{"schema":"vela.problem-packet.v0.1","problem":993,"statement":"The independent set sequence of any tree or forest is unimodal.In other words, if $i_k(G)$ counts the number of independent sets of vertices of size $k$ in a graph $G$, and $T$ is any tree or forest, then for some $m\\geq 0$$$i_{0}(T)\\leq i_{1}(T)\\leq\\cdots\\leq i_{m}(T)\\geq i_{m+1}(T)\\geq i_{m+2}(T)\\geq\\cdots.$$","status":"open","seam":"sealed","closureRoutes":[{"type":"verified_computation","verifierKind":"python:verify_993_result.py","note":"independence-polynomial unimodality aggregation (pinned tier: scripts/requirements-verifiers.txt)"},{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"}],"obligations":[{"findingId":"vf_7eba52ea8a9bc742","banked":"the banked unimodality result for the tree/forest case verified (frozen kernel)","open":"the general Alavi-Erdos unimodality conjecture is open.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_3710d5b389de35ed","kind":"dead_end","claim":"Erdos #993 (unimodality of independence polynomials of trees/forests, Alavi-Erdos-Malde-Schwenk 1987): a broadened structured search finds NO non-unimodal tree or forest. 112,916 generalized 'bush' trees (orders 26-60; 4,445 distinct non-log-concave, log-concavity defect down to -12.2) are all unimodal, and 253,695 forest objects (products/powers up to 20/path-smoothings over the 80 most-severe non-log-concave seeds) are all unimodal. The order-26 non-log-concave trees T_3,4,4 and T*_3,3,4 are reproduced exactly. This extends the verified-unimodal frontier into the forest/product surface that prior tree-only searches (incl. the 2025 PatternBoost sweep to 101 vertices) left untouched. By Hoggar's theorem a forest counterexample requires a non-log-concave component, so the non-log-concave families are the only place one can live.","grade":"extends_prior_work","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}