64, 60, 64), signed -8; N1(n) = N3(n) for ALL shells proof-grade via x -> -x (deg(-x) = -deg(x) mod 4). S2 exact DFT at shell 1: r-hat, j-hat as Gaussian integers. S3 all-shell extension: per-class enumeration n = 1..6 (norm <= 12), theta-constant series Th0..Th3 to O(q^NSER = 64) cross-checked against enumeration; per-shell DFT table; Cartan closure of all four character series; the K4 vanishing test with first-failure grade; the closed-quartic eta identity. S4 the packet-channel identities K1, K2, K3; adjudication of the composite claim "the surviving quadratic character IS the C2 packet channel with the exact -8": which character carries the -8 f8 cusp packet, which carries the surviving -8 constant. S5 controls (must fire): (a) scrambled sector assignment (seeded) breaks the quartic cancellation at shell 1; (b) wrong Cartan completion (4, 0, 0, 4) breaks j-hat = (248, 0, -8, 0). VERDICTS (frozen): MU4-COLLAPSE-EXACT (quartic vanish ALL shells after closure + quadratic = the packet channel with the exact -8), MU4-PARTIAL (shell-1 collapse exact + at least one frozen identity K1-K3 exact, but not all-shell vanishing), MU4-DEAD (shell-1 anchors fail or controls do not fire). FENCES: [E-candidate] readings are exact replications/identities; extensions typed as measured; v775 ROOTCLASS-MIXED cited -- the glue census is representation-level, no root-level matter reading and NO physics claim. Firewall: experiments/ discovery sandbox; writes nothing; no verification/, ledger, paper or website surface touched. Exact integer/Fraction/Gaussian-integer arithmetic; sympy for exact linear algebra; numpy only for integer enumeration; RNG only inside the negative control (fixed seed). Sources (read-only): e8_glue_chi4_signed_theta_probe.py (part 11), v492/v505 (root census), v535/v536/v537/v538 (Hecke/Eichler/f8 layer), v775 (ROOTCLASS-MIXED fence). """ import itertools import time from collections import Counter from fractions import Fraction as Fr import numpy as np from sympy import Matrix, ZZ from sympy.matrices.normalforms import smith_normal_form CHECKS = [] QMAX = 48 # q-series order (frozen) NCAP = 40 # direct-enumeration cap for the 256 R(n) identity SHELLS_INFO = 3 # class-theta census shells (info) TARGET_DIV = [1, 1, 1, 1, 2, 2, 2, 2] def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (" -- " + detail) if detail else ""), flush=True) def info(msg): print(" %s" % msg, flush=True) def section(title): print("=" * 78) print(title) print("=" * 78, flush=True) # ---------------------------------------------------- integer q-series def pmul(a, b, order): res = [0] * (order + 1) for i, ai in enumerate(a): if ai: top = order - i for j in range(min(len(b) - 1, top) + 1): if b[j]: res[i + j] += ai * b[j] return res def eta_pow(d, e, order): """prod_{n>=1} (1 - q^{d n})^e, no q-prefactor.""" s = [0] * (order + 1) s[0] = 1 for n in range(1, order // d + 1): f = [0] * (d * n + 1) f[0], f[d * n] = 1, -1 for _ in range(e): s = pmul(s, f, order) return s def pinv(a, order): assert a[0] == 1 res = [0] * (order + 1) res[0] = 1 for n in range(1, order + 1): res[n] = -sum(a[k] * res[n - k] for k in range(1, min(n, len(a) - 1) + 1)) return res def shift(a, k, order): return ([0] * k + a)[: order + 1] def rescale(a, d, order): res = [0] * (order + 1) for i, ai in enumerate(a): if ai and i * d <= order: res[i * d] = ai return res def main(): # PART 1 -- hecke_snf_theta_probe.py (top-level executable # statements moved unchanged into this function; the banner print # of the probe docstring is served by the local __doc__ binding) T0 = time.time() __doc__ = """hecke_snf_theta_probe -- HECKE.SNF_THETA.01 (positive-protocol round, strand C, probe 1 of 2). THE CLAIM UNDER TEST [C neu -> decidable]: the correction theta H(q) = q^3 psi(q^2)^4 psi(q^4)^4 (psi = Ramanujan's theta, psi(q) = sum_{k>=0} q^{k(k+1)/2}) has the quadratic form Q(u, v) = u_1^2 + ... + u_4^2 + 2(v_1^2 + ... + v_4^2) on ODD vectors: 256 R(n) = #{(u,v) in (2Z+1)^8 : Q(u,v) = 4n}, where R(n) is the q^n coefficient of H. Its Gram matrix diag(1,1,1,1,2,2,2,2) has Smith signature (1,1,1,1,2,2,2,2) -- EXACTLY the elementary divisors of the ramified inclusion (1+i)L c L of the unimodular hermitian E8 lattice (v689 machinery, read-only replication). THE DECIDER: does an isometry exist between the correction-theta lattice pair (M = Z^4 (+) sqrt2 Z^4 with the odd-coset structure, i.e. M c M* with M*/M = F2^4) and the ramified inclusion pair ((1+i)L c L with L/(1+i)L = F2^4), under ADMISSIBLE basis changes only? ADMISSIBLE CLASS (predeclared, frozen): maps definable from the Z[i]/transversal structure -- (i) ambient: R-linear isometries T (a global scale factor lambda > 0 allowed) with T(L) = lambda M* and T((1+i)L) = lambda M, composed with U(L,h)-automorphisms and the ramified twist x -> (1+i)x; (ii) quotient: F2-linear maps V = L/(1+i)L -> D = M*/M intertwining the induced finite bilinear structures (the symplectic / Fourier / Weil-representation class); INADMISSIBLE (the brake): any map whose definition REQUIRES a J/sigma-equivariant 4+4 coordinate split of the 8 real coordinates. The corpus has PROVEN no such split exists: v638_code_semantics.py S1.2 ("Z6-invariant 4-subsets: 0 => NO equivariant block-systematic 4+4 split exists"); replicated independently in S3c below. The naive coordinatewise map (first 4 coords -> u-block, last 4 -> v-block) is therefore inadmissible BY THEOREM, and the decider must census the transversal-coded maps only. SLICES: S1 the representation-number identity: H(q) two routes (psi-series and eta-quotient eta(4)^4 eta(8)^8 / eta(2)^4), then 256 R(n) = #{odd (u,v) : Q = 4n} by DIRECT lattice enumeration (meet-in-the-middle over the two 4-blocks) for n <= NCAP = 40. S2 Smith side: SNF(Gram_M) = (1,1,1,1,2,2,2,2); v689 replication (C* placement, L = Construction A, A = coords of (1+J)b_i), SNF(A) = (1,1,1,1,2,2,2,2); EXACT elementary-divisor match. Census info: per-class theta of L (shells 1..3) vs the odd-coset theta of M (normalized) -- the two 16-element quotient structures side by side. S3 THE DECIDER, three levels: (a) ambient isometry census: scale-invariant obstructions (kissing number, det x min^{-8}) for L vs M* and (1+i)L vs M -> census count at level (i); (b) quotient census: b_V = hbar (v752 convention, alternating -- replicated from the concrete lattice) vs b_D = identity form on F2^4 (disc form of M, NON-alternating); FULL GL(4,F2) census (20160 invertible maps): # intertwiners, |Aut(b_V)|, |Aut(b_D)|; sigma/mu4 equivariance of any intertwiner found; (c) the brake demonstration: all 70 coordinate 4-subsets tested for (pi_J, pi_sigma)-invariance (expect 0 -> the naive coordinatewise map FAILS the admissibility gate). S4 controls: wrong-signature lattice diag(1,1,1,2,2,2,2,2) must FAIL the SNF match (divisors (1^3,2^5), disc order 32 != 16); the naive map must fail the gate (S3c). Verdict adjudication. VERDICTS (frozen): SNF-ISOMETRIC (admissible isometry exists), SNF-FINGERPRINT (SNF match exact but NO admissible isometry -- the match is typed as a fingerprint), SNF-FALSE (theta identity or SNF match fails). FENCES: [C neu] readings typed; v775 ROOTCLASS-MIXED cited -- the census is lattice/character-level, NO root-level matter reading and NO physics claim is attached. Firewall: experiments/ discovery sandbox; writes nothing; no verification/, ledger, paper or website surface touched. Exact integer/Fraction arithmetic; sympy only for SNF; numpy only for integer enumeration. Deterministic (no RNG). Sources (read-only): v689_gaussian_code_bridge.py (C*, L, J, A, SNF), v752_projective_hamming_incidence.py (hbar convention), v638_code_semantics.py (no-4+4-split theorem), v775 (ROOTCLASS-MIXED). """ print(__doc__.split("SLICES")[0]) print("FROZEN SPEC: QMAX = %d, NCAP = %d, target divisors %s;" % (QMAX, NCAP, TARGET_DIV)) print("verdicts SNF-ISOMETRIC / SNF-FINGERPRINT / SNF-FALSE (frozen);") print("admissible class = transversal-coded (see docstring), naive 4+4") print("coordinatewise maps excluded by v638 S1.2 (replicated in S3c).") print() # ====================================================================== S1 section("S1: the correction theta H(q) and the representation identity") psi = [0] * (QMAX + 1) k = 0 while k * (k + 1) // 2 <= QMAX: psi[k * (k + 1) // 2] += 1 k += 1 # classical product form psi(q) = prod (1-q^{2n})^2 / (1-q^n) psi_prod = pmul(eta_pow(2, 2, QMAX), pinv(eta_pow(1, 1, QMAX), QMAX), QMAX) check("S1.1 psi(q) = sum q^{k(k+1)/2} = prod (1-q^{2n})^2/(1-q^n) " "termwise to O(q^%d)" % QMAX, psi == psi_prod) psi2_4 = [0] * (QMAX + 1) psi2_4[0] = 1 for _ in range(4): psi2_4 = pmul(psi2_4, rescale(psi, 2, QMAX), QMAX) psi4_4 = [0] * (QMAX + 1) psi4_4[0] = 1 for _ in range(4): psi4_4 = pmul(psi4_4, rescale(psi, 4, QMAX), QMAX) H = shift(pmul(psi2_4, psi4_4, QMAX), 3, QMAX) H_eta = shift(pmul(pmul(eta_pow(4, 4, QMAX), eta_pow(8, 8, QMAX), QMAX), pinv(eta_pow(2, 4, QMAX), QMAX), QMAX), 3, QMAX) info("H(q) coefficients R(n), n = 0..16: %s" % (H[:17],)) check("S1.2 two routes agree: H = q^3 psi(q^2)^4 psi(q^4)^4 = " "eta(4)^4 eta(8)^8 / eta(2)^4 (with q^3 prefactor) to O(q^%d)" % QMAX, H == H_eta) # direct lattice enumeration, meet in the middle over the two 4-blocks NORM_CAP = 4 * NCAP odd_u = [u for u in range(-13, 14, 2)] hist_u = Counter() # sum of u_i^2 over odd 4-vectors for quad in itertools.product(odd_u, repeat=4): s = sum(x * x for x in quad) if s <= NORM_CAP: hist_u[s] += 1 odd_v = [v for v in range(-9, 10, 2)] hist_v = Counter() # 2 * sum of v_j^2 for quad in itertools.product(odd_v, repeat=4): s = 2 * sum(x * x for x in quad) if s <= NORM_CAP: hist_v[s] += 1 rep_count = Counter() for su, cu in hist_u.items(): for sv, cv in hist_v.items(): if su + sv <= NORM_CAP: rep_count[su + sv] += cu * cv ok_rep = all(rep_count.get(4 * n, 0) == 256 * H[n] for n in range(NCAP + 1)) tab_rep = [(n, H[n], rep_count.get(4 * n, 0)) for n in range(3, 11)] info("representation table (n, R(n), #odd vectors with Q = 4n): %s" % tab_rep) check("S1.3 REPRESENTATION IDENTITY by direct enumeration: 256 R(n) = " "#{(u,v) in (2Z+1)^8 : sum u_i^2 + 2 sum v_j^2 = 4n} for ALL " "n <= %d (odd 4-blocks enumerated exhaustively, %d x %d norm " "histograms convolved)" % (NCAP, len(hist_u), len(hist_v)), ok_rep) theta_ok = psi == psi_prod and H == H_eta and ok_rep # ====================================================================== S2 section("S2: Smith side -- Gram of M vs ramified inclusion (1+i)L c L") GRAM_M = [[0] * 8 for _ in range(8)] for i in range(8): GRAM_M[i][i] = 1 if i < 4 else 2 D_M = smith_normal_form(Matrix(GRAM_M), domain=ZZ) div_M = sorted(abs(int(D_M[i, i])) for i in range(8)) check("S2.1 SNF(Gram_M) with Gram_M = diag(1,1,1,1,2,2,2,2): elementary " "divisors %s (disc group F2^4, order 16)" % div_M, div_M == TARGET_DIV) # ---- v689 replication (read-only, verbatim recipe) -------------------- G_NAIVE = [(1, 0, 0, 0, 0, 1, 1, 1), (0, 1, 0, 0, 1, 0, 1, 1), (0, 0, 1, 0, 1, 1, 0, 1), (0, 0, 0, 1, 1, 1, 1, 0)] C_NAIVE = frozenset(tuple(sum(m[k] * G_NAIVE[k][j] for k in range(4)) % 2 for j in range(8)) for m in itertools.product((0, 1), repeat=4)) PI_J = (1, 0, 3, 2, 5, 4, 7, 6) PI_SIG = (4, 5, 0, 1, 2, 3, 6, 7) def apply_perm(c, p): return tuple(c[p[k]] for k in range(8)) def code_image(code, p): return frozenset(apply_perm(c, p) for c in code) def J_vec(x): out = [] for kk in range(0, 8, 2): out += [-x[kk + 1], x[kk]] return tuple(out) def sig_vec(x): return (x[4], x[5], x[0], x[1], x[2], x[3], x[6], x[7]) def mat_det_inv(rows): n = len(rows) A = [[Fr(v) for v in r] for r in rows] I = [[Fr(1 if i == j else 0) for j in range(n)] for i in range(n)] det = Fr(1) for col in range(n): piv = next((r for r in range(col, n) if A[r][col] != 0), None) assert piv is not None if piv != col: A[col], A[piv] = A[piv], A[col] I[col], I[piv] = I[piv], I[col] det = -det det *= A[col][col] inv = 1 / A[col][col] A[col] = [a * inv for a in A[col]] I[col] = [a * inv for a in I[col]] for r in range(n): if r != col and A[r][col] != 0: f = A[r][col] A[r] = [a - f * b for a, b in zip(A[r], A[col])] I[r] = [a - f * b for a, b in zip(I[r], I[col])] return det, I def f2_rref(words): rows = [list(w) for w in sorted(words, reverse=True) if any(w)] basis, pivots = [], [] for r in rows: r = r[:] for b, pv in zip(basis, pivots): if r[pv]: r = [(a + c) % 2 for a, c in zip(r, b)] if any(r): basis.append(r) pivots.append(next(i for i, a in enumerate(r) if a)) return basis, pivots all_placements = set() for p in itertools.permutations(range(8)): all_placements.add(code_image(C_NAIVE, p)) both_inv = [c for c in sorted(all_placements, key=lambda c: sorted(c)) if code_image(c, PI_J) == c and code_image(c, PI_SIG) == c] W0246 = tuple(1 if i in (0, 2, 4, 6) else 0 for i in range(8)) CSTAR = [c for c in both_inv if W0246 in c][0] cb, pivots = f2_rref(CSTAR) B_ROWS = [tuple(r) for r in cb] B_ROWS += [tuple(2 if i == j else 0 for i in range(8)) for j in range(8) if j not in pivots] detB, Binv = mat_det_inv(B_ROWS) def in_L(x): return tuple(v % 2 for v in x) in CSTAR def coords(x): c = [sum(x[i] * Binv[i][j] for i in range(8)) for j in range(8)] assert all(v.denominator == 1 for v in c) return tuple(int(v) for v in c) A_INC = [coords(tuple(a + b for a, b in zip(b_, J_vec(b_)))) for b_ in B_ROWS] D_A = smith_normal_form(Matrix(A_INC), domain=ZZ) div_A = sorted(abs(int(D_A[i, i])) for i in range(8)) detA, _ = mat_det_inv(A_INC) check("S2.2 v689 replication: |det B| = 16, [L : (1+i)L] = |det A| = 16 " "= N(1+i)^4; SNF(A) elementary divisors %s" % div_A, abs(detB) == 16 and abs(int(detA)) == 16 and div_A == TARGET_DIV) snf_match = (div_M == TARGET_DIV and div_A == TARGET_DIV) check("S2.3 THE SNF MATCH (the claim's arithmetic core): " "divisors(Gram_M) == divisors((1+i)L c L) == (1,1,1,1,2,2,2,2)", snf_match) # roots + guards ROOTS = [x for x in itertools.product(range(-2, 3), repeat=8) if sum(v * v for v in x) == 4 and in_L(x)] check("S2.4 guards: 240 Construction-A roots; J, sigma preserve the root " "set", len(ROOTS) == 240 and all(J_vec(r) in set(ROOTS) for r in ROOTS[:60]) and all(sig_vec(r) in set(ROOTS) for r in ROOTS[:60])) # --- quotient labelling via parity functionals ------------------------- A2 = [[v % 2 for v in row] for row in A_INC] h_funcs = [] for cand in itertools.product((0, 1), repeat=8): if any(cand) and all(sum(a * c for a, c in zip(row, cand)) % 2 == 0 for row in A2): h_funcs.append(cand) # pick 4 independent functionals H4, piv4 = f2_rref(h_funcs) H4 = [tuple(r) for r in H4] check("S2.5 quotient functionals: null space of (A mod 2) has F2-rank 4 " "(V = L/(1+i)L = F2^4 read off linearly)", len(H4) == 4) def label4(x): c = coords(x) return tuple(sum(h[i] * c[i] for i in range(8)) % 2 for h in H4) # reps of all 16 classes from 0/1 combinations of the basis rows reps = {} for eps in itertools.product((0, 1), repeat=8): x = tuple(sum(e * b[i] for e, b in zip(eps, B_ROWS)) for i in range(8)) lb = label4(x) if lb not in reps: reps[lb] = x check("S2.6 all 16 labels realized; zero class = lattice vectors " "(1+i)L (label of (1+J)b_i is 0 for all rows)", len(reps) == 16 and all(label4(tuple(a + b for a, b in zip(b_, J_vec(b_)))) == (0, 0, 0, 0) for b_ in B_ROWS)) # class-theta census (info): shells 1..3 of L per class vs odd coset of M Binv16 = np.array([[int(16 * Binv[i][j]) for j in range(8)] for i in range(8)], dtype=np.int64) H4np = np.array(H4, dtype=np.int64) rng = np.arange(-3, 4, dtype=np.int16) grid = np.array(np.meshgrid(*[rng] * 8, indexing="ij")).reshape(8, -1).T nrm = np.einsum("ij,ij->i", grid.astype(np.int32), grid.astype(np.int32)) mask = (nrm >= 4) & (nrm <= 4 * SHELLS_INFO) Xv = grid[mask].astype(np.int64) nv = nrm[mask] code_arr = np.array(sorted(CSTAR), dtype=np.int64) inC = np.zeros(256, dtype=bool) for w in CSTAR: inC[int("".join(map(str, w)), 2)] = True key = ((Xv % 2) * (2 ** np.arange(7, -1, -1, dtype=np.int64))).sum(axis=1) selL = inC[key] XL, nL = Xv[selL], nv[selL] cL16 = XL @ Binv16 assert np.all(cL16 % 16 == 0) lab = ((cL16 // 16) @ H4np.T) % 2 labcode = (lab * (2 ** np.arange(3, -1, -1, dtype=np.int64))).sum(axis=1) info("per-class theta of L (norm 4, 8, 12 = shells 1..3), 16 classes:") class_theta = {} for cls in range(16): row = [int(np.sum((labcode == cls) & (nL == 4 * s))) for s in range(1, SHELLS_INFO + 1)] class_theta[cls] = row zero_row = class_theta[0] nz_rows = [tuple(class_theta[c]) for c in range(1, 16)] info(" zero class: %s ; the 15 nontrivial classes: %s" % (zero_row, dict(Counter(nz_rows)))) info("odd-coset theta of M (256 R(n) normalized to min-vector count): " "min norm 4n = 12, count 256; L nontrivial classes: min norm 4, " "count 16 each -- quotient-structure thetas DIFFER (fingerprint " "witness at series level)") check("S2.7 census guard: zero class carries 0 roots at shell 1 and the " "15 nontrivial classes carry 16 each (v689 I2 replication)", zero_row[0] == 0 and all(r[0] == 16 for r in nz_rows)) # ====================================================================== S3 section("S3: THE DECIDER -- isometry census under admissible maps") # ---- (a) ambient level: scale-invariant obstructions ------------------ # L (doubled-integer model): min norm 4, kissing 240, det Gram = 256. # M = Z^4 (+) sqrt2 Z^4: min 1, kissing 8, det 16. # M* = Z^4 (+) (1/sqrt2) Z^4: min 1/2, kissing 8, det 1/16. detGramL = (Matrix([[sum(a * b for a, b in zip(u, v)) for v in B_ROWS] for u in B_ROWS])).det() kiss_L = len(ROOTS) kiss_M = 8 inv_L = Fr(int(detGramL)) / Fr(4) ** 8 # det * min^-8 for L inv_Mstar = Fr(1, 16) / Fr(1, 2) ** 8 # det * min^-8 for M* inv_M = Fr(16) / Fr(1) ** 8 inv_ram = Fr(int(detGramL) * 2 ** 8) / Fr(8) ** 8 # (1+i)L: det*2^8, min 8 info("scale-invariant pairs (kissing, det*min^-8):") info(" L : (%d, %s) vs M* : (%d, %s)" % (kiss_L, inv_L, kiss_M, inv_Mstar)) info(" (1+i)L : (%d, %s) vs M : (%d, %s)" % (kiss_L, inv_ram, kiss_M, inv_M)) obstructed = (kiss_L != kiss_M and inv_L != inv_Mstar and inv_ram != inv_M) check("S3a.1 AMBIENT CENSUS = 0: kissing 240 != 8 and det*min^-8 " "mismatch on BOTH pair members -- no R-linear isometry (at ANY " "scale lambda, admissible or not) can carry L -> lambda M* or " "(1+i)L -> lambda M; the ambient isometry census is EMPTY, " "proof-grade", obstructed) # ---- (b) quotient level: full GL(4,F2) census ------------------------- def hbar(x, y): """h(x,y) = ( + i )/2 in Z[i]; reduce a+bi -> (a+b) mod 2 (v752 convention).""" re2 = sum(a * b for a, b in zip(x, y)) im2 = sum(a * b for a, b in zip(x, J_vec(y))) assert re2 % 2 == 0 and im2 % 2 == 0 return (re2 // 2 + im2 // 2) % 2 # F2 basis of V: 4 labels with a single 1 VBAS = [] for i in range(4): lb = tuple(1 if j == i else 0 for j in range(4)) VBAS.append(reps[lb]) B_V = [[hbar(u, v) for v in VBAS] for u in VBAS] alt_ok = all(hbar(r, r) == 0 for r in reps.values()) wd_ok = True for lb, r in list(reps.items())[:8]: r2 = tuple(a + b + c for a, b, c in zip(r, B_ROWS[0], J_vec(B_ROWS[0]))) # + (1+i) b_0 wd_ok &= all(hbar(r, VBAS[j]) == hbar(r2, VBAS[j]) for j in range(4)) nondeg = int(Matrix(B_V).det() % 2) == 1 info("b_V Gram (basis of V): %s" % (B_V,)) B_D = [[1 if i == j else 0 for j in range(4)] for i in range(4)] info("b_D Gram (disc form of M): %s (identity form, b(x,x) != 0)" % (B_D,)) check("S3b.1 b_V replicated from the concrete lattice: well-defined mod " "(1+i)L, ALTERNATING on all 16 reps, nondegenerate (v752); b_D = " "identity form on F2^4 = disc(M*/M), NON-alternating", alt_ok and wd_ok and nondeg) mats = [] for rows in itertools.product(range(16), repeat=4): M4 = [[(rows[i] >> (3 - j)) & 1 for j in range(4)] for i in range(4)] # F2 rank R = [r[:] for r in M4] rank = 0 for col in range(4): pv = next((r for r in range(rank, 4) if R[r][col]), None) if pv is None: continue R[rank], R[pv] = R[pv], R[rank] for r in range(4): if r != rank and R[r][col]: R[r] = [(a + b) % 2 for a, b in zip(R[r], R[rank])] rank += 1 if rank == 4: mats.append(M4) def congr(T, G): """T^T G T over F2.""" TG = [[sum(T[k][i] * G[k][l] for k in range(4)) % 2 for l in range(4)] for i in range(4)] return [[sum(TG[i][k] * T[k][j] for k in range(4)) % 2 for j in range(4)] for i in range(4)] n_intertw = sum(1 for T in mats if congr(T, B_D) == B_V) n_aut_V = sum(1 for T in mats if congr(T, B_V) == B_V) n_aut_D = sum(1 for T in mats if congr(T, B_D) == B_D) info("GL(4,F2) census over all %d invertible maps:" % len(mats)) info(" intertwiners b_D -> b_V : %d" % n_intertw) info(" |Aut(b_V)| (symplectic) : %d" % n_aut_V) info(" |Aut(b_D)| (orthogonal) : %d" % n_aut_D) check("S3b.2 QUOTIENT CENSUS: %d/20160 GL(4,F2) maps intertwine the " "disc form of M with the ramified form b_V (alternating vs " "non-alternating: b_V(x,x) = 0 for all x, b_D(e_i,e_i) = 1 -- " "no isomorphism of the finite bilinear structures exists); " "|Aut(b_V)| = %d = |Sp(4,F2)|" % (n_intertw, n_aut_V), len(mats) == 20160 and n_intertw == 0 and n_aut_V == 720) info("value-group observation: any quadratic refinement of b_V is " "Z/2-valued (Arf class, v776 selector q*); disc(M) is intrinsically " "(1/2)Z/2Z-valued (q(g_j) = 1/2) -- ODD type: the finite quadratic " "forms live in different categories, matching the census 0.") # sigma/mu4 equivariance of found intertwiners (vacuous if none) sig_perm = {lb: label4(sig_vec(r)) for lb, r in reps.items()} SIG_MAT = [list(sig_perm[tuple(1 if j == i else 0 for j in range(4))]) for i in range(4)] sig_lin = all(sig_perm[lb] == tuple( sum(lb[i] * SIG_MAT[i][j] for i in range(4)) % 2 for j in range(4)) for lb in reps) J_fixes = all(label4(J_vec(r)) == lb for lb, r in reps.items()) found_equiv = [T for T in mats if congr(T, B_D) == B_V] check("S3b.3 equivariance census: sigma acts F2-linearly on V (matrix " "%s), mu4 = J acts as the IDENTITY on V ((i-1) = i(1+i), v752); " "intertwiners to test: %d -> the sigma/mu4-equivariance census " "is VACUOUS (no isometry exists to be equivariant)" % (SIG_MAT, len(found_equiv)), sig_lin and J_fixes and len(found_equiv) == 0) # ---- (c) the brake: the naive coordinatewise 4+4 map ------------------ inv4 = [s for s in itertools.combinations(range(8), 4) if set(PI_J[i] for i in s) == set(s) and set(PI_SIG[i] for i in s) == set(s)] check("S3c.1 THE BRAKE FIRES: (pi_J, pi_sigma)-invariant coordinate " "4-subsets: %d of 70 -- the naive coordinatewise map (coords " "1-4 -> u-block, 5-8 -> v-block) REQUIRES such a split and is " "inadmissible BY THEOREM (v638 S1.2 replicated: no equivariant " "4+4 coordinate split exists)" % len(inv4), len(inv4) == 0) naive_gram = [[sum(a * b for a, b in zip(u, v)) for v in B_ROWS] for u in B_ROWS] check("S3c.2 the naive map is not even a candidate isometry: Gram(L) in " "the fixed basis is NOT diag(1,1,1,1,2,2,2,2)-congruent " "coordinatewise (Gram(L) != Gram_M as matrices)", naive_gram != GRAM_M) # ====================================================================== S4 section("S4: controls and verdict") WRONG = [[0] * 8 for _ in range(8)] for i in range(8): WRONG[i][i] = 1 if i < 3 else 2 D_W = smith_normal_form(Matrix(WRONG), domain=ZZ) div_W = sorted(abs(int(D_W[i, i])) for i in range(8)) check("S4.1 CONTROL FIRES (wrong signature): diag(1,1,1,2,2,2,2,2) has " "divisors %s != %s and disc order 32 != 16 -- the SNF match " "criterion rejects it" % (div_W, TARGET_DIV), div_W != TARGET_DIV and div_W == [1, 1, 1, 2, 2, 2, 2, 2]) controls_ok = (div_W != TARGET_DIV) and (len(inv4) == 0) iso_exists = (not obstructed) or (n_intertw > 0) if not (theta_ok and snf_match): VERDICT = "SNF-FALSE" elif iso_exists: VERDICT = "SNF-ISOMETRIC" else: VERDICT = "SNF-FINGERPRINT" check("S4.2 verdict adjudication (frozen logic): theta identity %s, SNF " "match %s, ambient census empty %s, quotient census 0 %s, brake " "fired %s => %s" % (theta_ok, snf_match, obstructed, n_intertw == 0, len(inv4) == 0, VERDICT), VERDICT == "SNF-FINGERPRINT" if (theta_ok and snf_match and not iso_exists) else True) print() print("FENCES: [C neu -> decided] typing only; v775 ROOTCLASS-MIXED " "cited: the census is lattice/character-level, no root-level " "matter reading, no physics claim.") print() n_pass = sum(1 for _, ok in CHECKS if ok) n_fail = len(CHECKS) - n_pass print("VERDICT: %s" % VERDICT) print("TOTAL: %d passed, %d failed (%.1fs)" % (n_pass, n_fail, time.time() - T0)) return 0 if n_fail == 0 else 1 _run_part1 = main def _run_part2(): # PART 2 -- hecke_mu4_packet_probe.py (verbatim; module-level names # local to this function scope; check() bookkeeping global -> # nonlocal plus a CHECKS record -- print/bookkeeping only, no gate; # the banner print of the probe docstring is served by the local # __doc__ binding) import itertools import random import time from fractions import Fraction import numpy as np import sympy as sp from sympy import Matrix, Rational from sympy.matrices.normalforms import smith_normal_form T0 = time.time() PASS = 0 FAIL = 0 CHECKS = [] NSER = 64 # q-series order (frozen) NENUM = 6 # enumeration shells (norm <= 12, frozen) TMAX = 8 * NSER # t = q^{1/8} order SEED = 20260805 # control RNG seed (frozen) AP_EXPECTED = {3: -4, 5: -2, 7: 24, 11: -44, 13: 22} def check(name, ok): nonlocal PASS, FAIL CHECKS.append((name, bool(ok))) tag = "PASS" if ok else "FAIL" print(" [%s] %s" % (tag, name), flush=True) if ok: PASS += 1 else: FAIL += 1 return bool(ok) def info(msg): print(" %s" % msg, flush=True) __doc__ = """hecke_mu4_packet_probe -- HECKE.MU4_PACKET.01 (positive-protocol round, strand C, probe 2 of 2): the mu4 -> C2 collapse [E-candidate]. CORPUS ANCHORS (located): first-shell glue sector counts r = (52, 64, 60, 64) (part-11 probe e8_glue_chi4_signed_theta_probe.py, v492/v505 replications); weight-1 current sectors j = (60, 64, 60, 64) = r + (8, 0, 0, 0) (Cartan; the beta-equivariant route gives the graded character (248, 0, -8, 0)); signed root census 52 - 60 = -8 = -rank(E8); f8 = q eta(2t)^4 eta(4t)^4 (weight-4 newform, level 8); v537: Sh_{t=2}(g) = -8 f8; v536 Eichler reading a_p = -c(p)/8. DFT CONVENTION (frozen): characters chi_t(c) = i^{t c} on the glue group mu4 = Z4 = E8/(D5 (+) A3); ordering t = (0, 1, 2, 3) = (trivial, quartic i, quadratic -1, quartic i^3). r-hat = (240, -8, -16, -8); j-hat = (248, 0, -8, 0). CARTAN CLOSURE (frozen, all shells): the level-1 current-algebra grading -- multiply each sector theta by the 8-fold oscillator factor 1/phi(q)^8, phi(q) = prod (1 - q^n) (lattice-VOA grading: e^lambda (x) oscillators; oscillators are glue-neutral). Grade 1 reproduces j = (60, 64, 60, 64) with the 8 Cartan modes in the trivial sector. FROZEN CANDIDATE IDENTITIES (declared BEFORE computing): K1 quadratic channel (unclosed) IS pure Eisenstein: Th0 - Th1 + Th2 - Th3 = theta4(q)^8 = (16 E4(q^2) - E4(q))/15 termwise to O(q^NSER); in particular its f8-component is 0. K2 THE -8 f8 NORMALIZATION (the Section-12 claim, sharpened): the cusp content of the QUARTIC character series csig = Th0 - Th2 is EXACTLY -8 f8, with -8 = the shell-1 quartic DFT = signed census 52 - 60 = -rank(E8): csig + 8 f8 lies in the pure sigma3-Eisenstein span {E4(q^d) : d = 1,2,4,8,16} + Eis(chi4,chi4), exactly to O(q^NSER). Consequence tested separately: csig(n) = -8 f8(n) for ALL odd n. K3 prime anchors: a_p(f8) = (-4, -2, 24, -44, 22) at p = (3, 5, 7, 11, 13) (v536/v537 values) and csig(p) = -8 a_p(f8) for all odd primes p <= 47. K4 all-shell quartic vanishing (the user's collapse claim): the Cartan-closed quartic series csig(q)/phi(q)^8 vanishes in EVERY grade n >= 1 (not just grade 1). Exact eta form of the closed series: csig/phi^8 = prod (1 + q^{2n})^{-4} (machine-verified), so the claim is decidable termwise. TASKS: S1 glue machinery replication (part-11 D5 (+) A3 basis, linear classifier): census (52, 64, 60, 64), signed -8; N1(n) = N3(n) for ALL shells proof-grade via x -> -x (deg(-x) = -deg(x) mod 4). S2 exact DFT at shell 1: r-hat, j-hat as Gaussian integers. S3 all-shell extension: per-class enumeration n = 1..6 (norm <= 12), theta-constant series Th0..Th3 to O(q^NSER = 64) cross-checked against enumeration; per-shell DFT table; Cartan closure of all four character series; the K4 vanishing test with first-failure grade; the closed-quartic eta identity. S4 the packet-channel identities K1, K2, K3; adjudication of the composite claim "the surviving quadratic character IS the C2 packet channel with the exact -8": which character carries the -8 f8 cusp packet, which carries the surviving -8 constant. S5 controls (must fire): (a) scrambled sector assignment (seeded) breaks the quartic cancellation at shell 1; (b) wrong Cartan completion (4, 0, 0, 4) breaks j-hat = (248, 0, -8, 0). VERDICTS (frozen): MU4-COLLAPSE-EXACT (quartic vanish ALL shells after closure + quadratic = the packet channel with the exact -8), MU4-PARTIAL (shell-1 collapse exact + at least one frozen identity K1-K3 exact, but not all-shell vanishing), MU4-DEAD (shell-1 anchors fail or controls do not fire). FENCES: [E-candidate] readings are exact replications/identities; extensions typed as measured; v775 ROOTCLASS-MIXED cited -- the glue census is representation-level, no root-level matter reading and NO physics claim. Firewall: experiments/ discovery sandbox; writes nothing; no verification/, ledger, paper or website surface touched. Exact integer/Fraction/Gaussian-integer arithmetic; sympy for exact linear algebra; numpy only for integer enumeration; RNG only inside the negative control (fixed seed). Sources (read-only): e8_glue_chi4_signed_theta_probe.py (part 11), v492/v505 (root census), v535/v536/v537/v538 (Hecke/Eichler/f8 layer), v775 (ROOTCLASS-MIXED fence). """ print(__doc__.split("TASKS")[0]) print("FROZEN: NSER = %d, NENUM = %d, seed = %d, DFT ordering " "(triv, i, -1, i^3)." % (NSER, NENUM, SEED)) print() # ---------------------------------------------------- integer q-series def pmul(a, b, order): res = [0] * (order + 1) for i, ai in enumerate(a): if ai: top = order - i for j in range(min(len(b) - 1, top) + 1): if b[j]: res[i + j] += ai * b[j] return res def ppow(a, e, order): res = [0] * (order + 1) res[0] = 1 for _ in range(e): res = pmul(res, a, order) return res def padd(a, b): return [x + y for x, y in zip(a, b)] def psub(a, b): return [x - y for x, y in zip(a, b)] def phalf(a): assert all(x % 2 == 0 for x in a) return [x // 2 for x in a] def eta_pow(d, e, order): s = [0] * (order + 1) s[0] = 1 for n in range(1, order // d + 1): f = [0] * (d * n + 1) f[0], f[d * n] = 1, -1 for _ in range(e): s = pmul(s, f, order) return s def pinv(a, order): assert a[0] == 1 res = [0] * (order + 1) res[0] = 1 for n in range(1, order + 1): res[n] = -sum(a[k] * res[n - k] for k in range(1, min(n, len(a) - 1) + 1)) return res def shift(a, k, order): return ([0] * k + a)[: order + 1] def theta3_t(step): s = [0] * (TMAX + 1) s[0] = 1 n = 1 while step * 4 * n * n <= TMAX: s[step * 4 * n * n] += 2 n += 1 return s def theta4_t(step): s = [0] * (TMAX + 1) s[0] = 1 n = 1 while step * 4 * n * n <= TMAX: s[step * 4 * n * n] += 2 * (-1) ** n n += 1 return s def theta2_t(step): s = [0] * (TMAX + 1) o = 1 while step * o * o <= TMAX: s[step * o * o] += 2 o += 2 return s def t_to_q(ts): assert all(v == 0 for k, v in enumerate(ts) if k % 8 and v) return [ts[8 * n] for n in range(NSER + 1)] def theta4_q(step): s = [0] * (NSER + 1) s[0] = 1 n = 1 while step * n * n <= NSER: s[step * n * n] += 2 * (-1) ** n n += 1 return s # ================================================================ S1 print("S1 -- glue machinery replication (part-11 recipe, verbatim)") def col(v): return [2 * x for x in v] e8_basis = [ [1, -1, -1, -1, -1, -1, -1, 1], col([1, 1, 0, 0, 0, 0, 0, 0]), col([-1, 1, 0, 0, 0, 0, 0, 0]), col([0, -1, 1, 0, 0, 0, 0, 0]), col([0, 0, -1, 1, 0, 0, 0, 0]), col([0, 0, 0, -1, 1, 0, 0, 0]), col([0, 0, 0, 0, -1, 1, 0, 0]), col([0, 0, 0, 0, 0, -1, 1, 0]), ] BE = Matrix(e8_basis).T gram = (BE.T * BE) / 4 l0_basis = [ col([1, -1, 0, 0, 0, 0, 0, 0]), col([0, 1, -1, 0, 0, 0, 0, 0]), col([0, 0, 1, -1, 0, 0, 0, 0]), col([0, 0, 0, 1, -1, 0, 0, 0]), col([0, 0, 0, 1, 1, 0, 0, 0]), col([0, 0, 0, 0, 0, 1, -1, 0]), col([0, 0, 0, 0, 0, 0, 1, -1]), col([0, 0, 0, 0, 0, 1, 1, 0]), ] BL = Matrix(l0_basis).T M = BE.solve(BL) snf = smith_normal_form(M) check("S1.1 E8 basis unimodular, [E8 : D5 (+) A3] = 4, SNF diag " "(1,...,1,4) -> glue group Z4 = mu4 (part-11/v-suite replication)", sp.det(gram) == 1 and abs(M.det()) == 4 and sorted(abs(snf[i, i]) for i in range(8)) == [1] * 7 + [4]) Fmap = M.inv() * BE.inv() def glue_frac2(v2): fr = Fmap * Matrix(list(v2)) return tuple(Fraction(sp.Rational(e).p, sp.Rational(e).q) % 1 for e in fr) roots2 = [] for i in range(8): for j in range(i + 1, 8): for si in (2, -2): for sj in (2, -2): v = [0] * 8 v[i], v[j] = si, sj roots2.append(tuple(v)) for signs in itertools.product((1, -1), repeat=8): if signs.count(-1) % 2 == 0: roots2.append(signs) assert len(roots2) == 240 g1 = glue_frac2(roots2[112]) classes = {} for k in range(4): classes[tuple((k * c) % 1 for c in g1)] = k BEinv = BE.inv() I256 = np.array([[int(256 * BEinv[i, j]) for j in range(8)] for i in range(8)], dtype=np.int64) dvec = np.array([classes[glue_frac2([BE[j, i] for j in range(8)])] for i in range(8)], dtype=np.int64) def deg_bulk(V2): prod = V2.astype(np.int64) @ I256.T assert np.all(prod % 256 == 0) return ((prod // 256) @ dvec) % 4 deg_roots = deg_bulk(np.array(roots2, dtype=np.int64)) r_census = [int(np.sum(deg_roots == k)) for k in range(4)] info("shell-1 sector counts r = %s; signed 52 - 60 = %d" % (tuple(r_census), r_census[0] - r_census[2])) check("S1.2 CORPUS ANCHOR: r = (52, 64, 60, 64), signed census " "52 - 60 = -8 = -rank(E8)", r_census == [52, 64, 60, 64] and r_census[0] - r_census[2] == -8) deg_neg = deg_bulk(np.array([[-v for v in r] for r in roots2], dtype=np.int64)) check("S1.3 PROOF-GRADE symmetry: deg(-x) = -deg(x) mod 4 on all 240 " "roots => N1(n) = N3(n) for EVERY shell (x -> -x bijects class 1 " "onto class 3)", all((int(a) + int(b)) % 4 == 0 for a, b in zip(deg_roots, deg_neg))) # ================================================================ S2 print("S2 -- exact DFT at shell 1 (Gaussian integers)") def dft4(vec): """DFT with chi_t(c) = i^{tc}; returns 4 Gaussian integers as (real, imag) pairs.""" out = [] for t in range(4): re, im = 0, 0 for c in range(4): ph = (t * c) % 4 # i^ph if ph == 0: re += vec[c] elif ph == 1: im += vec[c] elif ph == 2: re -= vec[c] else: im -= vec[c] out.append((re, im)) return out r_hat = dft4(r_census) j_census = [r_census[0] + 8] + r_census[1:] j_hat = dft4(j_census) info("r-hat = %s" % (r_hat,)) info("j = %s (Cartan closure r + (8,0,0,0))" % (tuple(j_census),)) info("j-hat = %s" % (j_hat,)) check("S2.1 CORPUS ANCHOR verified exactly: r-hat = (240, -8, -16, -8) " "(all imaginary parts 0 by S1.3)", r_hat == [(240, 0), (-8, 0), (-16, 0), (-8, 0)]) check("S2.2 CORPUS ANCHOR verified exactly: j = (60, 64, 60, 64), " "j-hat = (248, 0, -8, 0) -- at the current level the quartic " "characters COLLAPSE and the quadratic -8 survives (mu4 -> C2)", j_census == [60, 64, 60, 64] and j_hat == [(248, 0), (0, 0), (-8, 0), (0, 0)]) # ================================================================ S3 print("S3 -- all-shell extension: enumeration, series, Cartan closure") rng7 = np.arange(-3, 4, dtype=np.int16) gi = np.array(np.meshgrid(*[rng7] * 8, indexing='ij')).reshape(8, -1).T ni = np.einsum('ij,ij->i', gi.astype(np.int32), gi.astype(np.int32)) mi = (gi.astype(np.int32).sum(axis=1) % 2 == 0) & (ni >= 2) & (ni <= 12) V2i = 2 * gi[mi].astype(np.int64) n_i = ni[mi] // 2 rng6 = np.array([-5, -3, -1, 1, 3, 5], dtype=np.int16) gh = np.array(np.meshgrid(*[rng6] * 8, indexing='ij')).reshape(8, -1).T nh = np.einsum('ij,ij->i', gh.astype(np.int32), gh.astype(np.int32)) mh = (gh.astype(np.int32).sum(axis=1) % 4 == 0) & (nh <= 48) V2h = gh[mh].astype(np.int64) n_h = nh[mh] // 8 del gi, gh, ni, nh V2 = np.vstack([V2i, V2h]) nsh = np.concatenate([n_i, n_h]) deg = deg_bulk(V2) counts = {} for n in range(1, NENUM + 1): for k in range(4): counts[(n, k)] = int(np.sum((nsh == n) & (deg == k))) info("per-class shell counts N_c(2n), n = 1..%d (direct enumeration):" % NENUM) for k in range(4): info(" class %d: %s" % (k, [counts[(n, k)] for n in range(1, NENUM + 1)])) sig3 = [0] + [int(sp.divisor_sigma(n, 3)) for n in range(1, NSER + 1)] check("S3.1 enumeration totals: sum_c N_c(2n) = 240 sigma3(n) and " "N1 = N3 shell by shell for n = 1..%d" % NENUM, all(sum(counts[(n, k)] for k in range(4)) == 240 * sig3[n] for n in range(1, NENUM + 1)) and all(counts[(n, 1)] == counts[(n, 3)] for n in range(1, NENUM + 1))) th3, th4, th2 = theta3_t(1), theta4_t(1), theta2_t(1) t3_5, t4_5 = ppow(th3, 5, TMAX), ppow(th4, 5, TMAX) t3_3, t4_3 = ppow(th3, 3, TMAX), ppow(th4, 3, TMAX) Th0 = t_to_q(pmul(phalf(padd(t3_5, t4_5)), phalf(padd(t3_3, t4_3)), TMAX)) Th2 = t_to_q(pmul(phalf(psub(t3_5, t4_5)), phalf(psub(t3_3, t4_3)), TMAX)) th2_8 = ppow(th2, 8, TMAX) assert all(x % 4 == 0 for x in th2_8) Th1 = t_to_q([x // 4 for x in th2_8]) Th3 = Th1[:] check("S3.2 theta-constant series (route 2) match the enumeration " "(route 1) per class for n = 1..%d; totals = E4 to O(q^%d)" % (NENUM, NSER), all(Th0[n] == counts[(n, 0)] and Th2[n] == counts[(n, 2)] and Th1[n] == counts[(n, 1)] == counts[(n, 3)] for n in range(1, NENUM + 1)) and all(Th0[n] + 2 * Th1[n] + Th2[n] == (1 if n == 0 else 240 * sig3[n]) for n in range(NSER + 1))) # per-shell DFT series (all real: quartic = Th0 - Th2 since Th1 = Th3) dft_triv = [Th0[n] + Th1[n] + Th2[n] + Th3[n] for n in range(NSER + 1)] dft_quart = [Th0[n] - Th2[n] for n in range(NSER + 1)] # csig dft_quad = [Th0[n] - Th1[n] + Th2[n] - Th3[n] for n in range(NSER + 1)] print(" per-shell DFT table (n, N_c(2n) c=0..3, triv, quartic, " "quadratic), n = 1..16:") for n in range(1, 17): print(" n=%2d N=(%7d,%7d,%7d,%7d) triv=%9d " "quart=%7d quad=%9d" % (n, Th0[n], Th1[n], Th2[n], Th3[n], dft_triv[n], dft_quart[n], dft_quad[n]), flush=True) # Cartan closure: multiply by 1/phi^8 phi8 = eta_pow(1, 8, NSER) inv_phi8 = pinv(phi8, NSER) cl_triv = pmul(dft_triv, inv_phi8, NSER) cl_quart = pmul(dft_quart, inv_phi8, NSER) cl_quad = pmul(dft_quad, inv_phi8, NSER) cl_classes = [pmul(t, inv_phi8, NSER) for t in (Th0, Th1, Th2, Th3)] check("S3.3 Cartan closure normalization: grade-1 closed sector counts " "= j = (60, 64, 60, 64) (the 8 oscillator modes sit in the " "trivial sector) and closed DFT grade 1 = (248, 0, -8)", [c[1] for c in cl_classes] == [60, 64, 60, 64] and (cl_triv[1], cl_quart[1], cl_quad[1]) == (248, 0, -8)) # K4: all-shell quartic vanishing nz = [n for n in range(1, NSER + 1) if cl_quart[n] != 0] info("closed quartic series (Th0 - Th2)/phi^8, grades 0..12: %s" % (cl_quart[:13],)) info("nonzero grades <= %d: %s%s" % (NSER, nz[:12], " ..." if len(nz) > 12 else "")) k4_vanish = len(nz) == 0 check("S3.4 K4 MEASURED: closed quartic vanishes at grade 1 (the " "corpus j-hat anchor holds); all-shell vanishing claim is %s " "(first nonzero grade: %s) -- adjudicated in the verdict" % ("TRUE" if k4_vanish else "FALSE", nz[0] if nz else "NONE"), cl_quart[1] == 0) # exact eta form: csig/phi^8 = prod (1+q^{2n})^{-4} one_plus = [0] * (NSER + 1) one_plus[0] = 1 for n in range(1, NSER // 2 + 1): f = [0] * (2 * n + 1) f[0], f[2 * n] = 1, 1 for _ in range(4): one_plus = pmul(one_plus, f, NSER) check("S3.5 exact eta identity of the closed quartic channel: " "(Th0 - Th2)/phi^8 = prod (1 + q^{2n})^{-4} termwise to O(q^%d) " "(so all-shell vanishing is decidably FALSE: the product is not " "identically 1)" % NSER, cl_quart == pinv(one_plus, NSER)) check("S3.6 the quadratic character SURVIVES closure: closed quadratic " "series nonzero at grade 1 (= -8) and at infinitely many visible " "grades (first 5 nonzero grades %s)" % ([n for n in range(1, NSER + 1) if cl_quad[n]][:5],), cl_quad[1] == -8 and sum(1 for n in range(1, NSER + 1) if cl_quad[n]) >= 5) # ================================================================ S4 print("S4 -- the packet-channel identities K1, K2, K3") th4q8 = ppow(theta4_q(1), 8, NSER) E4q = [1] + [240 * sig3[n] for n in range(1, NSER + 1)] E4q2 = [1] + [240 * sig3[n // 2] if n % 2 == 0 else 0 for n in range(1, NSER + 1)] k1_ok = (dft_quad == th4q8 and all(15 * th4q8[n] == 16 * E4q2[n] - E4q[n] for n in range(NSER + 1))) check("K1: quadratic channel = theta4(q)^8 = (16 E4(q^2) - E4(q))/15 " "PURE Eisenstein termwise to O(q^%d) -- the C2 channel carries " "NO cusp form, in particular no f8 component" % NSER, k1_ok) f8 = shift(pmul(eta_pow(2, 4, NSER), eta_pow(4, 4, NSER), NSER), 1, NSER) def E4d(d): return [1] + [240 * sig3[n // d] if n % d == 0 else 0 for n in range(1, NSER + 1)] chi4 = lambda d: (1 if d % 4 == 1 else -1 if d % 4 == 3 else 0) eis16 = [0] + [sum(chi4(d) * chi4(n // d) * d ** 3 for d in sp.divisors(n)) for n in range(1, NSER + 1)] target = [dft_quart[n] + 8 * f8[n] for n in range(NSER + 1)] cols = [E4d(1), E4d(2), E4d(4), E4d(8), E4d(16), eis16] names = ["E4(q)", "E4(q^2)", "E4(q^4)", "E4(q^8)", "E4(q^16)", "Eis(chi4,chi4)"] NEQ = 24 Aeq = Matrix([[cols[j][n] for j in range(len(cols))] for n in range(NEQ)]) bvec = Matrix([target[n] for n in range(NEQ)]) sol, params = Aeq.gauss_jordan_solve(bvec) if params: sol = sol.subs({p: 0 for p in params}) recon = [sum(sol[j] * cols[j][n] for j in range(len(cols))) for n in range(NSER + 1)] k2_ok = recon == [Rational(c) for c in target] info("Eisenstein decomposition of csig + 8 f8: " + ", ".join( "%s: %s" % (names[j], sp.nsimplify(sol[j])) for j in range(len(cols)) if sol[j] != 0)) check("K2 (THE -8 f8 NORMALIZATION, frozen): csig + 8 f8 lies EXACTLY " "in the pure Eisenstein span to O(q^%d) -- the cusp content of " "the QUARTIC mu4 character is EXACTLY -8 f8, with -8 = signed " "census 52 - 60 = -rank(E8)" % NSER, k2_ok) k2_odd = all(dft_quart[n] == -8 * f8[n] for n in range(1, NSER + 1, 2)) check("K2 consequence: csig(n) = -8 f8(n) for ALL odd n <= %d (the " "Eisenstein part is supported on even n)" % NSER, k2_odd) k2_ok = k2_ok and k2_odd primes = [p for p in range(3, 48, 2) if sp.isprime(p)] ap_ok = all(f8[p] == AP_EXPECTED[p] for p in AP_EXPECTED) k3_ok = ap_ok and all(dft_quart[p] == -8 * f8[p] for p in primes) info("prime anchors (p, a_p(f8), csig(p)): %s" % [(p, f8[p], dft_quart[p]) for p in primes]) check("K3: a_p(f8) = (-4, -2, 24, -44, 22) at p = (3, 5, 7, 11, 13) " "(v536/v537 corpus values) and csig(p) = -8 a_p(f8) for all odd " "primes p <= 47 (the v536 Eichler reading a_p = -c(p)/8)", k3_ok) info("ADJUDICATION of the composite Section-12 claim: the -8 f8 cusp " "packet sits in the QUARTIC (order-4) character channel (K2); " "the QUADRATIC (C2) channel is pure Eisenstein (K1); the -8 that " "SURVIVES Cartan closure at the current level sits in the " "quadratic slot of j-hat = (248, 0, -8, 0) and equals the same " "signed census -8. 'The -8 comes from the mu4 character " "structure' is CONFIRMED (both -8's are the signed census " "52 - 60); 'the quadratic series IS the packet channel' is NOT: " "the packet (cusp) channel is the quartic one.") # ================================================================ S5 print("S5 -- controls (must fire)") rng = random.Random(SEED) scr = [rng.randrange(4) for _ in range(240)] scr_census = [scr.count(k) for k in range(4)] scr_hat = dft4([scr_census[0] + 8] + scr_census[1:]) check("S5.1 CONTROL FIRES (scrambled sector assignment, seed %d): " "census %s != (52, 64, 60, 64) and the quartic cancellation " "breaks (closed quartic slot %s != 0)" % (SEED, tuple(scr_census), scr_hat[1]), scr_census != [52, 64, 60, 64] and scr_hat[1] != (0, 0)) wrong_j = [r_census[0] + 4, r_census[1], r_census[2], r_census[3] + 4] wrong_hat = dft4(wrong_j) check("S5.2 CONTROL FIRES (wrong Cartan completion (4,0,0,4)): " "j-hat = %s != (248, 0, -8, 0) -- the quartic slot no longer " "cancels and the quadratic slot is wrong" % (wrong_hat,), wrong_hat != [(248, 0), (0, 0), (-8, 0), (0, 0)] and wrong_hat[1] != (0, 0)) # ================================================================ verdict print() shell1_exact = (r_census == [52, 64, 60, 64] and r_hat == [(240, 0), (-8, 0), (-16, 0), (-8, 0)] and j_hat == [(248, 0), (0, 0), (-8, 0), (0, 0)]) if shell1_exact and k4_vanish and k1_ok and k2_ok: VERDICT = "MU4-COLLAPSE-EXACT" elif shell1_exact and (k1_ok or k2_ok or k3_ok): VERDICT = "MU4-PARTIAL" else: VERDICT = "MU4-DEAD" print("VERDICT ADJUDICATION (frozen logic): shell-1 collapse exact = %s;" % shell1_exact) print(" K1 (quadratic pure Eisenstein) = %s; K2 (-8 f8 exact) = %s;" % (k1_ok, k2_ok)) print(" K3 (prime anchors) = %s; K4 (all-shell quartic vanish) = %s" % (k3_ok, k4_vanish)) print() print("FENCES: [E-candidate] typing on exact identities only; v775 " "ROOTCLASS-MIXED cited: representation-level census, no " "root-level matter reading, no physics claim.") print() print("VERDICT: %s" % VERDICT) print("TOTAL: %d passed, %d failed (%.1fs)" % (PASS, FAIL, time.time() - T0)) rc = 0 if FAIL == 0 else 1 return rc, list(CHECKS) def run(): """run_all entry point (combined adjudication): part 1 must be 18/18 (SNF-FINGERPRINT: the SNF match is exact but the isometry census is empty at every admissible level), part 2 must be 17/17 (MU4-PARTIAL: shell-1 collapse and K1-K3 exact, the all-shell quartic vanishing honestly FALSE).""" rc1 = _run_part1() fails1 = [n for (n, ok) in CHECKS if not ok] part1_ok = (rc1 == 0 and not fails1 and len(CHECKS) == 18) print("\n[%s] PART-1 PATTERN GATE: expected 18/18 " "(SNF-FINGERPRINT) -- fails: %s" % ("PASS" if part1_ok else "FAIL", fails1 or "none")) rc2, chks2 = _run_part2() fails2 = [n for (n, ok) in chks2 if not ok] part2_ok = (rc2 == 0 and not fails2 and len(chks2) == 17) print("\n[%s] PART-2 PATTERN GATE: expected 17/17 " "(MU4-PARTIAL) -- fails: %s" % ("PASS" if part2_ok else "FAIL", fails2 or "none")) ok = part1_ok and part2_ok print("\nCOMBINED ADJUDICATION: %s -- SNF-FINGERPRINT + " "MU4-PARTIAL: the 256 R(n) representation identity and the " "SNF match (1,1,1,1,2,2,2,2) are exact but the isometry " "census is EMPTY at every admissible level (ambient kissing " "240 vs 8 proof-grade, quotient 0/20160 over GL(4,F2) -- " "alternating vs odd form, |Aut(b_V)| = 720 = |Sp(4,F2)| -- " "and the naive 4+4 split inadmissible by theorem): a " "fingerprint, not an isometry; the cusp part of the quartic " "mu4 character is EXACTLY -8 f8 (csig = (1/15) E4(q^2) - " "(6/5) E4(q^4) + (32/15) E4(q^8) - 8 f8) with -8 = 52 - 60 " "= -rank(E8), the C2 channel is pure Eisenstein " "(theta4(q)^8 = (16 E4(q^2) - E4(q))/15), and the all-shell " "quartic vanishing is honestly FALSE (closed series " "prod (1 + q^{2n})^-4, fails from grade 2). NO RH claim." % ("PASS" if ok else "FAIL")) return 0 if ok else 1 if __name__ == "__main__": raise SystemExit(run())