{"schema":"vela.problem-packet.v0.1","problem":307,"statement":"Are there two finite sets of primes $P,Q$ such that\\[1=\\left(\\sum_{p\\in P}\\frac{1}{p}\\right)\\left(\\sum_{q\\in Q}\\frac{1}{q}\\right)?\\]","status":"open","seam":"sealed","closureRoutes":[{"type":"verified_computation","verifierKind":"python:verify_307_barrier.py","note":"congruence-barrier construction re-check (pinned tier: scripts/requirements-verifiers.txt)"},{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"}],"obligations":[{"findingId":"vf_523a610fc3947249","banked":"the congruence-barrier construction verified","open":"the unconditional bound remains open.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_24ca4b78340c3bc9","kind":"reduction","claim":"Transfer from #306/#295: PROVEN equivalence -- #307 <=> a squarefree arithmetic-derivative 2-cycle M(P)'=M(Q), M(Q)'=M(P); exhaustive negative search (no cycle in first 17-24 primes).","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_8c4c8b60210f476f","kind":"reduction","claim":"Erdős #307 is PROVEN EQUIVALENT (Lean, lake-build green, no sorry/axiom) to the existence of a squarefree ARITHMETIC-DERIVATIVE 2-CYCLE: finite prime sets P,Q with M(P)'=M(Q) and M(Q)'=M(P), where M(S)=prod(S) and M(S)'=sum_{p in S} M(S)/p (the arithmetic derivative of squarefree M(S)). The reciprocal-sum witness condition is exactly A(S)/M(S)=sum 1/p, verified arithmetically. This is a VERIFIED REDUCTION, not a settlement: whether such a 2-cycle exists is OPEN (exhaustive negative search found none in the first 17-24 primes; no nonexistence proof). Known easy bound: sum_{p in P∪Q} 1/p >= 2, so |P∪Q| >= 60.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_498d0afeab612155","kind":"dead_end","claim":"Erdos #307: finite local obstructions (parity, mod-8, per-prime quadratic-residue, avoidance of primes 3 mod 4) provably cannot bound the primes in a squarefree arithmetic-derivative 2-cycle. Constructively (verified): for any target one builds an arbitrarily large prime set U with all r = 1 mod 8 plus a constructed prime t such that N/r (N=prod U) is a quadratic residue mod every r in U and A(U) = |U| mod 8 is free, with reciprocal mass > 2. Hence the decisive control point is the GLOBAL double-square lemma (A(U) +/- 2 M(U) both perfect squares), not its finite congruence shadows. This maps the barrier; it does not settle the problem.","grade":"obstruction_map","gateStatus":"verified","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos307.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos307.lean"}],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}