{"schema":"vela.problem-packet.v0.1","problem":463,"statement":"Is there a function $f$ with $f(n)\\to \\infty$ as $n\\to \\infty$ such that, for all large $n$, there is a composite number $m$ such that\\[n+f(n)&#60;m&#60;n+p(m)?\\](Here $p(m)$ is the least prime factor of $m$.)","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_f21c725904b8bee2","kind":"partial_proof","claim":"Reduced Erdos #463 to a clean self-contained criterion: the answer is YES iff g(n)->infinity, where g(n)=max{m-n : m composite, m-n<p(m)}. Gave an explicit construction (p-rough unique multiple of a p","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_erdos463_margin_prefix_obstruction","kind":"vela.erdos_problem_attempt_record_draft.v1","claim":"Erdos #463: lean/Vela/Erdos463.lean proves the square-root finite-search bound for qualifying offsets, and scripts/erdos463_margin_search.py exactly recomputes g(n)=max{d>0 : n+d is composite and d<minFac(n+d)} for n<=10000000; this is finite-prefix evidence only and leaves g(n)->infinity open.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_8e5212d0a85ae0c7","kind":"known_result","claim":"Erdos #463: literature scoping (GPT-Pro, Opus-verified) shows g(n)->infinity is OPEN and NOT implied by any known result, so #463 is not closeable by citation. The equivalent reformulation is N - F_c(N) -> infinity with F_c(N)=min_{m>N composite}(m-p(m)); Erdos's stronger recorded conjecture N - F_c(N) ~ c N^{1/2} would imply it (matches our Lean sqrt(n) offset bound) but is itself open. The obstruction: at scale z ~ H no pointwise 'every length-H interval contains an H-rough number' bound can hold, since Rankin/Ford-Green-Konyagin-Maynard-Tao long-gap constructions (Y(x) >> x log x log_3 x / (log_2 x)^2 > x) give intervals fully covered by small-prime classes; sieve gives only almost-all intervals. Sibling: #385. Decision: do not re-fire GPT-Pro at #463.","grade":"honest_null","gateStatus":"verified","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos463.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos463.lean"}],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}