{"schema":"vela.problem-packet.v0.1","problem":203,"statement":"Is there an integer $m\\geq 1$ with $(m,6)=1$ such that none of $2^k3^\\ell m+1$ are prime, for any $k,\\ell\\geq 0$?","status":"open","seam":"sealed","closureRoutes":[{"type":"witness","verifierKind":"crt_partial_cover","note":"a larger CRT partial cover re-checked by the frozen verifier"},{"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_0d4ac181db98ceaa","banked":"a partial CRT covering certificate (m + 20 prime rows, ~0.7467 density) verified (crt_partial_cover)","open":"extend to a full cover (the verified rows do not cover all residues).","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_500d3435c96d4548","kind":"partial_proof","claim":"Erdős #203 (is there m with (m,6)=1, no 2^k 3^l m+1 prime?): VERIFIED PARTIAL CRT cover + strategic reframing, Opus-verified. With m=8168305011630835886634520238999 (gcd(m,6)=1), 20 primes each kill an affine congruence class: p | 2^k3^l m+1 iff 2^k3^l == -m^{-1} mod p, ONE linear congruence alpha*k+beta*l == gamma mod h (h=lcm(ord_p2,ord_p3)=|<2,3> in F_p*|), NOT a single rectangle. All 20 rows independently verified (ord_p2/ord_p3/h, m mod p, T_p=-m^{-1}, and congruence <=> divisibility, 0 mismatches over all (k,l) mod h^2). Union kills density 87702779/117448695 ~= 0.7467 of the (k,l) lattice (Monte-Carlo-confirmed ~0.7468); surviving ~0.2533. This is a PARTIAL cover (NOT 100%), explicitly NOT a covering certificate and NOT a settlement of #203. Reframing: the 'one rectangle per prime' model is too weak (sum 1/(ord2 ord3) over p<=1e6 ~ 0.238 < 1); the quadratic-form/Iwaniec route applies to the q_i==1 mod 4 VARIANT (sum-of-two-squares), not the 2^k3^l semigroup; the '>=10^10' bound is heuristic not proven; Filaseta-Finch-Kozek argue 'finite covering or nothing' is too narrow. Verified partial certificate + obstruction map.","grade":"obstruction_map","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}