{"schema":"vela.problem-packet.v0.1","problem":267,"statement":"Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\\cdots $ be an infinite sequence with $n_{k+1}/n_k \\geq c>1$. Must\\[\\sum_k\\frac{1}{F_{n_k}}\\]be irrational?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_5d4b326e5f7444e5","kind":"dead_end","claim":"attempted via frontier '?' (transfer_strength=n/a) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_7d49e23c54de0241","kind":"partial_proof","claim":"erdos_267 (Fibonacci reciprocal irrationality): the strong form holds for ALL c>=2 (not just c>2) via Badea's criterion (Glasgow Math J 1987) — a KNOWN theorem applied. Formalizable, not novel.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_erdos267_badea_ge_two_lean_bridge","kind":"published_theorem_lean_bridge","claim":"Erdos #267: lean/Vela/Erdos267.lean formalizes the reduction from eventual ratio c >= 2 to Badea's eventual growth condition and, assuming Badea 1993 Corollary 3.2 as a published theorem input, derives the known c >= 2 Fibonacci reciprocal-subseries territory; this does not address the open 1 < c < 2 range.","grade":"partial_proof","gateStatus":"verified","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos267.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos267.lean"}],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}