{"schema":"vela.problem-packet.v0.1","problem":396,"statement":"Is it true that for every $k$ there exists $n$ such that\\[\\prod_{0\\leq i\\leq k}(n-i) \\mid \\binom{2n}{n}?\\]","status":"open","seam":"sealed","closureRoutes":[{"type":"verified_computation","verifierKind":"python:verify_396_sieve_certificate.py","note":"sieve lower-bound certificate 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_7cb1a5b07207120e","banked":"the GPT-Pro reduction is banked and re-verified (obstruction map: HONEST, no solve)","open":"the reduction does not solve #396; the solve remains open.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_07b78ee6f3be9a0b","kind":"reduction","claim":"Erdos #396: verified REDUCTION. For every k, the conjecture (exists n with prod_{0<=i<=k}(n-i) | C(2n,n)) follows from long consecutive runs in the governor set G={m : m | C(2m,m)}. Chain (GPT-Pro, Opus-verified): Kummer per-prime form prod|C(2n,n) iff for all p sum_i v_p(n-i)<=C_p(n); carry-invariance lemma (q odd prime, q|m, 0<=j<=(q-1)/2 => C_q(m+j)=C_q(m)) [verified 0/3000 violations]; small-prime barrier (in a k-witness every block term passes the governor test at primes q>=2k+1); reduction proposition (n..n-k in G + finite small-prime check => n in E_k); the small-prime set S_k has density 1 (Pomerance Lemma 2), so small primes are NOT the obstruction. Density: E_k subset {n: P+(n),P+(n-1)<=sqrt(2n)} giving upper-log-density <= (1-log2)^2 ~ 0.094 (Teravainen, arXiv 1710.01195) < governor density 0.11425 (Ford-Konyagin, arXiv 1909.03903) -- so #396 is a genuinely correlated-pattern problem, not the k=0 governor problem. Data corrected to OEIS A375077 (witnesses k<=14; k=5,6,7 = 3648841,7979090,101130029 independently re-verified). The carry-invariance lemma + barrier are independently formalized in the Dehorty Lean project (github jdehorty/erdos396). OPEN CORE: prove G contains runs of length k+1 for every k (Ford-Konyagin give positive density but not runs).","grade":"verified_reduction","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[{"id":"A375077","name":"Smallest k such that Product_{i=0..n} (k-i) divides C(2*k,k).","terms":"2,2480,8178,45153,3648841,7979090,101130029,339949252,1019547844,17609764994,1070858041585,5048891644646,18253129921842,","url":"https://oeis.org/A375077"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}