{
  "schemaVersion": "1.6",
  "slug": "horton-tokunaga-kingman-asymptotics",
  "title": "Beyond the branching exponent",
  "shortTitle": "Tokunaga and Kingman asymptotics",
  "url": "https://evidencepress.org/releases/horton-tokunaga-kingman-asymptotics/",
  "oneLine": "Exact conditioning reveals order fluctuations in one random-tree model; a positive prefactor sharpens the branching law in another.",
  "abstract": "For every critical Tokunaga parameter c>1, this candidate derives a phase-dependent conditional law for Horton–Strahler order at exactly n leaves, including absolute moments and a continuous periodic correction to log base 2c of n. For Kingman’s deterministic limiting branch densities it proves a positive geometric prefactor, with tree size tending to infinity before order. Written analytic proofs are supplemented by exact finite checks and stabilized numerical diagnostics, not certified decimal constants.",
  "datePublished": "2026-10-05",
  "dateModified": "2026-10-05",
  "version": "1.1.0-candidate",
  "doi": "10.5281/zenodo.23160583",
  "doiUrl": "https://doi.org/10.5281/zenodo.23160583",
  "conceptDoi": "10.5281/zenodo.23160582",
  "pdfUrl": "https://github.com/ipitchford/horton-tokunaga-kingman-asymptotics/releases/download/v1.1.0-candidate/paper.pdf",
  "altPdfUrl": "https://zenodo.org/records/23160583/files/paper.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/23160583",
  "repoUrl": "https://github.com/ipitchford/horton-tokunaga-kingman-asymptotics",
  "releaseUrl": "https://github.com/ipitchford/horton-tokunaga-kingman-asymptotics/releases/tag/v1.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/horton-tokunaga-kingman-asymptotics/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/horton-tokunaga-kingman-asymptotics/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/horton-tokunaga-kingman-asymptotics.mp3?v=e16a3927a8",
  "imageUrl": "https://evidencepress.org/assets/og/horton-tokunaga-kingman-asymptotics.png?v=74fb04a37f",
  "coverArtUrl": "https://evidencepress.org/assets/art/horton-tokunaga-kingman-asymptotics.svg?v=ba27af40a7",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/horton-tokunaga-kingman-asymptotics.mp3",
      "name": "Beyond the branching exponent — audio briefing",
      "description": "Synthetic explanation, not additional research evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/horton-tokunaga-kingman-asymptotics.txt"
    },
    {
      "type": "video",
      "url": "https://youtu.be/o7UImy0j0CY",
      "name": "Beyond The Branching Exponent",
      "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence."
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed analytical candidate with supplied review actioned and internal finite replay. No authenticated external specialist review, formal proof or independent reproduction."
  },
  "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/23160583",
      "note": "GitHub and Zenodo public: all four assets downloaded and matched byte-for-byte to the frozen local payload."
    },
    {
      "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 checks, numerical stability comparison and six corruption controls passed locally; archive replay recorded separately."
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "Not established by this publication workflow."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Not established by this publication workflow."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "Not established by this publication workflow."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Not established by this publication workflow."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "Not established by this publication workflow."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "note": "Pinned scientific Python environment; no unaffiliated reproduction."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "Supplied research package; original model identity not established",
      "Codex review response and publication preparation"
    ],
    "humanRole": "Publication authorization; no scientific contribution inferred.",
    "disclosure": "Anonymous is scholarly creator. Supplied review is not authenticated external assurance."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Refined Tokunaga order and strong geometric Kingman Horton asymptotics",
    "url": "https://doi.org/10.1214/19-PS331"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Derived from the supplied proof package and review, revised as 1.1.0-candidate; discovery metrics unknown.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "Horton-Strahler order",
    "Tokunaga trees",
    "Kingman coalescent",
    "local limit theorem",
    "branching processes"
  ],
  "keyResults": [
    "Exact-leaf-conditioned Tokunaga order law for every c>1, including phase-dependent fluctuations and moments.",
    "Positive geometric Kingman prefactor H, after taking the finite-size limit at each fixed order.",
    "Weighted local bounds and a Mobius comparison make the two limiting arguments explicit."
  ],
  "reviews": [],
  "evidencePackage": "Written local-limit and hierarchy-comparison proofs; exact finite rational and symbolic checks; separate 40/60-order numerical stability tests and six rejected error controls.",
  "openProblems": [],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Kovchegov and Zaliapin (2020). Random self-similar trees: A mathematical theory of Horton laws. Model definitions, inverse pruning and open questions.",
      "url": "https://doi.org/10.1214/19-PS331",
      "doi": "10.1214/19-PS331"
    },
    {
      "citation": "Kovchegov and Zaliapin (2017). Horton self-similarity of Kingman’s coalescent tree. Existing root-Horton and endpoint comparison framework.",
      "url": "https://doi.org/10.1214/16-AIHP748",
      "doi": "10.1214/16-AIHP748"
    },
    {
      "citation": "Flajolet, Raoult and Vuillemin (1979). The number of registers required for evaluating arithmetic expressions. Classical binary case.",
      "url": "https://doi.org/10.1016/0304-3975(79)90009-4",
      "doi": "10.1016/0304-3975(79)90009-4"
    },
    {
      "citation": "Brandenberger, Devroye and Reddad (2021). The Horton–Strahler number of conditioned Galton–Watson trees.",
      "url": "https://doi.org/10.1214/21-EJP678",
      "doi": "10.1214/21-EJP678"
    },
    {
      "citation": "Khanfir (2025). The Horton–Strahler number of Galton–Watson trees with possibly infinite variance.",
      "url": "https://doi.org/10.1214/25-AAP2204",
      "doi": "10.1214/25-AAP2204"
    },
    {
      "citation": "Khanfir (2025, v2). Fluctuations of the Horton–Strahler number of stable Galton–Watson trees.",
      "url": "https://arxiv.org/abs/2401.13771v2",
      "doi": "10.48550/arXiv.2401.13771"
    }
  ],
  "claimAssurance": null,
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:horton-tokunaga-kingman-asymptotics",
    "attemptIds": [
      "ep-attempt:horton-tokunaga-kingman-asymptotics-publication"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
      "structural-compression",
      "adversarial-controls",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "reusable-method",
      "description": "Weighted exact-conditioning argument and positive geometric normalization.",
      "scope": "Critical Tokunaga c>1; deterministic Kingman branch-density limits with size before order."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "Auxiliary immigration marginals, lattice inversion and ordered ODE comparisons are distinguished from model assumptions and finite diagnostics.",
      "remainingRisks": [
        "Written proofs not formalized.",
        "Bounded novelty comparison.",
        "No empirical applicability or joint-limit theorem."
      ]
    },
    "humanJudgmentGates": [
      "Publication authorized; no human research role inferred.",
      "Unaffiliated specialists may scrutinize local-limit and normalization arguments."
    ],
    "parentLinks": [
      {
        "relation": "depends-on-claim",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://doi.org/10.1214/19-PS331",
        "inheritedClaim": "Critical Tokunaga model and inverse pruning.",
        "inheritedAssuranceCeiling": "Published mathematical input; not independent assurance of the new conditioning theorem."
      },
      {
        "relation": "depends-on-claim",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://doi.org/10.1214/16-AIHP748",
        "inheritedClaim": "Kingman limiting density and ODE framework.",
        "inheritedAssuranceCeiling": "Published baseline; no inherited positive-prefactor conclusion."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "specialistReview",
        "formalVerification",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Independent analytic scrutiny and unaffiliated replay.",
      "claimCeiling": "Unrefereed analytic candidate with internal diagnostics."
    },
    "impactClaims": [
      {
        "id": "horton-tokunaga-kingman-asymptotics-no-impact",
        "aim": "science",
        "outcome": "Refined mathematical tree asymptotics",
        "setting": "AI-assisted mathematical research and publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched-assurance comparison is registered.",
        "estimand": "No discovery-speed, cost, adoption or impact effect is estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  },
  "researchMetrics": {
    "schemaVersion": "1.0",
    "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json",
    "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json",
    "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:horton-tokunaga-kingman-asymptotics-publication",
        "status": "result-ready",
        "resultClass": "positive",
        "metrics": {
          "schemaVersion": "1.0",
          "measurementScope": "publication-only",
          "scopeBoundary": "Remaining packaging and publication; earlier revision and discovery effort not measured.",
          "forecast": {
            "frozenAt": "2026-10-05T11:25:51.177Z",
            "procedureClass": "review-revision-and-full-publication",
            "targetOutcome": "An exact-version candidate archive with GitHub and Zenodo identities, standard page and media, and canonical Evidence Press readback.",
            "expectedActiveMinutes": 45,
            "plausibleLowMinutes": 25,
            "plausibleHighMinutes": 75,
            "expectedUnattendedWaitMinutes": 15,
            "referenceClass": {
              "label": "Remaining standard publication after bounded revisions",
              "sampleSize": 0,
              "basis": "Task decomposition, not a calibrated empirical benchmark."
            },
            "fermiComponents": [
              {
                "component": "Package and media",
                "count": 1,
                "lowMinutesPerUnit": 15,
                "centralMinutesPerUnit": 25,
                "highMinutesPerUnit": 40,
                "basis": "Short manuscript, rapid replay"
              },
              {
                "component": "Archive and two guarded site cycles",
                "count": 1,
                "lowMinutesPerUnit": 10,
                "centralMinutesPerUnit": 20,
                "highMinutesPerUnit": 35,
                "basis": "Existing tools; concurrent site publication may require reconciliation"
              }
            ],
            "probabilityPositiveSignal": 0.9,
            "probabilityTargetClosure": 0.85,
            "probabilityHorizonMinutes": 120,
            "assumptions": [
              "No new proof gap",
              "Providers available"
            ],
            "reforecastTriggers": [
              "A load-bearing proof, attribution or licensing defect changes the payload.",
              "Provider failure, archive mismatch or site preservation defect delays delivery."
            ],
            "stopRule": "Timing is planning telemetry. Continue authorized work unless an unresolved integrity, rights or provider-access defect blocks publication; never promote an unchecked claim to meet the forecast."
          },
          "outcome": null
        },
        "missingnessReason": null
      }
    ]
  }
}