"""Exact checks for numberdb.org/T562; run with sage -python checks.py. Uses the standard campaign check.py for exactness, size and independent comparison. The mathematical checks use fractions.Fraction independently of the generator's Sage arithmetic. See source-notes.md for theorem locators. Optional --stored reads the authenticated draft and T561 from NumberDB; otherwise T561.json is the archived companion input. Both paths compare all 25 values. Read a token from stdin with NUMBERDB_KEY_FROM_STDIN=1. """ import json import os import sys from pathlib import Path from fractions import Fraction as F import numberdb.sage as numberdb from sage.rings.integer_ring import ZZ from sage.rings.rational_field import QQ from sage.rings.real_arb import RealBallField from sage.env import SAGE_VERSION from generate import LyapunovSums, hyperelliptic_sum from check import exactness, measure, agrees_with, names_its_rings W = Path(__file__).resolve().parent def rational(value): v = F(str(value)) return QQ(v.numerator) / QQ(v.denominator) def kappa(mu): # The equivalent form m(m+2)/(m+1) = (m+1)-1/(m+1). return sum((F(m+1)-F(1,m+1) for m in mu), F(0))/12 def partitions(total, maximum=None): if not total: yield () return for head in range(min(total, maximum or total), 0, -1): for tail in partitions(total-head, head): yield (head,)+tail def components(g, mu): # KZ Theorems 1-2, independently enumerate all classified components. if g == 2: return ('hyp',) if g == 3: return ('hyp','odd') if mu in ((4,),(2,2)) else ('connected',) hyper = mu in ((2*g-2,), (g-1,g-1)) if all(m % 2 == 0 for m in mu): return (('hyp',) if hyper else ()) + ('even','odd') return ('hyp','nonhyp') if hyper else ('connected',) def signature(params): return (int(params['g']), tuple(map(int,params['mu'].split(','))), params['component']) def document_values(document): block=document['Numbers']; out={} if isinstance(block,list): for row in block: out[signature(row['params'])] = str(row['number']) else: for g, profiles in block.items(): for mu, comps in profiles.items(): for c, row in comps.items(): out[(int(g),tuple(map(int,str(mu).split(','))),c)] = str(row['number'] if isinstance(row,dict) else row) return out def hyperelliptic_cover(g, mu): # EKZ Corollary 1, before its closed forms: sphere quadratic orders. ds = [2*g-3]+[-1]*(2*g+1) if len(mu)==1 else [2*g-2]+[-1]*(2*g+2) assert sum(ds)==-4 return sum((F(1,d+2) for d in ds if d%2), F(0))/4 def divisor_sum(mu, a, d, marked=()): # CM Props. 4.5,4.8. Divide the vanishing divisor intersection by C.lambda. # a + sum b_i/((m_i+1)L) - d*(12 - 12*kappa/L) = 0. return (sum((F(b,m+1) for m,b in marked), F(0))+12*d*kappa(mu))/(12*d-a) def cylinder_sum(g, mu, rows): # EMZ's r=c*zeta(2); EKZ/Vorobets: L=kappa+2 sum(q*r)/(D-1). dimension=2*g+len(mu)-1 weighted=sum((q*F(r) for q,r in rows),F(0)) return kappa(mu)+2*weighted/(dimension-1) def run(values, companion): # Controls must pass before using the independent methods. assert kappa((2,))==F(2,9) assert hyperelliptic_cover(2,(2,))==F(4,3) assert divisor_sum((4,),9,1)==F(8,5) # EMZ Example 14.7: only the cylinder configuration contributes, r=5/3. assert cylinder_sum(2,(2,),[(0,'10/3'),(1,'5/3')])==F(4,3) # Positive control for Arb's containment and finite-width checks. field=RealBallField(512) known=field(1)/3 assert known.is_finite() and (QQ(1)/QQ(3) in known) and field(known.rad())rational(kappa(mu)) hyp={key:value for key,value in values.items() if key[2]=='hyp'} assert not agrees_with(hyp,lambda key:rational(hyperelliptic_cover(key[0],key[1]))) for key,v in hyp.items(): assert hyperelliptic_sum(key[0],len(key[1]))==v # Farkas Thm 4.6, r=1,s=2: normalize the psi coefficient to one. r,s=1,2 a=F((r+2)*(r*r*s**3-r*r*s+2*r*s**3+6*r*s*s-2*r*s-8*r+s**3+6*s*s+3*s-8),2*(s+r+1)*(r*s+s-2)) c=F(s+1,2) b=F((s-1)*(s+1)*(r+1)*(r+2)*(r*s+s+4),12*(s+r+1)*(r*s+s-2)) assert (a/c,b/c)==(8,1) assert divisor_sum((3,3),8,1,[(3,-1),(3,-1)])==F(7,4) # Eight divisor-intersection calculations plus two odd-spin divisor slopes. cm={ (3,(4,),'odd'): divisor_sum((4,),9,1), (3,(3,1),'connected'): divisor_sum((3,1),9,1), (3,(2,1,1),'connected'): divisor_sum((2,1,1),-1,0,[(2,1),(1,3)]), (4,(6,),'even'): divisor_sum((6,),30,4,[(6,60)]), (4,(6,),'odd'): divisor_sum((6,),8,1,[(6,4)]), (4,(5,1),'connected'): divisor_sum((5,1),8,1,[(5,4)]), # Farkas Thm 4.6 has positive psi coefficients; CM Eq. (8) misprints signs. (4,(3,3),'nonhyp'): divisor_sum((3,3),8,1,[(3,1),(3,1)]), (4,(3,2,1),'connected'): divisor_sum((3,2,1),-1,0,[(3,1),(2,1),(1,3)]), } for g,mu in [(3,(2,2)),(4,(2,2,2))]: slope=F(4*(g+8),g+2) cm[(g,mu,'odd')]=12*kappa(mu)/(12-slope) assert len(cm)==10 disagreements=agrees_with({key:values[key] for key in cm},lambda key:rational(cm[key])) assert not disagreements,disagreements # EMZ Appendix B.2 p.100: q counts double arrows cyclically (end repeats start). even_rows=[(0,'180/37'),(1,'45/74'),(1,'225/37'),(0,'225/296'),(2,'5/37')] odd_rows=[(0,'252/31'),(1,'63/62'),(1,'144/31'),(2,'9/155'),(2,'16/155'),(0,'4/93'),(3,'1/186')] even=cylinder_sum(4,(2,2,2),even_rows) assert even==F(737,333) and even!=F(166,75) assert F(166,75)-even==F(1,8325) assert rational(even)==values[(4,(2,2,2),'even')] assert rational(cylinder_sum(4,(2,2,2),odd_rows))==values[(4,(2,2,2),'odd')] # Separate configuration reconstruction for H(3,3)^nonhyp. nonhyp_rows=[(1,'3699/1120'),(0,'64/5'),(1,'64/35'),(1,'27/80'),(0,'1/5'),(1,'1/32')] assert rational(cylinder_sum(4,(3,3),nonhyp_rows))==values[(4,(3,3),'nonhyp')] # All companion decimal values denote exact center +/- one last-place unit. # T561 derives its constants from EKZ; this is consistency, not independence. area=document_values(companion) assert set(area)==set(values) max_radius=field(0) for key,text in area.items(): center=F(text); ulp=F(1,10**len(text.partition('.')[2])) interval=field(rational(center)); interval=interval.add_error(field(rational(ulp))) back=field(rational(kappa(key[1])))+field.pi()**2*interval/3 assert back.is_finite() and field(back.rad())