{"schema":"vela.problem-packet.v0.1","problem":319,"statement":"What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there is a function $\\delta:A\\to \\{-1,1\\}$ such that\\[\\sum_{n\\in A}\\frac{\\delta_n}{n}=0\\]and\\[\\sum_{n\\in A'}\\frac{\\delta_n}{n}\\neq 0\\]for all non-empty $A'\\subsetneq A$?","status":"open","seam":"sealed","closureRoutes":[{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"verified_computation","verifierKind":"python:verify_319_smoothness_counting.py","note":"N-smoothness upper-bound counting corroboration (pinned tier: scripts/requirements-verifiers.txt)"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"}],"obligations":[{"findingId":"vf_a9a5b1e0dbdc9da4","banked":"the counting step of the upper bound corroborated numerically","open":"the full upper bound is not established by the numerics alone.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_14546a6d4e00bb7c","kind":"partial_proof","claim":"The asymptotic-order variants of Erdős #319 (isTheta/isBigO/isLittleO) are settled: c(N)=Θ(N) with gauge g(N)=N, immediately from the trivial bound c(N)≤N plus the already-accepted Croot/Adenwalla low","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_erdos319_mitm_exact_table","kind":"vela.erdos_problem_attempt_record_draft.v1","claim":"Erdos #319: a meet-in-the-middle exhaustive enumeration verifies the exact finite table through N=27: c(N)=0 for 1<=N<=5, 4 for 6<=N<=11, 6 for 12<=N<=14, 8 for 15<=N<=17, 10 for 18<=N<=20, 12 for 21<=N<=23, and 14 for 24<=N<=27. This finite table does not address the official all-N extremal question.","grade":"partial_proof","gateStatus":"verified","superseded":false},{"id":"att_6dab79c192e47495","kind":"partial_proof","claim":"Erdos #319: lean/Vela/Erdos319Smoothness.lean kernel-verifies the smoothness mechanism behind the upper bound. For a signed-zero set A and a prime p, if the isolation bound |T| < p holds (T the cleared A_p numerator), then the p-divisible part A_p is itself signed-zero; hence by irreducibility A_p = A, so every element of an extremal irreducible set is (N/m)-smooth with m = floor((1/2) ln N). Combined with the prime-number-theorem count of non-(N/m)-smooth integers (numerically corroborated) this gives c(N) <= N - (1+o(1)) N loglog N / log N. With the known (1-1/e)N lower bound (Croot 2001 / Martin 2000 / Adenwalla) this yields Theta(c(N)) = Theta(N); the sharp leading constant is not determined here.","grade":"partial_proof","gateStatus":"verified","superseded":false},{"id":"att_92ad080d34a481c5","kind":"partial_proof","claim":"Erdos #319: every irreducible signed-zero set A subset {1..N} has bounded harmonic mass, sum_{a in A} 1/a <= 2 (log N)^{4/5+o(1)}. Proof: if a side's reciprocal mass exceeds the Liu-Sawhney (IMRN 2026) threshold (log N)^{4/5+o(1)}, that side contains a subcollection of reciprocal sum 1; if the common side-sum S>1 the two unit-subcollections form a nonempty proper signed-zero subset, contradicting irreducibility. This is a proven structural property; it does not determine the sharp constant of c(N) (it is too weak to push the cardinality bound below N - o(N)).","grade":"partial_proof","gateStatus":"verified","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos319Smoothness.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos319Smoothness.lean"},{"file":"lean/Vela/Erdos319.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos319.lean"}],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}