{"schema":"vela.problem-packet.v0.1","problem":155,"statement":"Let $F(N)$ be the size of the largest Sidon subset of $\\{1,\\ldots,N\\}$. Is it true that for every $k\\geq 1$ we have\\[F(N+k)\\leq F(N)+1\\]for all sufficiently large $N$?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_8e3d8efdf7516946","kind":"dead_end","claim":"attempted via frontier 'sidon/B2' (transfer_strength=partial) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_9b9f85b529e4243d","kind":"verified_witness","claim":"A309370 (max integer-sum Sidon subset of {0,1}^n): NINE verified NEW lower-bound records beating the current OEIS best-known, Opus-independently-verified. Each set is a genuine Sidon set (re-verified from scratch with a base-3 per-bit vector-sum verifier distinct from the search's (a&b,a|b) key; matches OEIS definition 'a+b=c+d trivial only', doubles included; reproduces official a(0..5)=1,2,3,5,7,12). Improvements over the current OEIS comments (incl. Blair Jun 2-3 2026 and Sievers Sep 2025): a(9)>=47 (was 46), a(11)>=92 (was 87), a(12)>=133 (was 120), a(13)>=185 (was 169), a(14)>=257 (was 237), a(15)>=364 (was 334), a(16)>=505 (was 503), a(19)>=1435 (was 1397), a(20)>=1989 (was 1941). These are verified LOWER BOUNDS (A309370 is keyword:hard/more; exact values for n>=7 unknown), to be submitted as OEIS comment updates with witnesses (same format as the existing Blair Link). a(7)=24, a(8)=33, a(17)=712 MATCH current best (not new); a(10)=65 is below the current 66.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_21929f0161c967b4","kind":"reduction","claim":"Erdős #155 (F(N+k)<=F(N)+1 eventually <=> g(k):=limsup(F(N+k)-F(N))<=1): the B_2/endpoint route does NOT prove it, Opus-verified. The elementary difference count gives only the Golomb lower bound s(s-1)<=2(L-1), not the sharp Sidon upper bound. The endpoint-deletion reduction gives F(N+k)<=F(N)+h(k), h(k)=max disjoint-difference Sidon-pair size, but h(3)>=2 (X={1,2},Y={1,3} Sidon, (X-X)∩(Y-Y)={0}, verified) so it yields +2 not +1. The clean exact obstruction: #155 <=> optimal Golomb-ruler gap growth G(m+1)-G(m)->infinity; the B_2 count gives no mechanism for gap-growth. Honest obstruction map.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_ea5ba1aa938d0e48","kind":"verified_witness","claim":"A309370 (max integer-sum Sidon subset of {0,1}^n): VERIFIED Sidon constructions giving lower bounds a(n) >= size for n=7..20, Opus-independently-verified by two methods. Sizes: a(7)>=24, a(8)>=33, a(9)>=47, a(10)>=65, a(11)>=92, a(12)>=133, a(13)>=185, a(14)>=257, a(15)>=364, a(16)>=505, a(17)>=712, a(19)>=1435, a(20)>=1989. Each set independently re-verified with a from-scratch base-3 per-bit vector-sum verifier (distinct from the workflow's (a&b,a|b) key): all elements distinct and in [0,2^n), all pairwise integer-vector sums distinct (incl. doubles); the verifier agrees with the known a(7)=24. SCOPE/HONESTY: OEIS A309370 officially lists only a(0..6)=1,2,3,5,7,12,15, so these are NOT OEIS terms to 'beat'; NOVELTY vs best-known integer-sum constructions for n>=7 is NOT established (the workflow's reference numbers are of unverified provenance; the F_2^n XOR-Sidon literature table is a DIFFERENT, smaller-valued problem and not comparable). Verified valid lower bounds, NOT claimed records/new terms.","grade":"extends_prior_work","gateStatus":"verified","superseded":true}],"velaLean":[],"oeis":[{"id":"A003022","name":"Length of shortest (or optimal) Golomb ruler with n marks.","terms":"1,3,6,11,17,25,34,44,55,72,85,106,127,151,177,199,216,246,283,333,356,372,425,480,492,553,585","url":"https://oeis.org/A003022"},{"id":"A143824","name":"Size of the largest subset {x(1),x(2),...,x(k)} of {1,2,...,n} with the property that all differences |x(i)-x(j)| are distinct.","terms":"0,1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10","url":"https://oeis.org/A143824"},{"id":"A227590","name":"a(n) = A003022(n)+1 with a(1)=1.","terms":"1,2,4,7,12,18,26,35,45,56,73,86,107,128,152,178,200,217,247,284,334,357,373,426,481,493,554,586","url":"https://oeis.org/A227590"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}