#!/usr/bin/env python3 """qqww-t4-sec481-chain.py -- verify the chain: the first-order factorisation of the sector-481 eps^0 block of row 35 (the nonplanar T4 qqbar -> WW family; rows 54-59 of the served connection, n = 6), exact over Q(var). For a first-order system Y' = A(var) Y with A in M_n(Q(var)), search for a hyperexponential solution vector Y = prod_i f_i(var)^{e_i} * P(var), e_i in Q, P a polynomial vector, with the candidate exponents e_i read from the residue eigenvalues at every singular factor f_i (the rational eigenvalues of the residue matrix in the residue field Q[var]/(f_i), all conjugate roots carrying the same e_i), the degree of P bounded by the exponents at infinity (N = rho - e_inf, rho a rational eigenvalue of lim var*A, e_inf = sum e_i deg f_i), the coefficients of P from the exact nullspace (FLINT fmpq_mat rref) of D P' - (D A - D u'/u) P = 0; quotient the solution out by the rational gauge T = [P | e_j (j != i0)] (det T = +-P_{i0}) and iterate on the (n-1)x(n-1) block. Complete reduction to size 0 <=> a chain of first-order factors with algebraic solutions (the criterion of the audited external computation's report, report_C3.md L135-142 and L508-511, the method this script names in its receipt's recipe field); a Fuchsian remainder block of rank k >= 2 with NO hyperexponential solution over the complete candidate set = NO_FIRST_ORDER_FACTOR (a block with an elliptic maximal-cut curve whose solutions are not d'Alembertian is such a remainder). Completeness: at a simple pole the residue eigenvalues are necessary local exponents (rigorous); a pole of order >= 2 (finite or at infinity) is NOT Fuchsian -- the candidate set is then a declared bounded set (half-integers |e| <= EB) and the degree bound NB, and the negative verdict is labelled BOUNDED. A REDUCIBLE verdict is exact whatever the candidate bounds: every factor in the chain is an exact hyperexponential solution (the quotient asserts A P - P' = c P exactly); the bounds only weaken a NEGATIVE verdict. THE OBJECT FACT this script re-establishes (the record: vendor_row35_sec481/lines/, /rays/): on both served lines and along two mass-scaling rays the sector-481 eps^0 block is REDUCIBLE_TO_FIRST_ORDER -- a chain of six exact hyperexponential factors with exponents in (1/2)Z at the sector's letters (the two conic quadratics carrying exponent -3/2, the rest integer), remainder rank 0, two rational solutions (the first two factors' rates are 0) -- while the sector's maximal-cut curve is elliptic (j non-constant in z5). The certified thimble column of the record lies in the chain's solution space (vendor_row35_sec481/thimble/, 39.89 digits). Usage (from the bundle directory or with --served-dir naming it): qqww-t4-sec481-chain.py --controls [--out DIR] [--tag TAG] the four built-in controls C1-C4 (C1 a planted reducible 3x3 chain conjugated by a polynomial gauge: size 0, the exponents recovered; C2 the Legendre Picard-Fuchs companion system, irreducible with SL2 monodromy: NO_FIRST_ORDER_FACTOR rank 2; C3 a 2x2 triangular block with half-integer exponents: size 0; C4 a direct sum Legendre (+) hyperexponential 1x1: the 1x1 factor found and the rank-2 remainder left); exit 0 iff all four read as expected, else 3. Seconds-class (the record's rehearsal: 2.9 s). qqww-t4-sec481-chain.py --line L1|L2 [--served-dir DIR] [--reference RECEIPT.json] [--out DIR] [--tag TAG] the served line's sector-481 block (rows 54-59 of row35_data.json on L1, (x, z) = (17/2, 5/3); of row35_L2/connection_L2.json on L2, (x, z) = (17/2, 7/4)) at d = 4 in the variable y, both files pinned by sha256 (a mismatch refuses by name, rc 2). Measured walls of the record (one core): 164.5 s on L1, 159.9 s on L2 (the bounds EB 3 / NB 12 / MAXCOMBO 20000). qqww-t4-sec481-chain.py --block RAY_BLOCK.json [--reference RECEIPT.json] [--out DIR] [--tag TAG] a ray block receipt (key A_eps0_block_rows, the variable lam): the two vendored ray objects vendor_row35_sec481/rays/ P2_SEC481_BLOCK_p2live_S_*.json (the ray P) and P2_SEC481_BLOCK_p2liveE_S_*.json (the ray E), pinned by sha256 (another file refuses by name unless --allow-unpinned-block). Measured walls of the record: 72.9 s (P), 70.3 s (E). --reference RECEIPT.json: a receipt of the same form (the vendored chain receipts under vendor_row35_sec481/lines/ and /rays/, or one this script wrote); the run's chain is compared factor by factor -- the exponent lists and the rates u'/u decide (exit 3 on any difference), the partial fractions, degree bounds, P vectors and solution counts are reported beside. --planted-served: NOT offered from this bundle; refuses by name (rc 2). The served-block (+) Legendre planted control of the record ran 1773.1 s under NB 8 / MAXCOMBO 400 on an 1800-s clock and returned NO_FIRST_ORDER_FACTOR_BOUNDED_REMAINDER_RANK_5 after 3 factors (its exit code 7: the expected rank-2 remainder after six factors was not reached inside the truncated candidate set); the seconds-class planted control is C1 of --controls. --out DIR: the receipt directory (default: a fresh directory sec481-chain-run_/ under the current working directory; a directory inside the bundle refuses by name, rc 2); --tag TAG names the receipt qqww-t4-sec481-chain-run__.json (default TAG = the mode: controls, L1, L2, or block-). env: EB (default 3), NB (default 12), MAXCOMBO (default 20000) -- the candidate bounds of the search. Exit codes: 0 = the verdict is REDUCIBLE_TO_FIRST_ORDER (and, with --reference, the chain equals the reference factor by factor) / --controls all as expected; 2 = refused by name (usage, a pin mismatch, a missing file, --out inside the bundle, --planted-served); 3 = a negative verdict, a chain differing from the reference, or a control not as expected. The receipt names this script and every input by basename + sha256; it is written O_EXCL and never overwritten.""" import sys, os, re, json, hashlib, subprocess, time, itertools import argparse import sympy as sp from fractions import Fraction import flint R = sp.Rational EB = int(os.environ.get('EB', '3')); NB = int(os.environ.get('NB', '12')); MAXCOMBO = int(os.environ.get('MAXCOMBO', '20000')) def stamp(f='+%Y-%m-%dT%H:%M:%SZ'): return subprocess.run(['date', '-u', f], capture_output=True, text=True, check=True).stdout.strip() def sha(p): return hashlib.sha256(open(p, 'rb').read()).hexdigest() SCHEMA = 'P2_REDUCIBILITY_v1' RECIPE = 'report_C3.md L135-142 (sys_factor): hyperexponential vector solutions prod f_i^{e_i} P, quotient by a rational gauge, iterate; L508-511 (blocktri)' HERE = os.path.dirname(os.path.abspath(__file__)) SERVED_PINS = {'row35_data.json': '6acf455def67385491ef9be8734c223129c811854503d810abbd4b71a8dea9fc', 'row35_L2/connection_L2.json': 'fd83c63b685adb69364b936f26672543581ad0aa45d2f6f8d85d31458596d9bb'} LINE_FILE = {'L1': 'row35_data.json', 'L2': 'row35_L2/connection_L2.json'} LINE_XZ = {'L1': '(17/2, 5/3)', 'L2': '(17/2, 7/4)'} RAY_PINS = {'P2_SEC481_BLOCK_p2live_S_20260910T011946Z.json': 'e316f787c2012917e65f139df7c8e854089da5b835b93bb7ba268bfed9345f79', 'P2_SEC481_BLOCK_p2liveE_S_20260910T012441Z.json': '08f55706a9db2ba5c0d51bcfeda4e49230c6522efd7ce758b809f5f8ff724b1b'} SEC481_MASTERS = ['wpairT4[1,0,0,0,0,1,1,1,1]', 'wpairT4[1,-1,0,0,0,1,1,1,1]', 'wpairT4[1,0,-1,0,0,1,1,1,1]', 'wpairT4[1,0,0,-1,0,1,1,1,1]', 'wpairT4[1,0,0,0,-1,1,1,1,1]', 'wpairT4[1,-2,0,0,0,1,1,1,1]'] CHAIN_KEYS_DECIDING = ('exponents', 'rate_u_prime_over_u') CHAIN_KEYS_REPORTED = ('rate_partial_fractions', 'degree_bound', 'P', 'n_solutions_at_this_exponent') PLANTED_SERVED = {'wall_s': 1773.1, 'bounds': {'EB': 3, 'NB': 8, 'MAXCOMBO': 400}, 'clock_s': 1800, 'verdict': 'NO_FIRST_ORDER_FACTOR_BOUNDED_REMAINDER_RANK_5', 'chain_len': 3, 'exit_code': 7} def refuse(msg, rc=2): print(f'qqww-t4-sec481-chain.py REFUSED: {msg}', file=sys.stderr) sys.exit(rc) def check_pin(path, want, label): if not os.path.isfile(path): refuse(f'{label}: {os.path.basename(path)} is not at {os.path.dirname(path) or "."} -- restore the bundle') got = sha(path) if got != want: refuse(f'{label}: {os.path.basename(path)} sha256 {got[:16]}... != pinned {want[:16]}... -- refusing to run on altered bytes') return got def load_reference(path): ref = json.load(open(path)) if ref.get('schema') != SCHEMA or 'result' not in ref or 'chain' not in ref['result']: refuse(f'--reference {os.path.basename(path)}: not a {SCHEMA} receipt with a result.chain') return ref def compare_chain(result, ref_result): """factor by factor: the deciding keys (exponent lists, rates) must be equal; the reported keys are listed either way. Both sides are normalised through a JSON round trip first (the run's chain carries tuples where a receipt read from disk carries lists).""" norm = lambda o: json.loads(json.dumps(o, default=str)) result, ref_result = norm(result), norm(ref_result) a, b = result['chain'], ref_result['chain'] rep = {'verdict_equal': result['verdict'] == ref_result['verdict'], 'size_equal': result['size'] == ref_result['size'], 'remainder_rank_equal': result['remainder_rank'] == ref_result['remainder_rank'], 'chain_len': [len(a), len(b)], 'factors_deciding_equal': [], 'factors_reported_equal': [], 'differing': []} for i in range(max(len(a), len(b))): if i >= len(a) or i >= len(b): rep['differing'].append(f'factor {i + 1}: present in one chain only'); continue dec = {k: a[i].get(k) == b[i].get(k) for k in CHAIN_KEYS_DECIDING} repd = {k: a[i].get(k) == b[i].get(k) for k in CHAIN_KEYS_REPORTED} rep['factors_deciding_equal'].append(dec); rep['factors_reported_equal'].append(repd) for k, v in dec.items(): if not v: rep['differing'].append(f'factor {i + 1}: {k}') rep['equal'] = bool(rep['verdict_equal'] and rep['size_equal'] and rep['remainder_rank_equal'] and len(a) == len(b) and not rep['differing']) return rep def write_receipt(rec, out_dir, tag, fs): os.makedirs(out_dir, exist_ok=True) rp = os.path.join(out_dir, f'qqww-t4-sec481-chain-run_{tag}_{fs}.json') fd = os.open(rp, os.O_WRONLY | os.O_CREAT | os.O_EXCL, 0o644) with os.fdopen(fd, 'w') as f: json.dump(rec, f, indent=1, default=str) return rp def singular_factors(A, x): n = A.rows; L = sp.Integer(1) for i in range(n): for j in range(n): if A[i, j] != 0: L = sp.lcm(L, sp.fraction(sp.cancel(A[i, j]))[1]) L = sp.Poly(L, x) facs = [] for f, m in sp.factor_list(L.as_expr(), x)[1]: fp = sp.Poly(f, x) if fp.degree() == 0: continue fp = fp.monic() order = 0 for i in range(n): for j in range(n): if A[i, j] == 0: continue dd = sp.Poly(sp.fraction(sp.cancel(A[i, j]))[1], x); o = 0 while dd.degree() > 0 and sp.rem(dd, fp).is_zero: dd = sp.quo(dd, fp); o += 1 order = max(order, o) facs.append((fp, order)) return facs, L def inf_order(A, x): o = -99 for i in range(A.rows): for j in range(A.cols): if A[i, j] == 0: continue nn, dd = sp.fraction(sp.cancel(A[i, j])) o = max(o, int(sp.Poly(nn, x).degree() - sp.Poly(dd, x).degree() + 2)) return o # <= 1 : regular (Fuchsian) at infinity def rational_roots(poly_x, xx): """rational roots of a polynomial in xx over Q (exact), as sympy Rationals""" P = sp.Poly(poly_x, xx) if P.degree() <= 0: return [] out = [] for f, m in sp.factor_list(P.as_expr(), xx)[1]: fp = sp.Poly(f, xx) if fp.degree() == 1: a, b = fp.all_coeffs(); out.append(sp.Rational(-b, a)) return sorted(set(out)) def residue_exponents(A, x, fp): """rational eigenvalues e of the residue matrix at the roots of the irreducible factor fp (simple pole), the same e at every root: M = (f*A mod f) * (f')^{-1} mod f in Q[x]/(f); chi(t) = det(t - M) mod f; N(t) = Res_x(chi, f); candidates = rational roots of N with chi(e) == 0 mod f verified.""" n = A.rows; f = fp.as_expr(); fd = sp.diff(f, x) g = sp.invert(fd, f, x) # (f')^{-1} mod f M = sp.zeros(n, n) for i in range(n): for j in range(n): if A[i, j] == 0: continue nn, dd = sp.fraction(sp.cancel(A[i, j])) q_den, r_ = sp.div(sp.Poly(dd, x), fp) # dd = f * q_den (simple pole) or dd coprime to f if not r_.is_zero: continue # no pole at f in this entry: contributes 0 to the residue inv_den = sp.invert(sp.rem(q_den.as_expr(), f, x), f, x) # the other factors are units mod f M[i, j] = sp.rem(sp.expand(nn * inv_den * g), f, x) t = sp.Symbol('t_chi') chi = (t * sp.eye(n) - M).det(method='berkowitz') chi = sp.Poly(sp.expand(chi), t) # reduce the coefficients mod f chi_red = sum(sp.rem(sp.expand(c), f, x) * t ** k for k, c in zip(range(chi.degree(), -1, -1), chi.all_coeffs())) if fp.degree() == 1: r0 = sp.solve(f, x)[0] Npoly = sp.expand(chi_red.subs(x, r0)) else: Npoly = sp.resultant(sp.expand(chi_red), f, x) cands = rational_roots(Npoly, t) ok = [] for e in cands: if sp.rem(sp.expand(chi_red.subs(t, e)), f, x) == 0: ok.append(e) return ok def inf_exponents(A, x): """rational eigenvalues rho of A_1 = lim x*A(x) (regular at infinity): solutions ~ x^rho""" A1 = (A * x).applyfunc(lambda e: sp.limit(sp.cancel(e), x, sp.oo)) t = sp.Symbol('t_chi') chi = (t * sp.eye(A.rows) - A1).det(method='berkowitz') return rational_roots(sp.expand(chi), t) def poly_solutions(A, x, L, facs, exps, N): """nullspace of D P' - (D A - D u'/u) P over Q with deg P <= N; D = L (the lcm denominator); u'/u = sum e_i f_i'/f_i""" n = A.rows; D = L.as_expr() ul = sum(e * sp.diff(f.as_expr(), x) / f.as_expr() for (f, o), e in zip(facs, exps)) B = (D * A - D * ul * sp.eye(n)).applyfunc(sp.cancel) for i in range(n): for j in range(n): assert sp.fraction(B[i, j])[1].is_number, 'D A not polynomial' # unknowns c[i][k], P_i = sum_k c[i][k] x^k, k = 0..N # equation row i: D * P_i' - sum_j B_ij P_j = 0, polynomial identity in x degD = sp.Poly(D, x).degree(); degB = max([sp.Poly(sp.expand(B[i, j]), x).degree() for i in range(n) for j in range(n) if B[i, j] != 0] + [0]) maxdeg = max(degD + N, degB + N) + 1 nun = n * (N + 1) rows = [] Dp = sp.Poly(D, x) Bp = [[sp.Poly(sp.expand(B[i, j]), x) if B[i, j] != 0 else None for j in range(n)] for i in range(n)] for i in range(n): # coefficient of x^m in D*P_i' : sum_k c[i][k] k * [x^m] (D x^(k-1)) for m in range(maxdeg + 1): row = [Fraction(0)] * nun for k in range(1, N + 1): # D * k x^(k-1): coefficient of x^m = k * coeff of x^(m-k+1) in D mm = m - k + 1 if 0 <= mm <= degD: row[i * (N + 1) + k] += Fraction(int(sp.Rational(Dp.coeff_monomial(x ** mm)).p), int(sp.Rational(Dp.coeff_monomial(x ** mm)).q)) * k for j in range(n): if Bp[i][j] is None: continue for k in range(N + 1): mm = m - k if 0 <= mm <= Bp[i][j].degree(): c = sp.Rational(Bp[i][j].coeff_monomial(x ** mm)) row[j * (N + 1) + k] -= Fraction(int(c.p), int(c.q)) if any(v != 0 for v in row): rows.append(row) if not rows: # no constraint at all (D A - D u'/u == 0 and N == 0): every constant vector is a solution -- return the unit vectors return [[sp.Integer(1) if i == ii else sp.Integer(0) for i in range(n)] for ii in range(n)] M = flint.fmpq_mat(len(rows), nun, [flint.fmpq(v.numerator, v.denominator) for r in rows for v in r]) rr, rank = M.rref() if rank == nun: return [] # nullspace basis from the rref pivots = [] for r in range(rank): for c in range(nun): if rr[r, c] != 0: pivots.append(c); break free = [c for c in range(nun) if c not in pivots] sols = [] for fcol in free: v = [Fraction(0)] * nun; v[fcol] = Fraction(1) for r, pc in enumerate(pivots): q = rr[r, fcol]; v[pc] = -Fraction(int(q.p), int(q.q)) P = [sum(sp.Rational(v[i * (N + 1) + k].numerator, v[i * (N + 1) + k].denominator) * x ** k for k in range(N + 1)) for i in range(n)] sols.append(P) return sols def find_hyperexp(A, x, log): n = A.rows facs, L = singular_factors(A, x) io = inf_order(A, x) fuchsian = all(o <= 1 for f, o in facs) and io <= 1 cand = [] for fp, o in facs: if o == 1: cand.append(residue_exponents(A, x, fp)) else: cand.append([sp.Rational(k, 2) for k in range(-2 * EB, 2 * EB + 1)]) if io <= 1: rhos = inf_exponents(A, x) else: rhos = None log.append({'size': n, 'singular_factors': [(str(f.as_expr()), o) for f, o in facs], 'pole_order_at_infinity': io, 'fuchsian': fuchsian, 'candidate_exponents': [[str(e) for e in c] for c in cand], 'infinity_rational_exponents': ([str(r) for r in rhos] if rhos is not None else 'irregular at infinity: degree bound NB'), 'n_combinations': int(sp.prod([len(c) for c in cand])) if cand else 1}) combos = list(itertools.product(*cand)) if cand else [()] if len(combos) > MAXCOMBO: log[-1]['note'] = f'combinations {len(combos)} > MAXCOMBO {MAXCOMBO}: truncated (BOUNDED search)' combos = combos[:MAXCOMBO] tried = 0 for exps in combos: e_inf = sum(e * f.degree() for (f, o), e in zip(facs, exps)) if rhos is not None: Ns = [int(r - e_inf) for r in rhos if (r - e_inf).is_integer and r - e_inf >= 0] if not Ns: continue N = max(Ns) else: N = NB tried += 1 sols = poly_solutions(A, x, L, facs, list(exps), N) if sols: P = sols[0] g = sp.Integer(0) for p in P: g = sp.gcd(g, sp.Poly(p, x).as_expr()) if p != 0 else g P = [sp.cancel(p / g) for p in P] if g != 0 else P log[-1]['combinations_tried'] = tried return {'exponents': [(str(f.as_expr()), str(e)) for (f, o), e in zip(facs, exps)], 'degree_bound': N, 'P': [str(p) for p in P], 'n_solutions_at_this_exponent': len(sols)}, P, list(exps), facs, fuchsian log[-1]['combinations_tried'] = tried return None, None, None, facs, fuchsian def quotient(A, x, P): n = A.rows degs = [sp.Poly(p, x).degree() if p != 0 else 10 ** 9 for p in P] i0 = degs.index(min(degs)) cols = [sp.Matrix(P)] + [sp.eye(n)[:, j] for j in range(n) if j != i0] T = sp.Matrix.hstack(*cols) assert sp.cancel(T.det()) != 0 Ti = T.inv().applyfunc(sp.cancel) At = (Ti * (A * T - T.diff(x))).applyfunc(sp.cancel) first = [sp.cancel(At[i, 0]) for i in range(1, n)] assert all(v == 0 for v in first), 'the first column of the transformed system is not (u/u) e1' return At[1:, 1:], str(sp.cancel(At[0, 0])) def reduce_chain(A, x, label): log = []; chain = []; t0 = time.time() B = A.applyfunc(sp.cancel); bounded = False while B.rows > 0: found, P, exps, facs, fuchsian = find_hyperexp(B, x, log) if not fuchsian: bounded = True if found is None: verdict = ('NO_FIRST_ORDER_FACTOR' if fuchsian else 'NO_FIRST_ORDER_FACTOR_BOUNDED') + f'_REMAINDER_RANK_{B.rows}' return {'label': label, 'size': A.rows, 'verdict': verdict, 'chain': chain, 'remainder_rank': B.rows, 'log': log, 'bounded_search_used': bounded, 'wall_s': round(time.time() - t0, 3), 'remainder_block': [[str(sp.cancel(B[i, j])) for j in range(B.rows)] for i in range(B.rows)]} B, rate = quotient(B, x, P) found['rate_u_prime_over_u'] = rate # the canonical exponents: the search's e_i are defined only up to integers absorbed into P (P is made primitive in the # quotient); the rate u'/u of the found solution is the invariant object -- its partial fractions give the exponents try: found['rate_partial_fractions'] = str(sp.apart(sp.sympify(rate, locals={str(x): x}), x)) except Exception as ex: found['rate_partial_fractions'] = f'apart failed: {ex}' found['exponents_note'] = 'search labels; the canonical exponents are the residues of rate_u_prime_over_u (rate_partial_fractions)' chain.append(found) # a REDUCIBLE verdict is exact whatever the candidate bounds: every factor in the chain is an exact hyperexponential solution # (the quotient asserts A P - P' = c P exactly); the bounds only weaken a NEGATIVE verdict return {'label': label, 'size': A.rows, 'verdict': 'REDUCIBLE_TO_FIRST_ORDER', 'chain': chain, 'remainder_rank': 0, 'log': log, 'bounded_search_used': bounded, 'wall_s': round(time.time() - t0, 3)} def controls(x): out = {} # C1: planted reducible 3x3 chain (exponents 1/2, -1/2, 1 at 0, 1, 2), conjugated by a polynomial gauge with a non-constant det Dg = sp.diag(R(1, 2) / x - R(1, 2) / (x - 1), R(1, 3) / (x - 2), -1 / x + R(3, 2) / (x - 1)) Dg[0, 1] = 1 / (x - 1); Dg[1, 2] = x / (x - 2) T = sp.Matrix([[1, x, 0], [0, 1, x + 1], [x - 3, 0, 1]]) A1 = (T * Dg * T.inv() + T.diff(x) * T.inv()).applyfunc(sp.cancel) out['C1_planted_reducible_3x3'] = reduce_chain(A1, x, 'C1 planted reducible 3x3, gauge det ' + str(sp.factor(T.det()))) out['C1_planted_reducible_3x3']['expected'] = 'REDUCIBLE_TO_FIRST_ORDER' # C2: Legendre Picard-Fuchs companion: x(1-x) y'' + (1-2x) y' - y/4 = 0 A2 = sp.Matrix([[0, 1], [R(1, 4) / (x * (1 - x)), -(1 - 2 * x) / (x * (1 - x))]]) out['C2_legendre_irreducible_2x2'] = reduce_chain(A2, x, 'C2 Legendre PF companion (K(x), K(1-x))') out['C2_legendre_irreducible_2x2']['expected'] = 'NO_FIRST_ORDER_FACTOR_REMAINDER_RANK_2' # C3: 2x2 triangular with half-integer exponents A3 = sp.Matrix([[R(1, 2) / x - R(1, 2) / (x - 1), 1 / (x * (x - 1))], [0, R(-3, 2) / (x - 1)]]) out['C3_halfinteger_triangular_2x2'] = reduce_chain(A3, x, 'C3 half-integer triangular 2x2') out['C3_halfinteger_triangular_2x2']['expected'] = 'REDUCIBLE_TO_FIRST_ORDER' # C4: Legendre (+) a 1x1 hyperexponential factor, conjugated by a constant gauge A4 = sp.diag(A2, sp.Matrix([[R(1, 2) / (x - 1)]])) C = sp.Matrix([[1, 2, 0], [0, 1, 1], [1, 0, 1]]) A4 = (C * A4 * C.inv()).applyfunc(sp.cancel) out['C4_legendre_plus_1x1'] = reduce_chain(A4, x, 'C4 Legendre (+) 1x1 half-integer, constant gauge') out['C4_legendre_plus_1x1']['expected'] = 'NO_FIRST_ORDER_FACTOR_REMAINDER_RANK_2 after one factor' def cls(v): return v.replace('_BOUNDED', '') # the verdict class; the BOUNDED label (a non-Fuchsian point in the companion/gauged form) is recorded beside ok = (cls(out['C1_planted_reducible_3x3']['verdict']) == 'REDUCIBLE_TO_FIRST_ORDER' and cls(out['C2_legendre_irreducible_2x2']['verdict']) == 'NO_FIRST_ORDER_FACTOR_REMAINDER_RANK_2' and cls(out['C3_halfinteger_triangular_2x2']['verdict']) == 'REDUCIBLE_TO_FIRST_ORDER' and cls(out['C4_legendre_plus_1x1']['verdict']) == 'NO_FIRST_ORDER_FACTOR_REMAINDER_RANK_2' and len(out['C4_legendre_plus_1x1']['chain']) == 1) return out, ok def served_L1_block(path, rows=(54, 55, 56, 57, 58, 59)): y, d = sp.symbols('y d') raw = json.load(open(path)) def ev(nd): return sum(R(c) * d ** k for k, c in enumerate(nd['num'])) / sum(R(c) * d ** k for k, c in enumerate(nd['den'])) idx = {r: i for i, r in enumerate(rows)} B = sp.zeros(len(rows), len(rows)) for key, e in raw['A_entries'].items(): i, j = (int(t) for t in key.split(',')) if i in idx and j in idx: N = sum(ev(nd) * y ** k for k, nd in enumerate(e['N'])); Q = sum(ev(qd) * y ** k for k, qd in enumerate(e['Q'])) B[idx[i], idx[j]] = sp.cancel(N / Q) masters = [raw['masters'][r] for r in rows] return B.subs(d, 4).applyfunc(sp.cancel), masters, y def main(): ap = argparse.ArgumentParser(prog='qqww-t4-sec481-chain.py', description='verify the chain: the first-order factorisation of the sector-481 eps^0 block of row 35 (see the module docstring)') mode = ap.add_mutually_exclusive_group(required=True) mode.add_argument('--controls', action='store_true', help='the four built-in controls C1-C4 (seconds-class); exit 0 iff all as expected') mode.add_argument('--line', choices=('L1', 'L2'), help='the served line block (rows 54-59 at d = 4, variable y); the connection pinned by sha256') mode.add_argument('--block', metavar='RAY_BLOCK.json', help='a ray block receipt (key A_eps0_block_rows, variable lam): the vendored ray objects, pinned by sha256') mode.add_argument('--planted-served', action='store_true', help='not offered from this bundle (refuses by name; see the module docstring)') ap.add_argument('--served-dir', default=HERE, help='the bundle directory holding row35_data.json and row35_L2/connection_L2.json (default: the directory of this script)') ap.add_argument('--reference', metavar='RECEIPT.json', help='a receipt of the same form; the chain must equal it factor by factor (exponent lists, rates) for exit 0') ap.add_argument('--allow-unpinned-block', action='store_true', help='with --block: run on a receipt that is not one of the two vendored ray objects (stated in the receipt)') ap.add_argument('--out', metavar='DIR', help='the receipt directory (default: a fresh sec481-chain-run_/ under the working directory; never inside the bundle)') ap.add_argument('--tag', help='the receipt name qqww-t4-sec481-chain-run__.json (default: the mode)') args = ap.parse_args() st = stamp(); fs = stamp('+%Y%m%dT%H%M%SZ'); t0 = time.time() if args.planted_served: refuse(f"--planted-served is not offered from this bundle: the served-block (+) Legendre planted control of the record ran {PLANTED_SERVED['wall_s']} s under " f"NB {PLANTED_SERVED['bounds']['NB']} / MAXCOMBO {PLANTED_SERVED['bounds']['MAXCOMBO']} on a {PLANTED_SERVED['clock_s']}-s clock and returned {PLANTED_SERVED['verdict']} after " f"{PLANTED_SERVED['chain_len']} factors (exit code {PLANTED_SERVED['exit_code']}: the expected rank-2 remainder after six factors was not reached inside the truncated candidate set); " f"the seconds-class planted control is C1 of --controls") out_dir = os.path.abspath(args.out) if args.out else os.path.join(os.getcwd(), f'sec481-chain-run_{fs}') bundle = os.path.realpath(HERE) if os.path.realpath(out_dir) == bundle or os.path.realpath(out_dir).startswith(bundle + os.sep): refuse(f'--out {out_dir} is inside the bundle directory {bundle}; the receipt directory must lie outside it') rec = {'schema': SCHEMA, 'stamp_utc': st, 'tag': None, 'mode': None, 'line': None, 'producer': os.path.basename(__file__), 'producer_sha256': sha(os.path.abspath(__file__)), 'recipe': RECIPE, 'bounds': {'EB': EB, 'NB': NB, 'MAXCOMBO': MAXCOMBO}, 'sympy': sp.__version__, 'python_flint': flint.__version__} rc = 0 if args.controls: rec['mode'] = '--controls'; rec['tag'] = args.tag or 'controls' x = sp.Symbol('lam') rec['controls'], ok = controls(x); rec['controls_all_pass'] = ok; rc = 0 if ok else 3 elif args.block: src = os.path.abspath(args.block) if not os.path.isfile(src): refuse(f'--block {src}: no such file') got = sha(src); base = os.path.basename(src) pinned = RAY_PINS.get(base) if pinned is None or pinned != got: if not args.allow_unpinned_block: refuse(f'--block {base} sha256 {got[:16]}... is not one of the two vendored ray objects ({", ".join(k[:24] + "... " + v[:16] + "..." for k, v in RAY_PINS.items())}); pass --allow-unpinned-block to run on it anyway') blk = json.load(open(src)); x = sp.Symbol('lam') rows = blk['A_eps0_block_rows']; n = len(rows) A = sp.Matrix(n, n, lambda i, j: sp.sympify(rows[i][j], locals={'lam': x})) rec['mode'] = '--block'; rec['tag'] = args.tag or f'block-{blk["config_receipt"]["ray"]["name"]}' rec['input'] = {'file': base, 'sha256': got, 'pinned': pinned == got, 'block_size': n, 'masters': blk['config_receipt']['masters_served_order'], 'ray': blk['config_receipt']['ray']} rec['result'] = reduce_chain(A, x, f'sector {blk["config_receipt"]["sector"]} eps^0 block on the ray {blk["config_receipt"]["ray"]["name"]}') else: line = args.line served = os.path.abspath(args.served_dir) rel = LINE_FILE[line]; src = os.path.join(served, rel) got = check_pin(src, SERVED_PINS[rel], f'--line {line}') A, masters, y = served_L1_block(src) if [m.replace(' ', '') for m in masters] != SEC481_MASTERS: refuse(f'--line {line}: rows 54-59 of {rel} are not the six sector-481 masters {SEC481_MASTERS}: {masters}') rec['mode'] = '--line'; rec['tag'] = args.tag or line; rec['line'] = line rec['input'] = {'file': rel, 'sha256': got, 'served_dir': os.path.basename(served), 'rows': [54, 55, 56, 57, 58, 59], 'masters': masters, 'note': f'the SERVED {line} line block in y at d = 4 ((x, z) = {LINE_XZ[line]}); NOT the ray object'} rec['result'] = reduce_chain(A, y, f'served {line} sector-481 block, d = 4, variable y') if 'result' in rec: if rec['result']['verdict'] != 'REDUCIBLE_TO_FIRST_ORDER': rc = 3 if args.reference: rp = os.path.abspath(args.reference) if not os.path.isfile(rp): refuse(f'--reference {rp}: no such file') ref = load_reference(rp) cmp_ = compare_chain(rec['result'], ref['result']) rec['reference'] = {'file': os.path.basename(rp), 'sha256': sha(rp), 'verdict': ref['result']['verdict'], 'compare': cmp_} if not cmp_['equal']: rc = 3 rec['wall_s'] = round(time.time() - t0, 3); rec['stamp_end_utc'] = stamp(); rec['exit_code'] = rc rp = write_receipt(rec, out_dir, rec['tag'], fs) summ = {'receipt': os.path.basename(rp), 'receipt_dir': out_dir, 'sha256': sha(rp), 'wall_s': rec['wall_s'], 'rc': rc} if 'controls' in rec: summ['controls'] = {k: (v['verdict'], v['expected'], v['wall_s']) for k, v in rec['controls'].items()}; summ['controls_all_pass'] = rec['controls_all_pass'] if 'result' in rec: summ['verdict'] = rec['result']['verdict']; summ['chain_len'] = len(rec['result']['chain']); summ['remainder_rank'] = rec['result']['remainder_rank']; summ['bounded'] = rec['result']['bounded_search_used'] if 'reference' in rec: summ['reference_equal_factor_by_factor'] = rec['reference']['compare']['equal']; summ['reference_differing'] = rec['reference']['compare']['differing'] print(json.dumps(summ, indent=1, default=str)) sys.exit(rc) if __name__ == '__main__': main()