"""Independent exact checks for T652, signed Monster Fourier coefficients. sage -python checks.py # generated values sage -python checks.py --stored # authenticated API readback Requires the attached generator and JSON files. Companion JSON documents T79.json, T80.json, T81.json and T635.json may be supplied beside this script; otherwise they are read through the public API. No writes are performed. The checker uses published coefficients, independent eta products, character sums and held-out replication relations, not the generating recurrence. """ import numberdb.sage as numberdb from sage.rings.integer_ring import ZZ from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from pathlib import Path from math import gcd import json import os import sys import copy import generate from check import exactness, measure, agrees_with, names_its_rings HERE = Path(__file__).resolve().parent R = PolynomialRing(ZZ,'X') X = R.gen() def document(tid): path = HERE / (tid+'.json') return json.loads(path.read_text()) if path.exists() else numberdb.table(tid) def records(doc): block = doc['Numbers'] if isinstance(block,list): return [(tuple(str(row['params'][p]) for p in doc.get('Parameters',{})), ZZ(row['number']) if doc['Data properties']['type']=='Z' else row['number']) for row in block if 'number' in row] result=[] def walk(node, keys): if isinstance(node,dict) and 'number' in node: result.append((tuple(keys), node['number'])) elif isinstance(node,dict) and 'equals' in node: return elif isinstance(node,dict): for k,v in node.items():walk(v,keys+[str(k)]) else: result.append((tuple(keys),node)) walk(block,[]) return result def finite_eta(p, nmax): """R_p=q^-1 A; A=prod over p∤m of (1-q^m)^(24/(p-1)).""" exponent,remainder=divmod(24,p-1) assert remainder==0 size=nmax+1 a=[ZZ(1)]+[ZZ(0)]*size inv=[ZZ(1)]+[ZZ(0)]*size for m in range(1,size+1): if m%p==0: continue for _ in range(exponent): for j in range(size,m-1,-1):a[j]-=a[j-m] for j in range(m,size+1):inv[j]+=inv[j-m] assert a[1]==-exponent factor=p**(12//(p-1)) return {'B':{n:a[n+1] for n in range(1,nmax+1)}, 'A':{n:a[n+1]+factor*inv[n-1] for n in range(1,nmax+1)}} def multiply(a,b,cap=30): result={} for i,x in a.items(): for j,y in b.items(): if i+j<=cap:result[i+j]=result.get(i+j,ZZ(0))+x*y return {i:x for i,x in result.items() if x} def laurent_faber(a, max_degree): """Build by triangular Laurent cancellation, no Faber recurrence.""" series={-1:ZZ(1),**a} powers=[{0:ZZ(1)}] polys={0:R(1)} evaluations={0:powers[0]} for m in range(1,max_degree+1): powers.append(multiply(powers[-1],series)) q=dict(powers[m]); polynomial=X**m for j in range(m-1,-1,-1): coefficient=q.get(-j,ZZ(0)) polynomial-=coefficient*polys[j] for k,v in evaluations[j].items():q[k]=q.get(k,ZZ(0))-coefficient*v assert q[-m]==1 and all(q.get(k,0)==0 for k in range(-m+1,1)) polys[m]=polynomial evaluations[m]={i:v for i,v in q.items() if v} return polys,evaluations def replication_errors(data): errors=[]; compared=0 for g in generate.CLASSES: pol,ev=laurent_faber(data[g],5) assert pol[2]==X**2-2*data[g][1] assert pol[3]==X**3-3*data[g][1]*X-3*data[g][2] for m in [3,5]: other=generate.CLASS_DATA['power_maps'][str(m)][g] for n in range(1,30//m+1): expected=m*data[g][m*n] if n%m==0:expected+=data[other][n//m] compared+=1 if ev[m].get(n,0)!=expected:errors.append([g,m,n]) return errors,compared def run(stored=False): generated=generate.coefficients() data=copy.deepcopy(generated) if stored: doc=document('T652') rows=records(doc) expected=[(str(p['g']),str(p['n'])) for p in generate.McKayThompsonCoefficients().enumerate()] assert [k for k,v in rows]==expected data={g:{} for g in generate.CLASSES} data['1A']=generated['1A'] for (g,n),v in rows:data[g][int(n)]=ZZ(v) values={(g,n):v for g in generate.CLASSES for n,v in data[g].items()} assert not exactness(values) assert not names_its_rings(str(HERE/'generate.py')) # Known-answer controls execute before comparisons. e2=finite_eta(2,30) assert (e2['A'][1],e2['A'][2],e2['B'][1],e2['B'][2])==(4372,96256,276,-2048) pol,ev=laurent_faber({n:ZZ(0) for n in range(1,31)},5) assert pol[5]==X**5 and ev[5]=={-5:1} pol,_=laurent_faber(data['2A'],3) assert pol[2]==X**2-8744 and pol[3]==X**3-13116*X-288768 # Negative controls make sure the tests reject wrong signs and shifts. wrong=copy.deepcopy(data);wrong['2B'][30]=-wrong['2B'][30] assert wrong['2B'][30]!=finite_eta(2,30)['B'][30] wrong=copy.deepcopy(data);wrong['6A'][30]+=1 assert replication_errors(wrong)[0] shifted=copy.deepcopy(data);shifted['3A']={n:data['3A'].get(n+1,0) for n in range(1,31)} assert replication_errors(shifted)[0] # Published first-ten control rows, including all four seeds per class. controls=json.loads((HERE/'published-controls.json').read_text())['coefficients'] for g,expected in controls.items(): for n,v in enumerate(expected,1):assert data[g][n]==v,(g,n,data[g][n],v) # Independent character decompositions in Conway–Norton Table 1a. decomposition={1:[1,1],2:[1,1,1],3:[2,2,1,1],4:[2,3,2,1,0,1],5:[4,5,3,2,1,1,1]} chars=generate.CLASS_DATA['first15_characters'] assert chars['1A'][:7]==[1,196883,21296876,842609326,18538750076,19360062527,293553734298] for g in data: for n,multiplicities in decomposition.items(): assert sum(a*b for a,b in zip(multiplicities,chars[g]))==data[g][n],(g,n,'character') eta_values={} for p in [2,3,5,7]: for suffix,coefs in finite_eta(p,30).items(): for n,v in coefs.items():eta_values[(str(p)+suffix,n)]=v assert not agrees_with({k:values[k] for k in eta_values},lambda key:eta_values[key]) errors,replication_count=replication_errors(data) assert not errors,errors # All 30 identity coefficients agree with the separate existing expansion. jrows={int(k[0]):ZZ(v) for k,v in records(document('T79'))} for n in range(1,31):assert data['1A'][n]==jrows[n] # Formal E4 and Delta source controls supporting the identity-series calculation. erows={int(k[0]):ZZ(v) for k,v in records(document('T81'))} drows={int(k[0]):ZZ(v) for k,v in records(document('T80'))} delta=[ZZ(1)]+[ZZ(0)]*30 for m in range(1,31): for _ in range(24): for j in range(30,m-1,-1):delta[j]-=delta[j-m] for n in range(1,31): assert erows[n]==240*sum(ZZ(d)**3 for d in range(1,n+1) if n%d==0) assert drows[n]==delta[n-1] faber_rows=dict(records(document('T635'))) faber_count=0 for g in ['1A','2A','2B','3A','3B','5A','5B','7A','7B']: pol,_=laurent_faber(data[g],20) for m in range(2,21): assert pol[m]==R(str(faber_rows[(g,str(m))]).replace('^','**')),(g,m,'Faber') faber_count+=1 # Powers have the expected orders; identity plus chosen classes is closed. orders=dict(zip(generate.CLASS_DATA['class_names'],generate.CLASS_DATA['class_orders'])) for p,pm in generate.CLASS_DATA['power_maps'].items(): assert set(pm)==set(data) for g,h in pm.items():assert orders[h]==orders[g]//gcd(orders[g],int(p)) for g in generate.CLASSES: for n in range(1,31):assert abs(data[g][n])<=data['1A'][n] report={'table':'T652','mode':'stored' if stored else 'generated', 'skill_manifest':'agents/runs/20261101T140515Z-build-skills/manifest.json', 'skill_sha256':'a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278', 'all_checks_passed':True,'published_controls':330,'character_decompositions':165, 'independent_eta_coefficients':240,'held_out_replication_coefficients':replication_count, 'faber_companions':faber_count,'j_companions':30,'E4_companions':30,'Delta_companions':30, 'known_answer_controls':True,'wrong_sign_and_offset_controls_rejected':True, 'negative_entries':sum(v<0 for v in values.values()),'zero_entries':sum(v==0 for v in values.values()), 'nonzero_entries':sum(v!=0 for v in values.values()),'measurement':measure(values)} return report if __name__=='__main__': if os.environ.get('NUMBERDB_KEY_FROM_STDIN')=='1': os.environ['NUMBERDB_API_KEY']=sys.stdin.read().strip() print(json.dumps(run('--stored' in sys.argv or os.environ.get('CHECK_STORED')=='1'),indent=2))