{"schema":"vela.problem-packet.v0.1","problem":10,"statement":"Is there some $k$ such that every large integer is the sum of a prime and at most $k$ powers of 2?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_f141d525491844f5","kind":"dead_end","claim":"attempted via frontier '?' (transfer_strength=n/a) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_7199d8341d2c33b9","kind":"partial_proof","claim":"erdos_10 (every integer = prime + <=k powers of 2?) OPEN; not a settlement. Only formalizable result is the KNOWN grechuk_example (1117175146 not in sumPrimeAndTwoPows 3), re-verified exactly. Adversarial synthesis CAUGHT errors in agent reports (Mersenne off-by-one, wrong supply counts 132/52/9 vs correct 120/21/3, a logical gap in the covering->representability reduction).","grade":"partial_proof","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[{"id":"A387053","name":"a(n) is the least k such that n is the sum of a prime and k powers of 2.","terms":"0,0,1,0,1,0,1,1,1,0,1,0,1,1,2,0,1,0,1,1,2,0,1,1,2,1,2,0,1,0,1,1,1,1,2,0,1,1,2,0,1,0,1,1,2,0,1,1,2,1,2,0,1,1,2,1,2,0,1,0,","url":"https://oeis.org/A387053"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}