{"data":{"id":"lf-general-closed-form-for-the-network-count-difference-k","statement_oneline":"For every k≥2, Δ_k(ℓ) has the paper’s polynomial-times-double-factorial minus polynomial-times-factorial form, with the stated degree and integrality properties.","statement_full_latex":"For every k≥2, Δ_k(ℓ) has the paper’s polynomial-times-double-factorial minus polynomial-times-factorial form, with the stated degree and integrality properties. Open residue: k=2 and 3 proved; k=4 is the first new case.","source":{"title":"Exact Enumeration of Phylogenetic Networks: The Tree-Child, Reticulation-Visible and Orchard Hierarchy","arxiv":"2606.24325v2","url":"https://arxiv.org/abs/2606.24325","year":2026,"area":"Phylogenetic combinatorics"},"renown":{"score":1,"rationale":"Recent research-paper conjecture; renown scored at catalog level per Rubric v1.","scored_by":"rubric-v1"},"attackability":{"score":5,"rationale":"The formula is proved for k=2,3 and predicts k=4; exact enumeration at fixed k supplies enough values to fit and falsify the polynomial pattern.","scored_by":"rubric-v1","subscores":{"finite_witness":4,"oracle_cost":4,"freshness":5,"seedability":5}},"verification_tier":null,"ai_systems":[],"posed_by":null,"years_open":0,"notability":null,"source_trackers":[],"status":"open","ranking_score":5,"last_verified_open":"2026-07-27","added":"2026-07-27","resolution_type":null,"resolver":null,"resolution_date":null,"evidence_url":null,"evidence_note":null,"method_note":null,"submitted_by":"Luke Francis","verification_note":"Enumerate the network classes exactly, interpolate A_k and B_k over Q, then verify additional ℓ values. — Current arXiv revision checked; source-stated open/partial. Independent literature and author confirmation still required. Editorially screened candidate; not independently certified open. Author confirmation pending.","openconjecture_id":"4327","effective_status":"open","claims":[],"status_events":[{"id":"seed-lf-general-closed-form-for-the-network-count-difference-k","conjecture_id":"lf-general-closed-form-for-the-network-count-difference-k","status":"open","date":"2026-07-27T00:00:00.000Z","actor":"human-admin","rationale":"Imported Tier 1 contributor candidate from Luke Francis.","note":"Editorially screened candidate; author confirmation pending."}],"public_actions":[{"name":"claim","method":"POST","href":"/c/lf-general-closed-form-for-the-network-count-difference-k/claim","verification":"email magic link"},{"name":"report_resolution","method":"POST","href":"/c/lf-general-closed-form-for-the-network-count-difference-k/resolve","verification":"admin review"}],"resolutions":[]},"notice":"Status is community- and machine-tracked and may lag. Verify independently before investing effort."}