{"schema":"vela.problem-packet.v0.1","problem":301,"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_1}+\\cdots+\\frac{1}{b_k}\\]with distinct $a,b_1,\\ldots,b_k\\in A$?Estimate $f(N)$. In particular, is it true that $f(N)=(\\tfrac{1}{2}+o(1))N$?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_9b74c027f3570ca4","kind":"dead_end","claim":"Codex #301 global-LP bound 124691/154440~=0.8074 (claimed to beat the (FABRICATED, now-retracted) 'Wang 667/806' baseline) is NOT PROVEN: it uses the SAME aggregate dual format (capacity delta(M)/d) that GPT just invalidated for #302 -- necessary not sufficient for the layered dilation argument. Codex ran it in parallel, pre-correction. Whether the VALID (layered integer-prefix) #301 bound beats Wang is OPEN (heavy computation, timed out).","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_3117fc4e6e2c5007","kind":"partial_proof","claim":"#301 (HONEST, downgraded): our layered bound is Wang's OWN method (Xinjun Wang, preprint 2026-05-27, = van Doorn's finite block-dilation refined to Div(720)\\{1}) run at richer bases. Wang proves 667/806 at base 720; we execute the base-search Wang explicitly poses as future work and get 1801/2184~=0.824634 (base 1260), a ~0.003 improvement. NOT a new method, NOT independent -- cite Wang.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_39dc99caab0ce895","kind":"dead_end","claim":"RE-CORRECTION (supersedes the earlier 'fabrication' claim): Wang's 667/806 for #301 is REAL -- a recent preprint by Xinjun Wang, 'A 667/806 Upper Bound for Erdos Problem #301 on Unit-Fraction-Free Sets' (ResearchGate, 2026, not peer-reviewed, not yet on the official page). My literature agent's 'fabricated' verdict was a FALSE NEGATIVE.","grade":"honest_null","gateStatus":"verified","superseded":false},{"id":"att_782f4c4b2305fd63","kind":"dead_end","claim":"LESSON (agent literature blind spot): TWO independent Opus agents searched and both declared 'Wang 667/806 fabricated' -- both MISSED Xinjun Wang's real ResearchGate preprint. Agent negative literature findings are unreliable for non-indexed venues (ResearchGate, personal pages).","grade":"honest_null","gateStatus":"verified","superseded":false},{"id":"att_911dd113081f3e74","kind":"verified_witness","claim":"Erdős #301 (mixed-arity 1/a=sum 1/b_i identities): valid layered upper bounds at richer bases, PARTIALLY verified (NOT yet gate-passed). Wang's disjoint-dilation layered method f(N)/N<=1-delta(M)*sum_j(1/d_j-1/d_{j+1})*cover(D_j) extended to 2^a*3^b*5*7 bases gives base 840 (2^3*3*5*7)->793/960~=0.826042 and base 1260 (2^2*3^2*5*7)->901/1092~=0.825092, both beating Wang's published 667/806~=0.827543 (base 720). Opus INDEPENDENTLY verified: delta(M) values (120/403, 7/24, 15/52), divisor lists, and the bound ARITHMETIC recompute from the stored per-prefix covers all match exactly, and base 720 reproduces Wang's 667/806 exactly. NOT YET verified by Opus: the mixed-arity minimum-hitting-set covers themselves (the upfront-enumeration re-derivation does not scale past the base-720 apex=2 identity explosion; an independent lazy-cut reimplementation is still pending). Also corrected: the prior 1801/2184 is the aggregate global-LP relaxation, NOT a valid layered certificate.","grade":"improved_published_bound","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[{"id":"A390394","name":"a(n) is the maximum size of a subset A of {1,...,n} such that there are no solutions to 1/a = 1/b_1 + ... + 1/b_k with distinct a, b_1, ..., b_k in A.","terms":"1,2,3,4,5,5,6,7,8,9,10,10,11,12,12,13,14,15,16,16,17,18,19,19,20,21,22,22,23,23,24,25,26,27,27,27,28,29","url":"https://oeis.org/A390394"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}