"""Independent checks of Generalized log-sine integrals -- numberdb.org/T697. sage -python checks.py # check all computed entries, JSON to stdout sage -python checks.py --stored # additionally check the authenticated draft The numerical quadratures and finite polylogarithm/Gamma computations share no numerical code with the series generator. The generator supplies its enumeration and values only. Instructions: skill-manifest.json, originally archived at agents/runs/20261109T220207Z-build-skills/manifest.json. """ import os import sys import json from decimal import Decimal from fractions import Fraction from math import factorial import numberdb.sage as numberdb from numberdb._write import to_text from sage.rings.real_arb import RealBallField from sage.rings.complex_arb import ComplexBallField from sage.rings.rational_field import QQ import mpmath as mp from generate import LogSine, evaluate, WORKING_GUARD_BITS, TAIL_GUARD_DIGITS MANIFEST = 'agents/runs/20261109T220207Z-build-skills/manifest.json' SKILL_SHA256 = '886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b' def quadrature(n,k,t,dps): with mp.workdps(dps): theta=mp.pi*mp.mpf(t) if n-1-k==0: return -mp.quad(lambda v:v**k,[0,theta/2,theta]) return -mp.quad(lambda v:v**k*mp.log(2*mp.sin(v/2))**(n-1-k),[0,theta/2,theta]) def gamma_coefficients(R): c=[R(0),R(0)]+[(-1)**j*(1-R(2)**(1-j))*R(j).zeta()/j for j in range(2,10)] f=[R(1)] for m in range(1,10): f.append(sum((j*c[j]*f[m-j] for j in range(1,m+1)),R(0))/m) return [-R.pi()*factorial(m)*f[m] for m in range(10)] def one_log(k,t,C): z=(C.pi()*C(QQ(t))*C.gen(0)).exp() theta=C.pi()*C(QQ(t)); i=C.gen(0) out=sum(((-1)**j*factorial(k)//factorial(k-j)*theta**(k-j)*i**(-j-1)*z.polylog(j+2) for j in range(k+1)),C(0)) out-=(-1)**k*factorial(k)*i**(-k-1)*C(k+2).zeta() return out.real() def interval(text): # Exact decimal parsing, including its specified one-last-place uncertainty. d=Decimal(str(text)); center=Fraction(d) if '.' not in str(text) and 'e' not in str(text).lower():return QQ(center),QQ(center) radius=Fraction(10)**d.as_tuple().exponent return QQ(center-radius),QQ(center+radius) def contains(text,ball): lo,hi=interval(text) if not hasattr(ball,'lower'):return lo<=ball<=hi return lo<=ball.lower() and ball.upper()<=hi def records(document): block=document['Numbers'] if isinstance(block,list):return block out=[] for n,ks in block.items(): for k,ts in ks.items(): for t,v in ts.items(): row={'params':{'n':str(n),'k':str(k),'t':str(t)}} row.update(v if isinstance(v,dict) else {'number':v});out.append(row) return out def main(stored=False): mp.mp.dps=180 R=RealBallField(700);C=ComplexBallField(700) # Controls precede measurements, including a deliberate wrong-answer rejection. for t in ('1/3','1/2','2/3','5/6','1'): for n in (1,4,10): val=quadrature(n,n-1,t,130);theta=mp.pi*mp.mpf(t) assert abs(val+theta**n/n)600) assert expected.contains_zero() if key==(2,0,'1') else vh.overlaps(expected) gamma_count+=1 if n-1-k==1: expected=one_log(k,t,C) assert expected.is_finite() if key==(2,0,'1'):assert expected.contains_zero() and R(expected.rad())600 and vh.overlaps(expected),(key,'polylog moment') one_count+=1 rows.append({'params':p,'number':text,'accuracy_bits':int(v.accuracy()) if hasattr(v,'accuracy') else None, 'radius':str(v.rad()) if hasattr(v,'rad') else '0', 'quadrature_130':mp.nstr(q1,125),'quadrature_170':mp.nstr(q2,165), 'relative_quadrature_difference':mp.nstr(e,8)}) if (index+1)%30==0:print('checked',index+1,'entries',file=sys.stderr,flush=True) # Known low-order values, independently supplied by Borwein–Straub Example 1. assert high[(3,0,'1')].overlaps(-R.pi()**3/12) assert high[(4,0,'1')].overlaps(3*R.pi()*R(3).zeta()/2) # Specific companion relationships, using their current 100-digit intervals. companions={tid:numberdb.table(tid) for tid in ('T175','T681','T685')} def val(d,*keys): v=d['Numbers'] for key in keys:v=v[str(key)] return v['number'] if isinstance(v,dict) else v companion_count=0 for t in ('1/3','1/2','2/3'): assert contains(val(companions['T175'],'2',t),high[(2,0,t)]);companion_count+=1 for r in (2,3): assert contains(val(companions['T681'],'1',r),-high[(r+1,0,'1')]/R.pi());companion_count+=1 for r in range(1,10): expected=(high[(r+1,0,'1/3')]-high[(r+1,0,'1')])/R.pi() assert contains(val(companions['T685'],'3',r),expected);companion_count+=1 stored_count=0 stored_identity_checks=0 if stored: document=numberdb.table('T697');rs=records(document) assert [r['params'] for r in rs]==list(LogSine().enumerate()) back={} for row in rs: p=row['params'];key=(int(p['n']),int(p['k']),p['t']) assert contains(row['number'],high[key]),key # Read-back values are separately compared to the independent integral. lo,hi=interval(row['number']);q=mp.mpf(next(r['quadrature_170'] for r in rows if r['params']==p)) if key!=(2,0,'1'): assert mp.mpf(str(lo.numerator()))/int(lo.denominator())280 assert v.lower()<=hi and lo<=v.upper();stored_identity_checks+=1 evidence={'table':'T697','skill_manifest':MANIFEST,'skill_sha256':{'local':SKILL_SHA256,'live':SKILL_SHA256}, 'entries':len(rows),'minimum_accuracy_bits':min(r['accuracy_bits'] for r in rows if r['accuracy_bits'] is not None), 'working_bits':numberdb.bits(100,losing=WORKING_GUARD_BITS),'series_degree':numberdb.bits(100+TAIL_GUARD_DIGITS,losing=0), 'higher_precision_digits':140,'quadrature_working_digits':[130,170], 'maximum_scaled_quad_error':mp.nstr(worst_quad,10),'maximum_scaled_quad_precision_difference':mp.nstr(worst_precision,10), 'controls':15,'negative_control_rejected':True,'exact_zero_raw_radius':str(rawzero.rad()), 'gamma_checks':gamma_count,'finite_polylog_checks':one_count,'companion_checks':companion_count, 'stored_intervals_checked':stored_count,'stored_identity_checks':stored_identity_checks,'versions':{'numberdb':numberdb.__version__,'mpmath':mp.__version__},'rows':rows} print(json.dumps(evidence,indent=2)) if __name__=='__main__': if os.environ.get('NUMBERDB_KEY_FROM_STDIN')=='1':os.environ['NUMBERDB_API_KEY']=sys.stdin.read().strip() main('--stored' in sys.argv or os.environ.get('LOGSINE_STORED')=='1')