{
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  "slug": "ptolemaic-negative-type",
  "title": "Sharp negative type of finite Ptolemaic metrics",
  "shortTitle": "Ptolemaic metrics: the sharp exponent",
  "url": "https://evidencepress.org/releases/ptolemaic-negative-type/",
  "oneLine": "An analytic proof candidate gives the exact distance-power threshold guaranteed for every finite Ptolemaic metric, with strictness below it and explicit examples showing that it cannot be raised.",
  "abstract": "Ptolemy's inequality constrains every four-point configuration in a metric space. How much must its distances be compressed by a power before the result is guaranteed to embed in Euclidean space? This anonymous, unrefereed manuscript claims the sharp universal negative-type exponent P(m) = log_2(1 + m/floor((m-1)^2/4)) for every m >= 6, together with strict negative type below that threshold. Its analytic proof combines star packing, normalised inverse correlations and an induction whose strictness assumptions concern only smaller metrics. Complete split metrics show sharpness. The argument recovers the known six-point case; the advance beyond the inspected earlier results begins at seven points. Adding the cited smaller endpoints gives the all-cardinality formula and the rank-two uniform threshold q(n)=P(n-1), not every matroid threshold. Exact and symbolic replay checks identities and witnesses, not the universal proof; formal verification, unaffiliated reproduction and external specialist review are not claimed.",
  "datePublished": "2026-09-26",
  "dateModified": "2026-09-26",
  "version": "1.0.0-candidate",
  "doi": "10.5281/zenodo.22983333",
  "doiUrl": "https://doi.org/10.5281/zenodo.22983333",
  "conceptDoi": "10.5281/zenodo.22983332",
  "pdfUrl": "https://github.com/ipitchford/ptolemaic-negative-type/releases/download/v1.0.0-candidate/ptolemaic.pdf",
  "altPdfUrl": "https://zenodo.org/records/22983333/files/ptolemaic.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22983333",
  "repoUrl": "https://github.com/ipitchford/ptolemaic-negative-type",
  "releaseUrl": "https://github.com/ipitchford/ptolemaic-negative-type/releases/tag/v1.0.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/ptolemaic-negative-type/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/ptolemaic-negative-type/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/ptolemaic-negative-type.mp3?v=8a68556666",
  "imageUrl": "https://evidencepress.org/assets/og/ptolemaic-negative-type.png?v=78054b7e76",
  "coverArtUrl": "https://evidencepress.org/assets/art/ptolemaic-negative-type.svg?v=b963602aaa",
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      "url": "https://evidencepress.org/assets/audio/ptolemaic-negative-type.mp3",
      "name": "Plain-English audio briefing",
      "description": "OpenAI synthetic voice; communication only.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/ptolemaic-negative-type.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed analytic proof candidate. The universal theorem rests on the written argument, not on finite computation or numerical search. Producer-side intake replay passed the full nine-check route; the exact five-check route reproduced its compared outputs byte-for-byte. Optimized Python was deliberately refused, not passed as an optimized science replay. Eleven scoped rejection controls passed. Numerical convergence differences are retained separately. No unaffiliated reproduction, end-to-end formalisation, authenticated external specialist endorsement, journal peer review, exhaustive priority finding, or measured impact is claimed."
  },
  "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/22983333",
      "note": "Published GitHub and Zenodo assets were fully downloaded and matched to the local SHA-256 inventory. DOI/archive availability does not validate the analytic theorem."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Producer-side fresh intake: full nine-check replay passed; exact five-check route reproduced its compared outputs byte-for-byte; eleven scoped rejection controls passed. Python -O was refused. Numerical convergence differences are retained separately, not treated as exact matches or proof."
    },
    {
      "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 is evidenced; coordinated agents and supplied replay records do not establish independence."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Auxiliary symbolic and exact checks are implementation diversity within the producing 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 verification of this analytic theorem. Formalisation claims in prior work concern different results."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The supplied favourable review is retained as a document, not an authenticated external specialist endorsement."
    },
    {
      "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 external venue peer review is claimed."
    },
    {
      "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/ptolemaic-negative-type/actions/runs/36274292333",
      "note": "Pinned Linux full-replay CI passed at commit ce622e0053f287d6184baa0fbbfcd1f776d1d20c. The public receipt separately records the fresh macOS exact replay, reused semantic controls and differing local dependency versions. Numerical convergence can vary; PDF build is not hermetic."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI-assisted mathematical research and manuscript preparation; exact originating model identities are not asserted",
      "Producer-side Python exact, symbolic and numerical checks",
      "Codex publication-intake source comparison and release authoring"
    ],
    "humanRole": "Publication authority and supplied materials; no human mathematical contribution is inferred. Scholarly creator remains Anonymous.",
    "disclosure": "The supplied review, internal checks and publication-intake replay belong to a producer-coordinated workflow. They do not establish unaffiliated reproduction, authenticated specialist review or formal verification."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Baker–Huh–Kummer–Lorscheid Conjecture 6.4: the rank-two uniform threshold",
    "url": "https://arxiv.org/html/2607.15375v2"
  },
  "orderedProductAverage": null,
  "corrections": [],
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  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
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  "recordMaturity": "current",
  "metadataProvenance": "The supplied 25 September bundle is preserved. The 26 September intake source comparison adds the Ideosphere attribution without changing the theorem or inferring external review. GitHub v1.0.0-candidate and Zenodo record 22983333 are published; all six downloadable assets were fetched from both services and matched to local SHA-256 hashes on 26 September. Research assurance is separate from subsequent site-publication readback in the publication ledger.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "Ptolemaic metrics",
    "negative type",
    "metric embedding",
    "snowflake",
    "Schoenberg matrix",
    "metric inversion",
    "Lorentzian matrices",
    "triangular hyperfields",
    "star packing",
    "complete split metrics"
  ],
  "keyResults": [
    "For every integer m >= 6, every m-point Ptolemaic metric has p_m-negative type, where p_m = log_2(1 + m/floor((m-1)^2/4)); every exponent 0 < p < p_m gives strict negative type.",
    "For each m, a complete split metric on m points fails p-negative type for every p > p_m, proving that the proposed universal threshold is sharp.",
    "The first thresholds beyond the inspected established cases are P(7)=log_2(16/9) and P(8)=log_2(5/3); P(6)=1 is independently recovered rather than claimed as a new value.",
    "The cited three-, four- and five-point endpoints extend the formula to all m >= 3. Through the established identity q(n)=P(n-1), it gives the conjectured threshold for rank-two uniform matroids.",
    "The proposed analytic mechanism is a dimension-independent star-packing bound and a normalised inverse-correlation induction. The anchor constants and complete-split thresholds have prior antecedents and are not claimed as new."
  ],
  "reviews": [],
  "evidencePackage": "Analytic manuscript with explicit hypotheses and a two-induction dependency map; exact rational and symbolic checks; complete split witnesses; separately labelled numerical exploration; source and evidence ledgers; supplied review and revision response; immutable-input replay wrapper and rejection controls.",
  "openProblems": [],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Baker, Huh, Kummer and Lorscheid (2026), version 2: Conjecture 6.4 gives the parity formula; Proposition 7.5 identifies q(n)=P(n-1); Theorem 7.16 and Corollary 7.17 provide the four-point endpoint.",
      "url": "https://arxiv.org/abs/2607.15375v2",
      "doi": "10.48550/arXiv.2607.15375"
    },
    {
      "citation": "Ercan (2026), version 1, Theorem 1.1: P(5)=log_2(9/4) and P(6)=1, with partial correlations, copositivity and coefficient transport under inversion. The external five-point certificate is not a premise of the new m >= 6 proof.",
      "url": "https://arxiv.org/abs/2608.20606v1",
      "doi": "10.48550/arXiv.2608.20606"
    },
    {
      "citation": "Korea Superintelligence Labs, Sharp constants for Ptolemaic negative type, version 3 (source snapshot 7 September 2026): Sections 11–12 derive general anchor constants, split thresholds and an obstruction above exponent one; Section 15 leaves the general low-exponent estimate unresolved.",
      "url": "https://ideosphere.ai/papers/ptolemaic-negtype-v3.pdf",
      "doi": null
    },
    {
      "citation": "Schoenberg (1938), Metric spaces and positive definite functions: the classical negative-type and Euclidean-embedding background. The finite Gram identity used here is derived in the manuscript.",
      "url": "https://doi.org/10.1090/S0002-9947-1938-1501980-0",
      "doi": "10.1090/S0002-9947-1938-1501980-0"
    },
    {
      "citation": "Li and Weston (2010), Strict p-negative type of a metric space: strict descent supports the optional below-endpoint statement for the imported small cases, not the new cardinality induction.",
      "url": "https://doi.org/10.1007/s11117-009-0035-2",
      "doi": "10.1007/s11117-009-0035-2"
    },
    {
      "citation": "Sánchez (2012), On the supremal p-negative type of finite metric spaces: classical inverse-distance criteria; no new metric M-constant is claimed here.",
      "url": "https://doi.org/10.1016/j.jmaa.2011.11.041",
      "doi": "10.1016/j.jmaa.2011.11.041"
    }
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