A verified partial result (a case, a bound, an exact value)

Verdict: VERIFIED — the claim as stated, k(18) ≤ 34, is true, and with our standing lower bound it gives 33 ≤ k(18) ≤ 34, one better than the 35 our page carried since 11 August.

What we checked. The 34 denominators 18, 19, …, 36, 38, 39, 40, 43, 44, 45, 46, 48, 50, 52, 54, 63, 275, 14143068, 10457416992 are distinct, strictly increasing, all at least 18, and their reciprocals sum to exactly 1. We recomputed this three ways with our own code (none of the submitter's scripts were run): an exact rational sum (fractions.Fraction), an integer route (L = lcm = 6209352774424800 and Σ L/d = L), and the same two facts again in PARI/GP, a second language. A negative control (last denominator +1) is rejected by both routes. The five-for-four surgery on our 35-term identity — 1/42 + 1/640 + 1/337280 + 1/1735650 + 1/68479214496 = 1/46 + 1/275 + 1/14143068 + 1/10457416992, both sides 577164227639/22744881957600 — holds exactly, the five removed terms are in our set and the four added ones are not. Verifier, logs, a frozen copy of the release files (every fetched file matches the author's own sha256 manifest) and SHA256SUMS: verification/295/reader-k18-34.

Novelty. OEIS A192881 (shortest expansion whose first term is 1/n) ends at a(16)=30, last revised 2026-05-30; erdosproblems.com/295 records no value of k(18); the newest unit-fraction papers on arXiv and the OpenAI 2026-10-06 release say nothing about it. As far as we can find, k(18) ≤ 34 appears nowhere before this note. Since the certificate starts at 18, and every expansion counted by A192881 has all denominators at least its first, it also gives 33 ≤ a(18) ≤ 34 for A192881.

What is his and what is ours. The 34-term construction is the submitter's (his note says OpenAI Codex agents found it working for him, which he discloses). The lower bound k(18) ≥ 33 is our August exhaustive search; he reports reproducing the 32-term exclusion with a search of his own that uses two extra pruning lemmas (a p-adic forced-denominator condition and a quadratic bound on the two-term tail), so his node counts are not comparable to ours. Our referee wrote its own search from the lemma text in his proofs.md (not from his code) and reproduced his counters exactly — 814,918 states, 648,193 two-term states, 134,999 prime prunes, 31,726 failed states — and, with his extra pruning switched off, reproduced our published counters exactly (14,678,224 / 11,753,049 / 114,603,091 at 32 terms; 20,293 / 13,139 / 69,727 at 31). So his search is our search tree plus two proved lemmas, not an independent method; the note's "independently reproduced" overstates that, and its "integrality is checked exactly, without an imposed denominator cap" describes the code rather than the logged run (every two-term state there fell to the quadratic inequality; his own logs show 0 candidates tested). Neither point touches the certificate. The referee's plain-mode runs are, incidentally, a third implementation of our own 31/32-term exclusion agreeing with the published counters.

Labels. The upper bound is (a) elementary — a finite identity anyone can check by hand with a big-integer calculator. The lower bound remains (d) computer-assisted, as before.

What is left at 18. Whether a 33-term expansion exists. Nobody has found one or excluded it; our 32-term exclusion visited 14.7 million states, and the 33-term search will cost far more.

Referee. A separately briefed hostile referee (Claude Opus, with tools, told to refute) wrote its own certificate checker with five negative controls, re-proved the two pruning lemmas (§2 p-adic forced denominator, §3 the quadratic two-term bound; one wording gap in §2 — the replacement step is for the two smallest multiples — with the bound correct under that reading), ran 300,000 random controls on §3 and 113,224 on §2 with zero violations, compared full solution sets with and without his pruning on 23,095 random residual states (zero mismatches), found no prune in the code that could discard a valid branch, and returned VERIFIED; its objections are four sentences of wording in the note, none mathematical. Report and scripts: referee/REFEREE.md.