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  "slug": "ptolemaic-split-stability",
  "title": "Sharp local stability of Ptolemaic split metrics",
  "shortTitle": "How sharply does a metric leave equality?",
  "url": "https://evidencepress.org/releases/ptolemaic-split-stability/",
  "oneLine": "Exact first-order rates describe how the negative-type gap opens near every specified split metric, with a different, quadratic transition at four points.",
  "abstract": "This unrefereed candidate determines sharp lower and upper first-order stability constants at the binding complete split metrics of every cardinality at least six, and at the five-point split. A nonlinear realisation lemma identifies the exact tangent cone; 18 rational-function dual families and 36 small certificates establish sharp cone constants. The paper derives explicit quadratic remainders and fixed-cardinality sensitivity of the supremal negative-type exponent. Unequal arms at the four-point star instead give a quadratic gap. Global equality completeness and collision consequences retain their separate companion/source dependencies.",
  "datePublished": "2026-09-27",
  "dateModified": "2026-09-27",
  "version": "0.3.1-candidate",
  "doi": "10.5281/zenodo.23000081",
  "doiUrl": "https://doi.org/10.5281/zenodo.23000081",
  "conceptDoi": "10.5281/zenodo.23000080",
  "pdfUrl": "https://github.com/ipitchford/ptolemaic-split-stability/releases/download/v0.3.1-candidate/ptolemaic_split_stability.pdf",
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  "repoUrl": "https://github.com/ipitchford/ptolemaic-split-stability",
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  "authors": [
    "Anonymous"
  ],
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  "status": "unrefereed-candidate",
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    "detail": "Unrefereed candidate. Producer-side checks cover exact certificates and selected numerical examples, not formal verification of the analytic arguments. The supplied review checker reconstructs constraints separately but its local replay is not unaffiliated reproduction. Historical priority, external specialist review and journal peer review remain unestablished."
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      "state": "passed",
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      "note": "Six manifest-listed assets fully downloaded and SHA-256 matched from GitHub and Zenodo; both providers also carry the checksum list. Availability does not validate the mathematics."
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      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "All exact/symbolic checks and numerical corroboration pass locally; 70 five-mode controls and the original fault test reject semantic corruption. Separate supplied reconstruction checker passes locally."
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    {
      "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."
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    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
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      "note": "No proof-assistant verification of this analytic theorem. Formalisation claims in prior work concern different results."
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      "dimension": "specialistReview",
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    "name": "Sharp stability near complete split equality metrics",
    "url": "https://evidencepress.org/releases/ptolemaic-negative-type/"
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  "metadataProvenance": "Received v0.3 and separate reviewer audit preserved by hash. Review repairs harden optimisation safety, expand source comparison and clarify normalisation and dependency scope. Producer-side replay and archive availability are distinct from independent mathematical validation. Final public GitHub and Zenodo assets were fully downloaded and matched to the local release checksum inventory on 27 September 2026.",
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  "keywords": [
    "Ptolemaic metrics",
    "negative type",
    "sharp stability",
    "tangent cone",
    "rational certificates",
    "critical exponent",
    "metric geometry"
  ],
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    "Exact attained lower and upper first-order gap coefficients under mean-cross-distance-one and infinity-norm normalisation.",
    "Three stable parity-dependent lower-coefficient formulas for cardinality at least nine; six small exceptions are handled exactly.",
    "Every normalised linearised feasible direction is realised by a Ptolemaic curve with a quadratic inward correction.",
    "Fixed-cardinality first variation of the supremal negative-type exponent; four-point unequal-arm perturbations give a quadratic gap instead.",
    "Conditional collision-kernel characterisation and an explicit subcritical separated-metric gap bound; no general global stability constant is claimed."
  ],
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  "relatedWorks": [
    {
      "citation": "Anonymous, Sharp negative type of finite Ptolemaic metrics: companion global threshold candidate. The present direct local calculation does not depend on its universal proof; global completeness/collision interpretations retain its premises.",
      "url": "https://doi.org/10.5281/zenodo.22983333",
      "doi": "10.5281/zenodo.22983333"
    },
    {
      "citation": "Sánchez, On the supremal p-negative type of finite metric spaces: Theorem 2.3 and Corollary 2.4 give the earlier finite distance-matrix criterion.",
      "url": "https://arxiv.org/abs/1108.0451",
      "doi": "10.48550/arXiv.1108.0451"
    },
    {
      "citation": "Li and Weston, Strict p-negative type of a metric space: quantitative extension and strict descent; gap normalisation differs from this restricted L2 eigenvalue.",
      "url": "https://doi.org/10.1007/s11117-009-0035-2",
      "doi": "10.1007/s11117-009-0035-2"
    },
    {
      "citation": "Ercan, Ptolemaic negative type and the values q(6) and q(7), v1: small endpoints and inversion transport credited in the small-cardinality arguments.",
      "url": "https://arxiv.org/abs/2608.20606v1",
      "doi": "10.48550/arXiv.2608.20606"
    },
    {
      "citation": "Baker, Huh, Kummer and Lorscheid, v2: Theorem 7.16 supplies the four-point threshold.",
      "url": "https://arxiv.org/abs/2607.15375v2",
      "doi": "10.48550/arXiv.2607.15375"
    },
    {
      "citation": "Robertson, arXiv:2504.14840: a comparison with spectral/Gramian gap conventions in an ultrametric setting.",
      "url": "https://arxiv.org/abs/2504.14840",
      "doi": "10.48550/arXiv.2504.14840"
    }
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