{ "schemaVersion": "1.6", "slug": "lotka-volterra-tree-presentations", "title": "Linear coordinate presentations of Lotka–Volterra tree systems and a proof of van der Kamp's tree conjecture", "shortTitle": "Two trees, never the same equation", "url": "https://evidencepress.org/releases/lotka-volterra-tree-presentations/", "oneLine": "A proof of van der Kamp's conjecture that Lotka–Volterra systems built on different trees are never the same equation in disguise: for generic parameters, and with genericity needed on one side only. The proof rests on an exact list of a tree system's invariant planes, which also yields every coordinate system in which the equation keeps its form, closed counts for paths and caterpillars, and a test for recognising a tree system.", "abstract": "A Lotka–Volterra system is a quadratic differential equation used for interacting populations, chemical reactions and integrable systems. A tree system, introduced by Quispel, Tapley, McLaren and van der Kamp, has 3n − 2 parameters and one extra invariant hyperplane (a linear Darboux polynomial) for each edge of a tree on n vertices. After a linear change of variables the same equation can show a hypergraph instead of a tree, and van der Kamp (J. Dyn. Diff. Equat. 2026; arXiv:2411.18264, Conjecture 12) conjectured that nonisomorphic trees are never related in this way, for generic parameters. This paper proves the conjecture in a stronger form: a generic system of one tree family is never linearly conjugate to any system, generic or not, of the family of a nonisomorphic tree. An extension of the statement to all parameter values fails, because two tree families can share a hyperplane of special systems. The key invariant is the complete set of linear Darboux polynomials. For a tree system it consists of exactly 2n − 1 rays, forming the graphic arrangement of the cone over the tree, precisely when an explicit polynomial in the parameters is nonzero. From this the paper derives every linear coordinate system in which the equation keeps Lotka–Volterra form (one for each spanning tree of the cone), the symmetry group that identifies them, a reduction to rooted forests on the internal vertices of the tree, closed counts for paths and caterpillars, a description of the normalised family as the quadratic vector fields tangent to the arrangement, a recognition test, and transfer to Kahan discretisations. Exact computations check every statement for small trees and reproduce van der Kamp's counts for n ≤ 8. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-22", "dateModified": "2026-10-22", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23564794", "doiUrl": "https://doi.org/10.5281/zenodo.23564794", "conceptDoi": "10.5281/zenodo.23564793", "pdfUrl": "https://github.com/ipitchford/lotka-volterra-tree-presentations/releases/download/v1.0.0-candidate/lotka-volterra-tree-presentations-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23564794", "repoUrl": "https://github.com/ipitchford/lotka-volterra-tree-presentations", "releaseUrl": "https://github.com/ipitchford/lotka-volterra-tree-presentations/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/lotka-volterra-tree-presentations/index.md", "bibtexUrl": "https://evidencepress.org/releases/lotka-volterra-tree-presentations/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/lotka-volterra-tree-presentations.mp3", "imageUrl": "https://evidencepress.org/assets/og/lotka-volterra-tree-presentations.png?v=53eff000a0", "coverArtUrl": "https://evidencepress.org/assets/art/lotka-volterra-tree-presentations.svg?v=6646cc3b93", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/lotka-volterra-tree-presentations.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/lotka-volterra-tree-presentations.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Every theorem has a written proof; exact computations and one exploratory example are labelled. Internal AI-assisted review, AI-assisted adversarial review and a verification re-review only. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23564794", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "verify_all.py regenerates every saved output byte for byte. A clean full replay of the final package in a separate copy reproduced every output; the frozen archive was bound to it by hash, and the checks affected by publication changes were rerun." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "No independent reimplementation; the core-forest count is an algorithmically distinct enumeration written in the same project, not an independent replication." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Exact checks need Python 3 and SymPy only (tested with Python 3.13; versions listed, not locked in a container). No external data are used." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, verification re-review and adversarial review (including OpenAI gpt-5.6-sol)", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the approved final manuscript and its exact check outputs. Versioned GitHub and Zenodo research archives are checked against the frozen release assets on 22 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Lotka–Volterra systems", "Darboux polynomials", "superintegrable systems", "hyperplane arrangements", "graphic matroids", "spanning trees", "linear conjugacy", "Kahan discretisation" ], "keyResults": [ "Complete invariant-plane list (Theorem 2.3, proof): a tree system has exactly 2n − 1 homogeneous linear Darboux rays, the n coordinates and one per edge, if and only if an explicit polynomial Δ_T in its parameters is nonzero; Proposition 2.5 describes what changes where exactly one factor of Δ_T vanishes.", "Van der Kamp's Conjecture 12 proved, in a one-sided form (Theorem 3.6, Corollary 3.7, proof): a generic system of one tree family is never linearly conjugate to any system, generic or special, of the family of a nonisomorphic tree. An extension to all parameter values fails, because two tree families can share a hyperplane of special systems (Proposition 3.8, proof).", "Every Lotka–Volterra coordinate system (Theorem 4.2, proof): the linear changes of variables that keep Lotka–Volterra form correspond to spanning trees of the cone over the tree, det(I + L) of them, and their support hypergraphs are read off as paths in the chosen spanning tree.", "Counting distinct presentations (Theorems 5.2, 6.3, 7.1, 7.2, proof): isomorphic support pictures are orbits of an explicit symmetry group; the count reduces to rooted forests on the internal vertices; paths give (F_{2n−2} + F_{n−1})/2 and caterpillars a closed formula. Exact computation reproduces van der Kamp's Table 7 for n ≤ 8 and extends it to n = 10.", "Consequences (proof unless stated): the normalised family is exactly the space of quadratic vector fields tangent to the cone arrangement (dimension 2n − 1); a recognition test decides whether a given quadratic system is a disguised generic tree system; all 2n − 1 planes survive Kahan's discretisation; only the original presentation preserves positivity. Applying the test to van der Kamp's 13-parameter system at five sample points is exploratory." ], "reviews": [], "evidencePackage": "Written proofs of every theorem for all trees; exact-arithmetic checks of every statement for small trees (all 94 trees with n ≤ 9 by cone bases, all 200 trees with n ≤ 10 by a separate core-forest enumeration, direct hypergraph canonicalisation for n ≤ 6); every saved output regenerated byte for byte by verify_all.py.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "van der Kamp (2026), Hypergraphs and Lotka–Volterra systems with linear Darboux polynomials. Journal of Dynamics and Differential Equations 38(3), 1331–1352 (arXiv:2411.18264v4; source of Conjecture 12).", "url": "https://doi.org/10.1007/s10884-025-10446-2", "doi": "10.1007/s10884-025-10446-2" }, { "citation": "Quispel, Tapley, McLaren and van der Kamp (2023), Linear Darboux polynomials for Lotka–Volterra systems, trees and superintegrable families. Journal of Physics A 56, 315201.", "url": "https://doi.org/10.1088/1751-8121/ace0e9", "doi": "10.1088/1751-8121/ace0e9" }, { "citation": "van der Kamp, Quispel and McLaren (2024), Trees and superintegrable Lotka–Volterra families. Mathematical Physics, Analysis and Geometry 27, 25.", "url": "https://doi.org/10.1007/s11040-024-09496-7", "doi": "10.1007/s11040-024-09496-7" }, { "citation": "van der Kamp, McLachlan, McLaren and Quispel (2024), Measure preservation and integrals for Lotka–Volterra tree-systems and their Kahan discretisation. Journal of Computational Dynamics 11(4), 468–484.", "url": "https://doi.org/10.3934/jcd.2024011", "doi": "10.3934/jcd.2024011" }, { "citation": "Celledoni, Evripidou, McLaren, Owren, Quispel, Tapley and van der Kamp (2019), Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps. Journal of Physics A 52, 31LT01.", "url": "https://doi.org/10.1088/1751-8121/ab294b", "doi": "10.1088/1751-8121/ab294b" }, { "citation": "Proudfoot and Ramos (2019), Functorial invariants of trees and their cones. Selecta Mathematica 25, 62.", "url": "https://doi.org/10.1007/s00029-019-0509-4", "doi": "10.1007/s00029-019-0509-4" }, { "citation": "Reiner and Smith (2024), Sandpile groups for cones over trees. Research in the Mathematical Sciences 11, 59.", "url": "https://doi.org/10.1007/s40687-024-00471-w", "doi": "10.1007/s40687-024-00471-w" }, { "citation": "Evripidou, Kassotakis and Vanhaecke (2022), Morphisms and automorphisms of skew-symmetric Lotka–Volterra systems. Journal of Physics A 55, 325201.", "url": "https://doi.org/10.1088/1751-8121/ac7e90", "doi": "10.1088/1751-8121/ac7e90" }, { "citation": "Whitney (1933), 2-isomorphic graphs. American Journal of Mathematics 55, 245–254.", "url": "https://doi.org/10.2307/2371127", "doi": "10.2307/2371127" }, { "citation": "OEIS Foundation, sequence A051450 (the path counts, with index shifted).", "url": "https://oeis.org/A051450", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:lotka-volterra-tree-presentations", "attemptIds": [ "ep-attempt:lotka-volterra-tree-presentations-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "A proof of van der Kamp's Conjecture 12 (nonisomorphic trees are not LV-equivalent) for generic parameters in a one-sided form, with the complete linear Darboux list and its exact genericity condition, the classification of all Lotka–Volterra coordinate presentations, their symmetry quotient and counts, a recognition test and consequences for Kahan maps and positivity.", "scope": "Homogeneous (and constant-growth) Lotka–Volterra tree systems over the reals or complexes under invertible constant linear changes of variables; genericity required on at least one side; points where two or more factors of the genericity polynomial vanish, quasi-monomial equivalence and completeness for Kahan maps are not classified." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Van der Kamp's definition of LV-equivalence, his generic-parameter convention, his computational procedure and Conjecture 12 are quoted from arXiv:2411.18264v4 with locations; three readings of the conjecture are distinguished and the generic (and one-sided) readings are proved.", "remainingRisks": [ "The journal version was compared through its arXiv v4 numbering.", "Identification of the counts with the source's equivalence classes uses the source's stated procedure; that the general class attached to each hypergraph consists only of tree presentations is open for general n.", "The bounded literature check is not exhaustive priority certification." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "independentRerun", "specialistReview", "formalVerification", "priorityAssessment" ], "nextAction": "Have a specialist in integrable Lotka–Volterra systems check the complete Darboux list (Theorem 2.3) and the one-sided rigidity argument (Lemma 3.5 and Theorem 3.6).", "claimCeiling": "Proof of van der Kamp's tree conjecture for generic parameters in one-sided form, with the stated classification and counting results. 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