{ "schemaVersion": "1.6", "slug": "infinite-birth-death-thinned-counts", "title": "Sharp variance and functional limits for thinned counts in infinite birth–death chains", "shortTitle": "When event counts have a Brownian limit", "url": "https://evidencepress.org/releases/infinite-birth-death-thinned-counts/", "oneLine": "An exact edge-energy criterion separates finite from divergent stationary variance rates, with functional limits and a joint-count covariance extension.", "abstract": "This unrefereed candidate treats independently retained jumps of nonexplosive, irreducible, positive recurrent continuous-time birth–death chains with finite retained intensity. A cumulative edge-energy series gives the exact finite-versus-infinite stationary variance-rate boundary without requiring square-integrable intensity or finite mean raw activity. Finite energy yields a functional central limit theorem from every fixed initial distribution; variance convergence requires additional control of that distribution. A multivariate extension gives explicit covariance for correlated detections at the same jump, and worked examples separate weak convergence from convergence of moments.", "datePublished": "2026-10-18", "dateModified": "2026-10-18", "version": "1.0.0", "doi": "10.5281/zenodo.23448519", "doiUrl": "https://doi.org/10.5281/zenodo.23448519", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/infinite-birth-death-thinned-counts/releases/download/v1.0.0-candidate/infinite-birth-death-thinned-counts.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23448519", "repoUrl": "https://github.com/ipitchford/infinite-birth-death-thinned-counts", "releaseUrl": "https://github.com/ipitchford/infinite-birth-death-thinned-counts/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/infinite-birth-death-thinned-counts/index.md", "bibtexUrl": "https://evidencepress.org/releases/infinite-birth-death-thinned-counts/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/infinite-birth-death-thinned-counts.mp3", "imageUrl": "https://evidencepress.org/assets/og/infinite-birth-death-thinned-counts.png?v=ae143bca2e", "coverArtUrl": "https://evidencepress.org/assets/art/infinite-birth-death-thinned-counts.svg?v=fa32616789", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/infinite-birth-death-thinned-counts.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/infinite-birth-death-thinned-counts.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." }, "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/23448519", "note": "GitHub and Zenodo preserve the exact frozen ZIP, PDF and checksum manifest; availability is not correctness or 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": "Normal and optimized producer replay, semantic corruption controls and fresh archive checks pass. Exact finite tests support formula consistency but do not prove the infinite-state analytic arguments.", "evidenceUrl": "https://github.com/ipitchford/infinite-birth-death-thinned-counts/blob/v1.0.0-candidate/REPLAY_RECEIPT.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No external assurance 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": "No external assurance 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": "No external assurance established by this publication workflow." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "No external assurance 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": "No external assurance established by this publication workflow." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "not-assessed", "note": "No external assurance established by this 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": "Infinite-state extension of thinned birth–death counts", "url": "https://doi.org/10.1007/s11134-026-09974-5" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final version-1.0.0 manuscript, supplied substantive review and exact frozen archive. Research commit ee3e2d422bb70ce069d4de86796acb7580cb9474; archive SHA-256 25d07b8bd2892e56bb6df4781a57c1eb83021130af78ab8d62c07db9513eaa9e. Public GitHub and Zenodo assets verified byte-identical on 18 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "birth–death chain", "thinned counts", "functional central limit theorem", "variance", "correlated marks" ], "keyResults": [ "The stationary variance rate is finite exactly when a cumulative edge-energy series converges; otherwise variance divided by time diverges.", "Finite energy gives a Brownian functional limit from every fixed initial probability law.", "A heavy-tailed starting distribution can have infinite variance at every positive time despite the weak functional limit.", "Joint counts allow dependent marks at the same jump, with an explicit covariance matrix separating shared-chain effects from simultaneous detection." ], "reviews": [], "evidencePackage": "Self-contained analytic manuscript with exact producer-side computations and scoped internal assessment; finite checks do not establish the full analytic theorem.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Daley, Nazarathy and Wang (2026), Second order properties of thinned counts in finite birth–death processes.", "url": "https://doi.org/10.1007/s11134-026-09974-5", "doi": "10.1007/s11134-026-09974-5" }, { "citation": "Chojecki (2011), On the central limit theorem for some birth and death processes: close cumulative-square antecedent for discrete-time state observables.", "url": "https://doi.org/10.2478/v10062-011-0003-8", "doi": "10.2478/v10062-011-0003-8" }, { "citation": "Anantharam and Konstantopoulos (1995), A functional central limit theorem for the jump counts of Markov processes.", "url": "https://doi.org/10.2307/1427836", "doi": "10.2307/1427836" }, { "citation": "Wu (1999), Forward-backward martingale decomposition and compactness results for additive functionals.", "url": "https://doi.org/10.1016/S0246-0203(99)80008-9", "doi": "10.1016/S0246-0203(99)80008-9" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:infinite-birth-death-thinned-counts", "attemptIds": [ "ep-attempt:infinite-birth-death-thinned-counts-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "reusable-method", "description": "Sharp cumulative-energy variance criterion and scalar and joint functional limits for thinned crossing counts.", "scope": "Nonexplosive positive recurrent birth-death chains under stated finite-intensity and energy assumptions; weak limits do not imply unrestricted initial-law moment convergence." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Detailed balance links edge conductances to the counting martingale and finite-energy corrector; finite exact checks do not prove the infinite-state theorem.", "remainingRisks": [ "Written analytic arguments are not formally verified.", "Producer checks are not unaffiliated reproduction.", "Bounded source comparison is not exhaustive priority clearance." ] }, "humanJudgmentGates": [ "Publication is authorised; external specialist scrutiny of the infinite-state arguments and their integrability assumptions remains open." ], "parentLinks": [ { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://doi.org/10.1007/s11134-026-09974-5", "inheritedClaim": "The thinned-count variance framework and its infinite-state regularity question, compared with the candidate's explicit energy criterion.", "inheritedAssuranceCeiling": "Published source; the new candidate remains unrefereed." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "specialistReview", "formalVerification", "priorityAssessment" ], "nextAction": "Unaffiliated reconstruction of the written argument and certificate replay, with specialist scrutiny of the stated boundary cases.", "claimCeiling": "Nonexplosive positive recurrent birth-death chains under stated finite-intensity and energy assumptions; weak limits do not imply unrestricted initial-law moment convergence. Unrefereed candidate, not formal verification, independent reproduction or exhaustive priority clearance." }, "impactClaims": [ { "id": "infinite-birth-death-thinned-counts-no-impact", "aim": "science", "outcome": "Sharp cumulative-energy variance criterion and scalar and joint functional limits for thinned crossing counts.", "setting": "AI-assisted theoretical research", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None registered.", "estimand": "No causal research-speed or downstream scientific effect estimated.", "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:infinite-birth-death-thinned-counts-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Only remaining provider, media, site and readback stages from this registration; all earlier research and revision are left-censored.", "forecast": { "frozenAt": "2026-10-18T07:46:50.736Z", "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": "2026-10-18T08:20:28.761Z", "reason": "Selected gate timings are not complete scoped effort or resource measurements; no retrospective estimate or calibration update is made.", "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 } ] } }