"""One-form Laplacian determinants on lens spaces -- numberdb.org/T633. Run with SageMath: sage -pip install numberdb sage -python generate.py # verify every stored entry sage -python generate.py --fill # fill the draft with NUMBERDB_API_KEY The full one-form spectrum is the disjoint union of exact eigenvalues m^2-1 and coexact eigenvalues m^2, m>=2, on the radius-one quotient. The scalar generator of T632 and coexact generator of T631 are combined here. No companion decimals are used as numerical inputs. Exact SU(2) weight counts give quadratic multiplicity polynomials on residue classes modulo 2p. The shifted factors m-1 and m+1 have no multiplicative anomaly here, because the multiplicity Dirichlet series has no positive even pole. See continuation.md for the proof. Finite Hurwitz zeta sums and first derivatives are evaluated with Arb enclosures, then exponentiated. The 180 extra working bits are measured by checks.py across the full range. """ import os import sys from collections import Counter from functools import lru_cache from math import gcd import numberdb.sage as numberdb from sage.rings.integer_ring import ZZ from sage.rings.rational_field import QQ from sage.rings.complex_arb import ComplexBallField from sage.rings.real_arb import RealBallField from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing WORKING_GUARD = 180 def presentations(limit=20): for p in range(2, limit + 1): for q in range(1, p): if gcd(p, q) == 1: r = pow(q, -1, p) if q == min(q, p-q, r, p-r): yield p, q def cases(limit=20): for p, q in presentations(limit): for u in range(p//2 + 1): yield p, q, u @lru_cache(None) def scalar_coefficients(p, q, u): """Exact (m0,c0,c1,c2) with d(m0+2p*k)=c0+c1*k+c2*k^2.""" assert p >= 2 and 0 < q < p and gcd(p, q) == 1 and 0 <= u < p period = 2*p rows = [] for m0 in range(2, period+2): weights = Counter(a % period for a in range(1-m0, m0, 2)) c0 = c1 = c2 = 0 residues = range((m0-1) % 2, period, 2) for a in residues: for b in residues: exponent = a*(1+q) + b*(1-q) assert exponent % 2 == 0 if (exponent//2-u) % p == 0: c0 += weights[a]*weights[b] c1 += 2*(weights[a]+weights[b]) c2 += 4 rows.append((m0, ZZ(c0), ZZ(c1), ZZ(c2))) # Residue of F(z)=sum d(m)m^(-z) at z=2 vanishes exactly. assert sum(c1*period-2*c2*m0 for m0,c0,c1,c2 in rows) == 0 return tuple(rows) def scalar_multiplicity(p, q, u, m): if m == 1: return ZZ(u == 0) assert m >= 2 period = 2*p m0 = 2+(m-2) % period _, c0, c1, c2 = scalar_coefficients(p,q,u)[m0-2] k = (m-m0)//period return c0+c1*k+c2*k*k @lru_cache(None) def hurwitz_data(p, precision): C = ComplexBallField(precision) P = PolynomialRing(C, 's') s = P.gen() period = 2*p out = {} for numerator in range(1, period+3): a = QQ(numerator)/period for j in range(3): z = (s-j)._zeta_series(2, a=C(a)) assert z[0].imag().contains_zero() and z[1].imag().contains_zero() assert all(v.real().is_finite() and v.imag().is_finite() for v in (z[0],z[1])) out[numerator,j] = z[0].real(), z[1].real() return out @lru_cache(None) def scalar_logdet(p, q, u, digits=100): precision = numberdb.bits(digits, losing=WORKING_GUARD) R = RealBallField(precision) period = 2*p data = hurwitz_data(p, precision) lp = R(period).log() answer = R(0) for m0, c0, c1, c2 in scalar_coefficients(p,q,u): for shift in (-1, 1): a = QQ(m0+shift)/period alpha = (c0-c1*a+c2*a*a, c1-2*c2*a, c2) for j in range(3): z, dz = data[m0+shift,j] answer += alpha[j]*(lp*z-dz) assert answer.is_finite() return answer @lru_cache(None) def coexact_coefficients(p, q, u): """(m0, c0,c1,c2) with d(m0+2p*k)=c0+c1*k+c2*k^2, k>=0. Weight exponents are [a(1+q)+b(1-q)]/2. Both weights have the same parity, so the division by 2 is exact. Adding 2p to m adds two copies of each admissible residue to each SU(2) weight string. """ assert p >= 2 and 0 < q < p and gcd(p,q)==1 and 0 <= u < p period = 2*p rows = [] for m0 in range(2, period+2): large = Counter(a % period for a in range(-m0,m0+1,2)) small = Counter(b % period for b in range(2-m0,m0-1,2)) c0=c1=c2=0 residues=range(m0 % 2,period,2) for a in residues: for b in residues: e=a*(1+q)+b*(1-q) assert e % 2 == 0 if (e//2-u) % p == 0: c0 += large[a]*small[b]+small[a]*large[b] c1 += 2*(large[a]+small[b]+small[a]+large[b]) c2 += 8 rows.append((m0,ZZ(c0),ZZ(c1),ZZ(c2))) return tuple(rows) def coexact_multiplicity(p,q,u,m): if m < 2: return ZZ(0) period=2*p m0=2+(m-2)%period _,c0,c1,c2=coexact_coefficients(p,q,u)[m0-2] k=(m-m0)//period return c0+c1*k+c2*k*k @lru_cache(None) def coexact_logdet(p, q, u, digits=100): precision = numberdb.bits(digits, losing=WORKING_GUARD) R = RealBallField(precision) period = 2*p data = hurwitz_data(p, precision) lp = R(period).log() answer = R(0) for m0, c0, c1, c2 in coexact_coefficients(p,q,u): a = QQ(m0)/period alpha = (c0-c1*a+c2*a*a, c1-2*c2*a, c2) for j in range(3): z, dz = data[m0,j] answer += 2*alpha[j]*(lp*z-dz) assert answer.is_finite() return answer @lru_cache(None) def logdet(p, q, u, digits=100): """Full one-form log determinant from the two disjoint spectra.""" return scalar_logdet(p,q,u,digits)+coexact_logdet(p,q,u,digits) class OneFormLensDeterminants(numberdb.Generator): table = 'T633' parameters = ('p','q','u','convention') type = 'R' digits = 100 rigour = 'proven' files = ('generate.py','checks.py','independent.py','check.py', 'argument-selection.md','continuation.md','numerical-evidence.json', 'skill-evidence.json','source-evidence.json') def enumerate(self, limit=20): for p,q,u in cases(limit): for convention in ('det','logdet'): yield dict(p=str(p),q=str(q),u=str(u),convention=convention) def value(self, params, digits): p,q,u = (int(params[k]) for k in ('p','q','u')) L = logdet(p,q,u,digits) if params['convention'] == 'det': return L.exp() assert params['convention'] == 'logdet' return L if __name__ == '__main__': if os.environ.get('NUMBERDB_KEY_FROM_STDIN') == '1': key = sys.stdin.read().strip() if '=' in key and key.split('=',1)[0].replace('_','').isalnum(): key = key.split('=',1)[1].strip().strip("\"'") numberdb.configure(api_key=key) generator = OneFormLensDeterminants() if '--fill' in sys.argv or os.environ.get('NUMBERDB_PUBLISH') == '1': print(generator.publish(message='Fill full one-form lens-space draft with certified 100-digit values')) else: report = generator.verify(sample=None) print(report) sys.exit(0 if report.ok else 1)