{
  "schemaVersion": "1.6",
  "slug": "uniform-split-reliability",
  "title": "Uniformly most split reliable graphs: a complete answer outside two edge counts",
  "shortTitle": "When one split-network design wins",
  "url": "https://evidencepress.org/releases/uniform-split-reliability/",
  "oneLine": "An existence classification leaves two all-order boundary counts, certifies them through 51 vertices, and controls explicit probability intervals at every order.",
  "abstract": "This unrefereed mathematical candidate studies connected simple graphs whose independently retained edges must form exactly two components, one per designated terminal. For n at least seven it proves nonexistence of uniform maximisers in an intermediate edge range and existence in the dense range, leaving two boundary edge counts conditional at unrestricted orders. Exact cone-reliability certificates resolve those counts through n = 51. Analytic cone comparisons cover p at least one half and an explicit interval near zero at every order, without resolving the intervening interval. The result classifies existence, not all optimisers.",
  "datePublished": "2026-10-18",
  "dateModified": "2026-10-18",
  "version": "1.0.0",
  "doi": "10.5281/zenodo.23450265",
  "doiUrl": "https://doi.org/10.5281/zenodo.23450265",
  "conceptDoi": "10.5281/zenodo.23450264",
  "pdfUrl": "https://github.com/ipitchford/uniform-split-reliability/releases/download/v1.0.0/split_reliability.pdf",
  "altPdfUrl": null,
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/23450265",
  "repoUrl": "https://github.com/ipitchford/uniform-split-reliability",
  "releaseUrl": "https://github.com/ipitchford/uniform-split-reliability/releases/tag/v1.0.0",
  "markdownUrl": "https://evidencepress.org/releases/uniform-split-reliability/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/uniform-split-reliability/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/uniform-split-reliability.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/uniform-split-reliability.png?v=639bfb5e3f",
  "coverArtUrl": "https://evidencepress.org/assets/art/uniform-split-reliability.svg?v=7fc208df13",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/uniform-split-reliability.mp3",
      "name": "Plain-English audio briefing",
      "description": "Communication aid; not additional research evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/uniform-split-reliability.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed AI-assisted candidate. No external peer review, unaffiliated reproduction or proof-assistant formalisation established. Two all-order boundary counts remain conditional."
  },
  "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/23450265",
      "note": "Full public downloads of manuscript PDF, research ZIP and checksum manifest from GitHub and Zenodo matched the frozen local bytes. Archiving is not mathematical correctness or external 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": "Fresh normal and optimised Python runs each checked 2,024 cone triples and 1,354,694 scalar inequalities for N=7–50, plus 5,422 supplied base-case graphs. The written reduction gives finite release coverage through n=51; corpus completeness is inherited, not independently regenerated. This is producer-side replay, not formal proof or external reproduction.",
      "evidenceUrl": "https://github.com/ipitchford/uniform-split-reliability/blob/v1.0.0/REPLAY.md"
    },
    {
      "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 producer-coordinated 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 producer-coordinated 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 producer-coordinated publication workflow."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "Not established by this producer-coordinated 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 producer-coordinated publication workflow."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "not-assessed",
      "note": "Not established by this producer-coordinated publication workflow."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI-assisted research workflow",
      "OpenAI Codex: checking, revision and publication"
    ],
    "humanRole": "Publication authority only; no human research contribution inferred.",
    "reviewStatus": "Internal AI-assisted assessment, not external peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "Existence of uniformly most split reliable graphs",
    "url": "https://doi.org/10.1002/net.22166"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Prepared from the final revised manuscript and producer replay. Research commit b5318f9524ed0ef652af23ad8327c545b505b19c; ZIP SHA-256 91991c6fa860818abee46e15a858bd02df0b98e291faf2f92fdf878bdd0b7ee3. Full downloads of PDF, ZIP and checksum manifest from GitHub and Zenodo matched local bytes on 18 October 2026.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "split reliability",
    "extremal graphs",
    "uniform optimality",
    "reliability polynomials",
    "exact certificates"
  ],
  "keyResults": [
    "For n ≥ 7, no uniform maximiser exists for n ≤ m ≤ C(n,2) − n − 1; explicit dense maximisers exist for m ≥ C(n,2) − n + 2.",
    "The two boundary edge counts are certified for 7 ≤ n ≤ 51; at unrestricted orders they remain conditional.",
    "All-order cone comparisons hold for p ≥ 1/2 and on an explicit positive interval near zero; the intermediate interval remains unresolved."
  ],
  "reviews": [],
  "evidencePackage": "Written analytic arguments and exact finite cone-reliability certificates; all-order interval bounds are distinct from finite full-interval coverage.",
  "openProblems": [],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Brown and Mol (2017), introducing split reliability.",
      "url": "https://arxiv.org/abs/1703.10566",
      "doi": "10.48550/arXiv.1703.10566"
    },
    {
      "citation": "Brown and McMullin (2023), On split reliability of graphs. Networks 82(2), 177–185.",
      "url": "https://doi.org/10.1002/net.22166",
      "doi": "10.1002/net.22166"
    },
    {
      "citation": "Romero (2025), Characterization of locally most split reliable graphs. Theoretical Computer Science 1047, 115327.",
      "url": "https://doi.org/10.1016/j.tcs.2025.115327",
      "doi": "10.1016/j.tcs.2025.115327"
    }
  ],
  "claimAssurance": null,
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:uniform-split-reliability",
    "attemptIds": [
      "ep-attempt:uniform-split-reliability-publication"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "communication"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
      "certificate-first",
      "structural-compression",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "bound",
      "description": "Uniform split-reliability existence classification and certified boundary comparisons.",
      "scope": "n≥7, outside two boundary counts; finite full-interval certification n≤51 and all-order partial-probability comparisons remain distinct."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "Exactly two components, one terminal each; terminal identification preserves multiedges and coefficient loop contribution.",
      "remainingRisks": [
        "Written proofs are not formally verified.",
        "Finite certificates do not establish all-order full-probability comparison."
      ]
    },
    "humanJudgmentGates": [
      "Publication authorised; external specialist assessment remains open."
    ],
    "parentLinks": [
      {
        "relation": "depends-on-claim",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://doi.org/10.1016/j.tcs.2025.115327",
        "inheritedClaim": "Prior local split-reliability optimality characterisation.",
        "inheritedAssuranceCeiling": "Published source; this candidate remains unrefereed."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "specialistReview",
        "formalVerification",
        "priorityAssessment"
      ],
      "nextAction": "Unaffiliated proof scrutiny and exact finite replay.",
      "claimCeiling": "n≥7, outside two boundary counts; finite full-interval certification n≤51 and all-order partial-probability comparisons remain distinct. No external assurance inferred."
    },
    "impactClaims": [
      {
        "id": "uniform-split-reliability-no-impact",
        "aim": "science",
        "outcome": "Uniform split-reliability existence classification and certified boundary comparisons.",
        "setting": "AI-assisted theoretical research",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "None registered.",
        "estimand": "No comparative research-speed or scientific-impact effect measured.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  },
  "researchMetrics": {
    "schemaVersion": "1.0",
    "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json",
    "outcomePolicyUrl": "https://evidencepress.org/api/v2/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:uniform-split-reliability-publication",
        "status": "published",
        "resultClass": "positive",
        "metrics": {
          "schemaVersion": "1.0",
          "measurementScope": "assurance-through-publication",
          "scopeBoundary": "Remaining publication only; earlier work and initial media creation left-censored.",
          "forecast": {
            "frozenAt": "2026-10-18T10:01:11.865Z",
            "procedureClass": "remaining-candidate-site-and-archive-delivery",
            "targetOutcome": "Accurately scoped archived candidate and canonical site readback if every mandatory gate passes.",
            "expectedActiveMinutes": 70,
            "plausibleLowMinutes": 35,
            "plausibleHighMinutes": 140,
            "expectedUnattendedWaitMinutes": 25,
            "referenceClass": {
              "label": "Remaining Evidence Press candidate publication stages",
              "sampleSize": 0,
              "basis": "Procedural planning judgement, not an empirical calibration."
            },
            "fermiComponents": [
              {
                "component": "Provider identity, media, guarded site delivery and readback",
                "count": 1,
                "lowMinutesPerUnit": 35,
                "centralMinutesPerUnit": 70,
                "highMinutesPerUnit": 140,
                "basis": "Revised papers and producer replays exist; exact archive gates and provider/site delivery remain mandatory."
              }
            ],
            "probabilityPositiveSignal": 0.95,
            "probabilityTargetClosure": 0.85,
            "probabilityHorizonMinutes": 360,
            "assumptions": [
              "No provider outage or byte mismatch blocks delivery."
            ],
            "reforecastTriggers": [
              "Provider failure, immutable-asset mismatch or site preservation regression."
            ],
            "stopRule": "Complete every mandatory gate or retain a precise blocker; never weaken assurance to meet the forecast."
          },
          "outcome": {
            "status": "unavailable",
            "policyVersion": "2.0",
            "recordedAt": "2026-10-18T10:58:41.993Z",
            "reason": "Gate and delivery observations do not provide complete effort telemetry. No retrospective reconstruction or calibration.",
            "missingFields": [
              {
                "field": "calendarElapsedMinutes",
                "reason": "Observed opening and delivery times are retained as milestones; elapsed wall time is not active effort and no numeric metrics series is emitted."
              },
              {
                "field": "activeAgentMinutes",
                "reason": "The scoped activeAgentMinutes counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "unattendedWaitMinutes",
                "reason": "The scoped unattendedWaitMinutes counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "blockedMinutes",
                "reason": "The scoped blockedMinutes counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "reworkMinutes",
                "reason": "The scoped reworkMinutes counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "agentRuns",
                "reason": "The scoped agentRuns counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "maxParallelAgents",
                "reason": "The scoped maxParallelAgents counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "modelTurns",
                "reason": "The scoped modelTurns counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "researchCycles",
                "reason": "The scoped researchCycles counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "falsificationGatesRun",
                "reason": "The scoped falsificationGatesRun counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "candidateArchitecturesTested",
                "reason": "The scoped candidateArchitecturesTested counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "candidateArchitecturesRejected",
                "reason": "The scoped candidateArchitecturesRejected counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "substantiveReviewRounds",
                "reason": "The scoped substantiveReviewRounds counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "p0Findings",
                "reason": "The scoped p0Findings counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "p1Findings",
                "reason": "The scoped p1Findings counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "prepublicationClaimCorrections",
                "reason": "The scoped prepublicationClaimCorrections counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "resultState",
                "reason": "Published candidate delivery is observed in statusHistory; incomplete workflow telemetry does not negate publication."
              },
              {
                "field": "positiveSignalObserved",
                "reason": "Candidate publication evidence is retained separately; no calibrated signal metric is inferred."
              },
              {
                "field": "targetReached",
                "reason": "Delivery target is recorded in statusHistory; no calibrated target metric is inferred."
              },
              {
                "field": "forecastErrorMinutes",
                "reason": "Active effort was not measured; forecast error cannot be computed from wall time."
              },
              {
                "field": "forecastRatio",
                "reason": "Active effort was not measured; elapsed time is not a valid active-effort denominator."
              },
              {
                "field": "withinForecastInterval",
                "reason": "Active effort was not measured, so interval membership is unknown."
              },
              {
                "field": "activeHumanMinutes",
                "reason": "The scoped activeHumanMinutes counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "computeMinutes",
                "reason": "The scoped computeMinutes counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "deduplicatedModelTokens",
                "reason": "The scoped deduplicatedModelTokens counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              },
              {
                "field": "uncachedInputTokens",
                "reason": "The scoped uncachedInputTokens counter was not instrumented or aggregated; unknown is not zero, and no estimate is inferred from logs, elapsed time or left-censored work."
              }
            ],
            "calibration": null
          }
        },
        "missingnessReason": null
      }
    ]
  }
}