"""Independent checks for T622, run with sage -python checks.py. Default: compute the generator locally, then check it by a cotangent sum in 512-bit Arb, Dedekind reciprocity, and weighted lattice-point enumeration. --stored: read the table through the API and run every check on stored values. NUMBERDB_KEY_FROM_STDIN=1 selects --stored in the campaign wrapper. This file and the unchanged campaign check.py are attached to the draft. The output following EVIDENCE_JSON is a complete reproducible evidence object. """ import numberdb.sage as numberdb from sage.rings.rational_field import QQ from sage.rings.real_arb import RealBallField from sage.env import SAGE_VERSION from fractions import Fraction from functools import lru_cache from math import gcd import hashlib import json import os from pathlib import Path import sys import generate from check import exactness, measure, agrees_with, names_its_rings BITS = 512 R = RealBallField(BITS) def cotangent_eta(p, q, u): """Independent trigonometric representation, without sawtooth arithmetic.""" total = R(0) for j in range(1, p): a = R.pi()*R(j)/R(p) b = R.pi()*R(q*j)/R(p) c = R.pi()*R(2*u*j)/R(p) total += a.cos()/a.sin() * b.cos()/b.sin() * c.cos() return -total/R(p) @lru_cache(None) def dedekind_reciprocity(a, b): """Classical Dedekind sum from reciprocity, using only Python Fractions.""" if b == 1: return Fraction(0) a %= b assert a > 0 and gcd(a, b) == 1 return (-Fraction(1, 4) + (Fraction(a, b)+Fraction(b, a) +Fraction(1, a*b))/12 - dedekind_reciprocity(b, a)) def lattice_rho(p, q, u): """Count actual points of the Casson-Gordon triangle, including edges. The triangle has vertices (0,0), (k,0), (k,k*q/p), where k*q=u mod p. Interior, open-edge, nonorigin vertex weights are 1, 1/2, 1/4. k=0 is handled separately: the trivial rho is zero. """ if u == 0: return Fraction(0) k = (u*pow(q, -1, p)) % p weighted = Fraction(0) for x in range(k+1): for y in range((k*q)//p+1): if p*y > q*x or (x == 0 and y == 0): continue if x == k and (y == 0 or p*y == q*k): weighted += Fraction(1, 4) elif y == 0 or x == k or p*y == q*x: weighted += Fraction(1, 2) else: weighted += 1 area = Fraction(q*k*k, 2*p) return 4*(weighted-area) def independent_exact(key): p, q, u = key rational = -4*dedekind_reciprocity(q,p)+lattice_rho(p,q,u) return QQ(rational.numerator) / QQ(rational.denominator) def canonical_key(p, q, u): inverse = pow(q, -1, p) if inverse < q: q, u = inverse, (u*inverse) % p return p, q, min(u, (-u) % p) def control_checks(): known = {(2,1,0):QQ(0), (2,1,1):QQ(0), (3,1,0):-QQ(2)/9, (3,1,1):QQ(1)/9, (5,1,0):-QQ(4)/5, (5,1,1):-QQ(1)/5, (5,1,2):QQ(3)/5, (3,2,1):-QQ(1)/9} assert dedekind_reciprocity(1, 3) == Fraction(1,18) assert lattice_rho(3,1,1) == Fraction(1,3) assert lattice_rho(5,1,2) == Fraction(7,5) for key, expected in known.items(): ball = cotangent_eta(*key) assert ball.is_finite() and R(ball.rad()) < R(10)**(-130) assert ball.contains_exact(expected), (key, ball, expected) assert independent_exact(key) == expected, (key, independent_exact(key), expected) return {','.join(map(str,k)):str(v) for k,v in known.items()} def main(): controls = control_checks() # Must pass before using either checker. generator = generate.LensEta() expected_keys = [tuple(int(params[k]) for k in generator.parameters) for params in generator.enumerate()] stored_mode = '--stored' in sys.argv or os.environ.get('NUMBERDB_KEY_FROM_STDIN') == '1' verify_report = None if stored_mode: client = generate.client_from_environment() report = generator.verify(sample=None, client=client) assert report.ok, str(report) verify_report = str(report) doc = numberdb.table('T622', client=client) # The entries endpoint stores flat records; no Django database is needed. assert isinstance(doc['Numbers'], list) values = {tuple(int(row['params'][k]) for k in generator.parameters): QQ(row['number']) for row in doc['Numbers']} assert list(values) == expected_keys else: values = {key:generate.eta_exact(*key) for key in expected_keys} assert len(values) == 637 assert len(set(key[:2] for key in values)) == 87 assert not exactness(values) assert not names_its_rings(str(Path(__file__).with_name('generate.py'))) disagreements = agrees_with(values, independent_exact) assert not disagreements, disagreements[:10] ball_records = [] max_radius = R(0) min_relative_bits = BITS symmetry_count = 0 all_domain_count = 0 q1_count = 0 for key, value in values.items(): p,q,u = key ball = cotangent_eta(p,q,u) assert ball.is_finite() and R(ball.rad()) < R(10)**(-130), (key, ball) assert ball.contains_exact(value), (key, ball, value) if value: assert ball.accuracy() >= 430, (key, ball.accuracy()) min_relative_bits = min(min_relative_bits, int(ball.accuracy())) max_radius = max(max_radius, R(ball.rad())) ball_records.append({'p':p,'q':q,'u':u,'exact':str(value), 'cotangent_ball':str(ball), 'radius':str(ball.rad())}) if q == 1: formula = -QQ((p-1)*(p-2))/QQ(3*p) if u: formula += QQ(2*u*(p-u))/QQ(p)-1 assert formula == value q1_count += 1 # Check all omitted labels as well, using stored values for reconstruction. for p in range(2, generate.MAX_P+1): for q in range(1, p): if gcd(p,q) != 1: continue for u in range(p): key = (p,q,u) value = values[canonical_key(*key)] assert generate.eta_exact(*key) == value assert independent_exact(key) == value assert values[canonical_key(p,q,(-u)%p)] == value assert values[canonical_key(p,p-q,u)] == -value r=pow(q,-1,p) assert values[canonical_key(p,r,(u*r)%p)] == value symmetry_count += 3 all_domain_count += 1 assert sum(values[canonical_key(p,q,u)] for u in range(p)) == 0 evidence = {'table':'T622','mode':'stored' if stored_mode else 'local', 'archived_skill_manifest':json.loads(Path(__file__).with_name('skill-manifest.json').read_text()), 'sage_version':SAGE_VERSION,'python_version':sys.version, 'working_bits':BITS,'controls':controls,'measure':measure(values), 'selected_entries':len(values),'oriented_presentations':87, 'independent_exact_comparisons':len(values), 'ball_comparisons':len(values),'max_ball_radius':str(max_radius), 'minimum_nonzero_relative_accuracy_bits':min_relative_bits, 'full_domain_reconstruction_checks':all_domain_count, 'symmetry_comparisons':symmetry_count,'homogeneous_formula_checks':q1_count, 'character_sum_zero_checked':True,'verify_report':verify_report, 'generator_files':{name:hashlib.sha256(Path(__file__).with_name(name).read_bytes()).hexdigest() for name in ['generate.py','checks.py','check.py','argument-selection.md']}, 'q_representatives':{str(p):sorted({q for pp,q,u in values if pp==p}) for p in range(2,21)}, 'records':ball_records} print('EVIDENCE_JSON') print(json.dumps(evidence,indent=2)) if __name__ == '__main__': main()