{
  "schemaVersion": "1.6",
  "slug": "negative-square-energy-five-vertex-valley",
  "title": "A Minimal Strict Valley in Negative Square Energy under Edge Addition",
  "shortTitle": "Minimal negative-square-energy valley",
  "url": "https://evidencepress.org/releases/negative-square-energy-five-vertex-valley/",
  "oneLine": "A five-vertex graph gives the smallest exact decrease-then-increase in negative adjacency square energy, refuting universal prescribed-order peak-unimodality while leaving favourable-order existence open.",
  "abstract": "For a graph G, let S^-(G) be the sum of the squares of its negative adjacency eigenvalues. This anonymous, unrefereed candidate gives a connected five-vertex graph and two ordered missing edges for which S^- strictly decreases and then strictly increases, with exact endpoint energy 5 and middle energy below 3697/784. A structural proof excludes all connected graphs on at most four vertices, while an exhaustive exact census finds 840 labelled five-vertex valleys in nine simultaneous-relabeling orbits. The same starting tree also has an exactly certified favourable full ordering. The result therefore refutes only universal or arbitrary prescribed-order peak-unimodality; it does not settle either favourable-order existential formulation. Exact producer replay, four mutation controls, internal review, and a fully actioned user-supplied review pass at PASS_WITH_NOTES. No historical-priority, unaffiliated-reproduction, formal-verification, specialist-review, or peer-review claim is made.",
  "datePublished": "2026-08-30",
  "dateModified": "2026-08-30",
  "version": "0.2.0-candidate",
  "doi": "10.5281/zenodo.22169169",
  "doiUrl": "https://doi.org/10.5281/zenodo.22169169",
  "conceptDoi": "10.5281/zenodo.22169168",
  "pdfUrl": "https://github.com/ipitchford/negative-square-energy-five-vertex-valley/releases/download/v0.2.0-candidate/negative-square-energy-five-vertex-valley-v0.2.0-candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/22169169/files/negative-square-energy-five-vertex-valley-v0.2.0-candidate.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22169169",
  "repoUrl": "https://github.com/ipitchford/negative-square-energy-five-vertex-valley",
  "releaseUrl": "https://github.com/ipitchford/negative-square-energy-five-vertex-valley/releases/tag/v0.2.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/negative-square-energy-five-vertex-valley/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/negative-square-energy-five-vertex-valley/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/negative-square-energy-five-vertex-valley.mp3?v=72338f36e7",
  "imageUrl": "https://evidencepress.org/assets/og/negative-square-energy-five-vertex-valley.png?v=0a6a5a322a",
  "coverArtUrl": "https://evidencepress.org/assets/art/negative-square-energy-five-vertex-valley.svg?v=36c7ffb72b",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/negative-square-energy-five-vertex-valley.mp3",
      "name": "Audio briefing — minimal negative-square-energy valley",
      "description": "Plain-English OpenAI API synthetic-voice account of the exact valley, vertex-minimality, five-vertex census, favourable-order boundary, and assurance limits; a communication aid, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/negative-square-energy-five-vertex-valley.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed spectral-graph-theory candidate at internal PASS_WITH_NOTES. The witness and four-vertex minimality have short hand proofs; the 840-count and nine-orbit classification rely on exhaustive producer-side exact computation. A supplied review reports a separate Linux replay and numerical reimplementation, but reviewer identity and supporting artifacts were not authenticated, so independent assurance remains not assessed. The source order quantifier is unstated: the release refutes the universal or arbitrary prescribed-order formulation only and leaves both favourable-order existential formulations open."
  },
  "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/22169169",
      "note": "The tagged GitHub prerelease and published Zenodo record expose the canonical PDF, deterministic archive, checksum sidecar, claims, source map, review response, exact verifier, and replay files; independently downloaded public assets match local bytes."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Normal and optimized exact censuses reproduce all counts and favourable-order masks, the JavaScript witness check passes, all four semantic corruptions are rejected, and the DOI-bearing PDF passes deterministic structural and raw-TeX checks.",
      "evidenceUrl": "https://github.com/ipitchford/negative-square-energy-five-vertex-valley/actions/runs/33294630796"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "The user-supplied review reports a successful different-environment Linux rerun, but reviewer identity, environment receipt, and unaffiliated status were not authenticated by this workflow."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The supplied review reports a separate numerical enumerator reproducing headline counts, but its code, authorship, and independence were not authenticated; the released Python and JavaScript programs were produced within one coordinated workflow."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No proof assistant formalizes the witness, quantifier bridge, structural minimality proof, exhaustive enumeration, or orbit classification."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The five-role review, supplied full review, and confirmation are not authenticated unaffiliated spectral-graph-theory 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 journal or comparable venue has conducted editorial peer review to a decision."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/negative-square-energy-five-vertex-valley/blob/v0.2.0-candidate/ENVIRONMENT.txt",
      "note": "Commands, dependency versions, exact certificates, checksums, and Linux CI are public, but no declarative system image or authenticated unaffiliated clean-environment reconstruction is reported."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Python 3 with SymPy exact algebra",
      "Node.js standard library"
    ],
    "humanRole": "Research direction, candidate selection, review provision, claim-boundary decisions, publication authorization, and institutional responsibility by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.",
    "disclosure": "AI systems contributed mathematical search, exact computation, proof drafting, source and novelty auditing, internal role-separated review, review response, editing, replay, and release engineering within one producer-coordinated workflow. Cross-language agreement and internal review are not unaffiliated reconstruction, specialist review, or editorial peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "AIMPL Problem 1.4, item 2: unimodality of negative square energy",
    "url": "http://aimpl.org/spectralhypergraph/1/"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "reader-first-reference",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Authored prospectively for reader-first schema 2.0 against the reviewed theorem boundary and reserved DOI; deployment remains gated on immutable GitHub and Zenodo readbacks, final exact replay, and public asset identity.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "spectral graph theory",
    "negative square energy",
    "edge addition",
    "unimodality",
    "counterexample",
    "exact enumeration",
    "graph isomorphism",
    "unrefereed candidate"
  ],
  "keyResults": [
    "A connected five-vertex graph has negative square-energy sequence 5>S^-(F1)<5 under two prescribed edge additions, with the exact bound S^-(F1)<3697/784<5.",
    "No connected graph on at most four vertices has an ordered two-edge strict valley, so five vertices are necessary.",
    "Among all 728 connected labelled five-vertex graphs and 11,460 ordered two-edge sequences, exactly 840 sequences are strict valleys.",
    "The 840 labelled valleys form exactly nine orbits under simultaneous vertex relabelling with edge-addition order retained.",
    "The same starting tree has the favourable full order ub,vc,uv,ab,ac,bc with energy pattern 4<5<6>5>S^-(J4)>S^-(J5)>4, so neither existential formulation is refuted."
  ],
  "reviews": [],
  "evidencePackage": "A five-page canonical candidate manuscript and abridged Markdown; a four-claim machine-readable registry; an elementary witness proof and structural four-vertex minimality proof; an exhaustive Python/SymPy exact census with rational root isolation; a dependency-free JavaScript witness checker; a producer-generated recomputable census record; four mutation controls; source, citation, novelty, provenance, and licence records; a five-role producer-coordinated internal review; a fully actioned user-supplied full review; a SHA-bound confirmation; and a complete manifest. Local normal and optimized replay and GitHub Actions reproduce the declared counts and reject all mutations. GitHub and Zenodo release bytes are matched before deployment. These establish public availability, integrity, producer replay, and the declared finite computation—not historical priority, unaffiliated reproduction, formal verification, external specialist review, or peer review.",
  "openProblems": [
    "Prove or refute that every connected starting graph has at least one peak-unimodal full missing-edge ordering.",
    "Determine whether every target graph lies on a peak-unimodal full ordering beginning from one of its spanning trees.",
    "Independently reimplement the exact five-vertex census and orbit classification in a materially separate stack.",
    "Formalize the witness, four-vertex proof, exhaustive enumeration bridge, and nine-orbit classification.",
    "Obtain an unaffiliated spectral-graph-theory source, proof, and priority assessment."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "American Institute of Mathematics. AIMPL Problem 1.4: Positive and negative l2-mass on spectrum.",
      "url": "http://aimpl.org/spectralhypergraph/1/",
      "doi": null
    },
    {
      "citation": "Elphick, C., Farber, M., Goldberg, F., & Wocjan, P. (2016). Conjectured bounds for the sum of squares of positive eigenvalues of a graph. Discrete Mathematics 339, 2215–2223.",
      "url": "https://doi.org/10.1016/j.disc.2016.01.021",
      "doi": "10.1016/j.disc.2016.01.021"
    },
    {
      "citation": "Abiad, A., de Lima, L., Desai, D. N., Guo, K., Hogben, L., & Madrid, J. (2023). Positive and negative square energies of graphs. Electronic Journal of Linear Algebra 39, 307–326.",
      "url": "https://doi.org/10.13001/ela.2023.7827",
      "doi": "10.13001/ela.2023.7827"
    },
    {
      "citation": "Tang, Q., Liu, Y., & Wang, W. (2026). On the positive and negative p-energies of graphs under edge addition. Discrete Applied Mathematics 388, 25–33.",
      "url": "https://doi.org/10.1016/j.dam.2026.02.045",
      "doi": "10.1016/j.dam.2026.02.045"
    },
    {
      "citation": "Liu, Y., Tang, Q., & Zhang, S. (2026). The positive and negative square-energy conjecture. arXiv:2607.18031.",
      "url": "https://arxiv.org/abs/2607.18031",
      "doi": "10.48550/arXiv.2607.18031"
    }
  ],
  "claimAssurance": null,
  "operatingModel": {
    "version": "1.0",
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    "aims": [
      "science"
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    "artifactRoles": [
      "research-output",
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      "method-demonstration",
      "communication"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
      "counterexample-proxy-first",
      "certificate-first",
      "structural-compression",
      "adversarial-controls",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "counterexample",
      "description": "An exact five-vertex ordered strict valley that refutes universal prescribed-order peak-unimodality, plus structural vertex-minimality, a complete five-vertex classification, and a favourable-order boundary witness.",
      "scope": "The universal formulation U, exact five-vertex witness, structural minimality through four vertices, exhaustive 840-count and nine-orbit classification, and one favourable order for the same tree; not E_start, E_target, historical priority, or external validation."
    },
    "bottleneckTargeted": [
      "discovery",
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The package formalizes universal and two existential order statements, embeds the valley at positive indices, and supplies both a bad and favourable order from the same tree so the resolved and open formulations are mechanically and mathematically distinct.",
      "remainingRisks": [
        "The AIMPL source does not print an explicit order quantifier.",
        "The exhaustive classification remains producer-side computation rather than a standalone proof object.",
        "The supplied review's identity and independent status were not authenticated.",
        "The targeted novelty search may miss differently phrased, unindexed, or informal prior work."
      ]
    },
    "humanJudgmentGates": [
      "Keep U, E_start, and E_target distinct and state that only U is refuted.",
      "Check the rational root-isolation argument and four-vertex case split as mathematics rather than inferring truth from replay.",
      "Keep producer exact census replay separate from unaffiliated reproduction and proof-assistant formalization.",
      "Keep found novelty evidence separate from first, priority, solved-problem, or field-acceptance language."
    ],
    "parentLinks": [
      {
        "relation": "depends-on-claim",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://aimath.org/problemlists/",
        "inheritedClaim": "AIMPL Problem 1.4 item 2 asks for negative square energy to be unimodal along successive additions of the listed missing edges.",
        "inheritedAssuranceCeiling": "The source record fixes the question and attribution but leaves the order quantifier unstated and does not validate this candidate's witness, proof, classification, novelty, or priority."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "dataEnvironmentReproducibility",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Obtain an authenticated unaffiliated exact reconstruction and spectral-graph-theory literature assessment, then investigate E_start and E_target without promoting the present universal-only claim ceiling.",
      "claimCeiling": "An anonymous unrefereed counterexample to U with exact producer replay, structural minimality, exhaustive five-vertex classification, a favourable-order boundary witness, public immutable assets, and internal PASS_WITH_NOTES; not a resolution of E_start or E_target, historical priority, unaffiliated validation, formal verification, external specialist review, or editorial peer review."
    },
    "impactClaims": [
      {
        "id": "negative-square-energy-valley-science-no-impact",
        "aim": "science",
        "outcome": "Faster or more reliable resolution and assurance of spectral graph theory open problems",
        "setting": "AI-assisted counterexample search, exact enumeration, proof compression, source audit, review repair, and guarded candidate publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional research, theorem-review, or publication workflow was registered.",
        "estimand": "No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation, or field outcomes was estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
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    "claimCeiling": "These records describe scoped workflow inputs and outcomes. They support calibration and reference-class construction, but do not by themselves establish research acceleration, theorem correctness, novelty, independent assurance or impact.",
    "attempts": [
      {
        "attemptId": "ep-attempt:negative-square-energy-five-vertex-valley-assurance-publication",
        "status": "published",
        "resultClass": "positive",
        "metrics": {
          "schemaVersion": "1.0",
          "measurementScope": "assurance-through-publication",
          "scopeBoundary": "From prospective registration before actioning the supplied Minor Revision report through terminal public readback or a preserved stop. The earlier source search, theorem discovery, exact census, prepare-only review and package construction are excluded and remain recorded in the predecessor package metrics.",
          "forecast": {
            "frozenAt": "2026-08-30T04:45:13Z",
            "procedureClass": "minor-review-repair-through-full-mathematics-candidate-publication",
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            "expectedActiveMinutes": 110,
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              "label": "Minor review repair through full candidate publication",
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            },
            "fermiComponents": [
              {
                "component": "quantifier, favourable-order and assurance repair plus confirmation",
                "count": 1,
                "lowMinutesPerUnit": 20,
                "centralMinutesPerUnit": 30,
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                "basis": "One five-page exact note, one supplied Minor Revision report and a bounded exact favourable-order extension."
              },
              {
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                "count": 1,
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                "basis": "One compact anonymous candidate with standard licences and three principal assets."
              },
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                "component": "Evidence Press page, operating records, art, audio and thumbnail",
                "count": 1,
                "lowMinutesPerUnit": 20,
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                "basis": "One exact graph-theory release using established reader-first and deterministic media tooling."
              },
              {
                "component": "A/B and C/D seals, CI, guarded deployments and canonical readback",
                "count": 1,
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                "basis": "One new slug requiring two bounded merge and deployment cycles."
              }
            ],
            "probabilityPositiveSignal": 0.95,
            "probabilityTargetClosure": 0.8,
            "probabilityHorizonMinutes": 180,
            "assumptions": [
              "The supplied review's favourable ordering is exactly certifiable and introduces no conflict with the strict-valley theorem.",
              "The repaired manuscript and census package pass the bounded confirmation gate without a new load-bearing defect.",
              "GitHub API, Zenodo, OpenAI fable TTS and zero-cost Cloudflare authentication are recoverable without weakening a gate or exposing a secret.",
              "The current Evidence Press operating-model and preservation contracts admit the new bounded counterexample record."
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              "GitHub API authentication cannot be safely recovered or another required public service is unavailable.",
              "Origin main advances with overlapping operating-model, ledger or release changes requiring reconciliation."
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            "stopRule": "Stop on a fatal proof, semantic, novelty, licence, authentication, preservation, CI, accessibility or exact-readback failure. At 110 active agent minutes consolidate to load-bearing gates and report variance; at 180 active minutes pause rather than weaken a gate unless the user explicitly authorises continuation after seeing the exact blocker."
          },
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            "measuredAt": "2026-08-30T05:46:30Z",
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            "resultState": "target-closed",
            "resultSummary": "The supplied Minor Revision was fully actioned, the repaired universal-versus-existential claim boundary and favourable-order witness passed exact confirmation and four mutation controls, and byte-identical immutable GitHub and Zenodo assets reached canonical Evidence Press publication with provenance-bound media, composite CI, accessibility, preservation, protocol and browser readback. The fork-aware goal counter recorded 844,070 model tokens at first publication from a goal boundary three minutes earlier than site-ledger registration. The result remains an anonymous unrefereed candidate with no external-assurance or impact promotion.",
            "positiveSignalObserved": true,
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            "missingFields": [
              {
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}