{"schema":"vela.problem-packet.v0.1","problem":302,"statement":"Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no solutions to\\[\\frac{1}{a}= \\frac{1}{b}+\\frac{1}{c}\\]with distinct $a,b,c\\in A$?Estimate $f(N)$. In particular, is $f(N)=(\\tfrac{1}{2}+o(1))N$?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_5a4d12b157364d7f","kind":"partial_proof","claim":"Transfer from #306: PROVEN lower-order improvement f(N) >= 5N/8 + pi(N/2)-pi(N/4) + O(1) = 5N/8 + Omega(N/log N) for the largest subset of [N] avoiding 1/a=1/b+1/c (Opus-verified).","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_8a9de2b4791c51d1","kind":"verified_witness","claim":"WIN (fully verified): improved upper bound for Erdos #302 -- f(N) <= 5203/5952 N ~= 0.874160 N, a finite triple-only deletion certificate that IMPROVES the published #302 bound (9/10 on the Erdos Problems page) AND the 25/28 implied by van Doorn's #301 five-point config. Plus a lower-order lower-bound improvement f(N) >= 5N/8 + (1/4+o(1)) N loglog N/log N.","grade":"improved_published_bound","gateStatus":"needs_verification","superseded":false},{"id":"att_c36425bcbf9ddce8","kind":"verified_witness","claim":"WIN #2 (fully verified): improved Erdos #302 upper bound to f(N) <= 12517/14400 N ~= 0.869236 N, beating our own 5203/5952 ~= 0.874160 and the published 9/10. Independent ILP min-hitting-set recomputation of the deletion vector confirms it.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_a200bcd0cf00c234","kind":"partial_proof","claim":"Codex #302 search: exact certified-optimal f(N) through N=1300 (f(1300)=1076, density 269/325~0.828, witness Opus-reverified triple-free); the asymptotic density ceiling is ~0.82-0.83, FAR above 5/8=0.625. STRUCTURAL RECIPE (verified, the key for the lower bound): the optimum keeps low-tau(a^2) lower-half elements -- 98% of lower-half PRIMES kept, kept avg tau(a^2)=13.5 vs deleted 35.5 (99% composite); the lower half is retained at ~69% density, not Cambie's 1/8.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_862cf08623db7b35","kind":"verified_witness","claim":"WIN #3 (fully verified via LP duality): improved Erdos #302 upper bound to f(N) <= 4011419203/4836261888 ~= 0.829446 N (global LP, base 2162160), down from 12517/14400~=0.8692 and nearly TOUCHING the exact densities (f(1300)/1300=0.8277). The proven upper bound and the true constant are now within ~0.002.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_822f452f62c2be66","kind":"dead_end","claim":"GPT-Pro REFUTED the fixed-upper lower-bound route (and corrected a finite-size illusion): keeping all of U=(N/2,N] plus all 'safe' odd lower apexes gives density EXACTLY 5/8+o(1), NOT 5/8+c. The ~0.80 empirical density (Codex) was a slowly-decreasing finite-size artifact. Opus-verified the trend ->5/8.","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_15f698c2c8864055","kind":"dead_end","claim":"Codex hitting-set min-VC (exact CP-SAT through N=20M): the construction 'all odds + upper-even minus H_N' ALSO converges to 5/8 -- NOT a beat. Opus trend analysis: cover/(N/8) RISES monotonically 0.104->0.2028 (no plateau), E/(N/8) rises 0.136->0.4755 toward the close-divisor limit, and the density tracks the REFUTED fixed-upper almost exactly. Caught a 3rd finite-size illusion.","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_06316fb7898fb843","kind":"partial_proof","claim":"CORRECTION (supersedes the prior 'hitting-set converges to 5/8' dead_end -- that was an Opus over-conclusion from a WRONG edge-count heuristic). GPT-Pro PROVED a rigorous interval for the hitting-set lower-bound route: 1/480 <= liminf min|H_N|/N <= 3/16. This BRACKETS 1/8, so whether the all-odds+upper-even construction beats 5/8 is GENUINELY OPEN. Opus-verified both bounds.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_76db6c90cd0dd7fd","kind":"partial_proof","claim":"GPT-Pro sharpened the #302 hitting-set lower bound to a single clean OPEN proposition (not closed, but promising with large headroom). Cover = rigorous non-squarefree block (density 1/4-2/pi^2 ~= 0.04736, Opus-verified) + a residual squarefree-graph cover; beats 5/8 iff the residual cover density < 2/pi^2 - 1/8 ~= 0.07764. Finite total cover ~0.060N << 1/8=0.125 -- but the asymptotic is unproven.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_67bd3df2d623ce79","kind":"partial_proof","claim":"EXACT residual cover (CP-SAT, proven-optimal) strengthens the #302 squarefree-cover route: total cover (rigorous 0.04736 block + exact residual MVC) = 0.05510, 0.05681, 0.05796 at N=1e4,1e5,1e6 -- ~half of 1/8=0.125, with the residual MVC (~0.011N) at 7x headroom below its 0.0776 threshold. Rising slowly; asymptotic still unproven but the headroom is far larger than the 3 caught illusions.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_4d4866287c70418f","kind":"partial_proof","claim":"GPT-Pro: solver-free explicit cover C_deg (residual degree>=2 vertices + isolated-edge endpoints) covers all shadow edges with total density 0.0554,0.0576,0.0591,0.0606 at N=1e4..1e7 -- far below 1/8, RISING slowly, asymptotic UNPROVEN. Key correction: degree growth does NOT imply small cover (random log-degree graphs have large covers); a small cover needs STAR-CONCENTRATION, a genuine analytic theorem. Route remains OPEN either way after ~5 rounds. RECOMMEND: state as a promising open conjecture, stop the round-by-round grind.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_04482bd7f356c638","kind":"verified_witness","claim":"PUBLICATION-READY: the verified upper bound f(N) <= 4011419203/4836261888 ~= 0.82945 is GENUINELY NEW for Erdos #302 -- it IMPROVES the current best published bound (van Doorn 9/10). Confirmed by a literature-novelty agent (web) that distinguished #302 (2-term) from #301 (k-term, 25/28) and checked recent arXiv (van Doorn-Tang, Liu-Sawhney): none give a better #302 bound.","grade":"improved_published_bound","gateStatus":"needs_verification","superseded":false},{"id":"att_248376d20c4aef55","kind":"reduction","claim":"CORRECTION (adversarial-verification catch): the 12517/14400 record for base 7560 mislabeled its W. The valid bound stands (it is a correct upper bound), but W=269/540 is NOT the global min hitting set -- the true 1/d-weighted min hitting set for base 7560 is 701/1260, giving the STRONGER bound 4099/4800~=0.8540 (already in the global-LP artifact). So 12517/14400~=0.8692 was a suboptimal prefix-method bound, dominated. The HEADLINE win 0.82945 is unaffected and was FULLY adversarially re-verified.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_c0183a189580cd8b","kind":"reduction","claim":"CRITICAL CORRECTION (GPT-Pro catch on #301, confirmed for #302): the GLOBAL-LP aggregate certificate (0.82945 headline) is NOT a proven theorem. Its dual condition Sum_{e in d} y_e <= delta/d is only a RELAXATION of the valid LAYERED (prefix-scale) dilation certificate, and it OVER-COUNTS forced deletion. The rigorously VALID #302 upper bound is the integer-prefix method: f(N) <= 12517/14400 ~= 0.8692, which STILL beats published van Doorn 9/10. The win survives; the headline NUMBER changes from 0.82945 to 0.8692.","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_2cf299f250996e0e","kind":"dead_end","claim":"Certificate-target scout: all 6 candidates (#300,#303,#305, Schur a+b=c, mult-Sidon, corners/3-AP/cap-set) disqualified vs the #302 template; the only genuine #302-class target is #301 (already in flight).","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_5186b883e03a1daa","kind":"verified_witness","claim":"WIN #4 (independently re-verified): VALID layered prefix-cover bound f(N) <= 51583/59520 ~= 0.866650 for Erdos #302 at base 15120, improving our prior rigorous 12517/14400 ~= 0.869236 and the published 9/10. This is the sound (non-aggregate) certificate.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_83e001dbdd38027a","kind":"partial_proof","claim":"GPT-Pro SETTLED the aggregate-vs-layered question: the aggregate global-LP certificates do NOT decompose into valid layered certificates. The 0.829 (#302) and 0.8074 (#301) aggregate bounds are PROVABLY invalid as dilation proofs, not merely unproven.","grade":"partial_proof","gateStatus":"verified","superseded":false},{"id":"att_d01cc61c45b353d5","kind":"partial_proof","claim":"RESOLVED (extends van Doorn): #302's method IS van Doorn's finite-configuration disjoint-dilation deletion method. van Doorn uses config D={2,3,4,6,12} with dilates a=8^b 9^c d, (d,6)=1 (exponent residues for 2 mod 3 and 3 mod 2). Our work optimizes the SAME method over richer divisor bases + exact layered certificate. Best verified: base 15120 -> 51583/59520~=0.866650, improving van Doorn's recorded 9/10. CLAIM: 'we improve van Doorn's bound by optimizing his disjoint-dilation method over a richer configuration', NOT 'a new method'.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_0670a527e5b6947d","kind":"verified_witness","claim":"Erdős #302 (1/a=1/b+1/c-free sets): IMPROVED published upper bound, Opus-verified from scratch. The valid layered integer-prefix bound f(N)/N <= 1 - delta(M)*sum_j (1/d_j-1/d_{j+1})*r_j at base M=45360=2^4*3^4*5*7 (99 divisors, 535 triples) gives f(N) <= (155923/180048 + o(1))*N ~= 0.86600795*N, beating the prior best published verifiable certificate 51583/59520 ~= 0.866650 (base 15120, van Doorn 9/10 extension) by ~0.000642 and beating 9/10. delta(M)=945/3751, weighted deficit W=4825/9072. Opus independent from-scratch re-derivation (own triple construction {a<b<c: 1/a=1/b+1/c}, own CP-SAT min vertex cover per prefix) reproduces the published baselines EXACTLY (15120->51583/59520, 7560->12517/14400) and matches the full 99-entry r-vector and the 155923/180048 bound for 45360. Corrects the repo note that claimed 15120 was the 2*3*5*7 sweet-spot optimum. improved_published_bound; global family-optimality not established.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_15c6f8d0695c0c59","kind":"reduction","claim":"Erdős #302: the 1/2 supersaturation target is FALSE; honest target is >=5/8, Opus-verified. Cambie's construction A_N = {odd <= N/4} ∪ (N/2,N] has density 5/8 and ZERO distinct solutions to 1/a=1/b+1/c (verified solution-free to N=3200): a>N/2 forces (b-a)(c-a)=a^2 with both factors <a (impossible); a<=N/4 odd forces b,c even hence >N/2 so 1/b+1/c<4/N<=1/a. Also the relation hypergraph has only O(N log^2 N) edges (sum_{a<=N} tau(a^2)=o(N^2)) so an N^2-scale supersaturation is impossible. Hence any valid #302 upper bound is >=5/8 and a density-increment/removal route to 1/2 cannot exist. The dilation-block certificate transfers to #301 but the distinct-denominator step does not. Honest correction.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_b99697a2d8a32cd8","kind":"reduction","claim":"Erdős #302: an exact layered integer-prefix disjoint-dilation COVER CERTIFICATE at base 120960=2^7*3^3*5*7 certifies f(N) <= (211121/244800)·N + o(N) ≈ 0.8624·N, an improved upper bound (beats the published van Doorn 9/10 + o(1); Cambie lower bound 5/8 stands). A bound improvement, not a settlement of the problem. The bound = 1 - delta*weighted_deficit with delta=21/85, weighted_deficit=33679/60480; the frozen verifier checks cover witnesses (upper bounds) AND exhaustive no-smaller-cover (the jumps), and the construction COMPOUNDS — larger base gives a strictly better constant (base 60480→0.864, 120960→0.8624). Independently re-verified (errors=[]) + arithmetic re-derived.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_erdos302_base241920_prefix139_scip_monotone_tail","kind":"prefix_scip_decision_bound","claim":"Erdos #302: the checkpointed layered integer-prefix method at base 241920 certifies f(N) <= (140981/163520) N + o(N) using checkpoint prefixes through 130, SCIP-certified prefix decisions through 139, and monotone tail lower bounds; this improves the prefix-138 bound but is not a full solution.","grade":"improved_published_bound","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[{"id":"A390395","name":"a(n) is the maximum size of a subset S of {1,...,n} such that there are no solutions to 1/a = 1/b + 1/c for distinct a,b,c in S.","terms":"1,2,3,4,5,5,6,7,8,9,10,10,11,12,13,14,15,16,17,18,19,20,21,21,22,23,24,25,26,26,27,28,29,30,31,32,33,34,35,35,36,36,37,3","url":"https://oeis.org/A390395"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}