{ "schemaVersion": "1.6", "slug": "iid-multiunit-prophet-limits", "title": "Limits of online selection with a fixed number of choices", "shortTitle": "How close can online choices get to hindsight?", "url": "https://evidencepress.org/releases/iid-multiunit-prophet-limits/", "oneLine": "An unrefereed mathematical candidate characterises the long-horizon worst-case ratio for every fixed selection budget and certifies a narrow interval near 83.7% for two choices.", "abstract": "For independent, identically distributed nonnegative observations from a known law with finite positive mean, an online decision maker may accept at most k values. This candidate proves existence of the worst-case optimal-online-to-top-k-prophet ratio as the horizon tends to infinity, for each fixed positive integer k, and identifies it with a continuum minimax value. Written transfer and compactness arguments cover the general theorem. For k=2, explicit policy and value-profile witnesses certified with Arb give 0.83695 <= gamma_2* <= 0.8372914, within the rounded headline interval [0.83695, 0.83730]. A separate certified comparison establishes asymptotic non-tightness of the specified earlier nonlinear-system value. No equality with the infimum over all horizons, exact constant, uniform-in-k rate, or numerical certificate for k>=3 is claimed.", "datePublished": "2026-10-15", "dateModified": "2026-10-15", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23404235", "doiUrl": "https://doi.org/10.5281/zenodo.23404235", "conceptDoi": "10.5281/zenodo.23404234", "pdfUrl": "https://github.com/ipitchford/iid-multiunit-prophet-limits/releases/download/v0.1.0-candidate/paper.pdf", "altPdfUrl": null, "altPdfLabel": null, "repoUrl": "https://github.com/ipitchford/iid-multiunit-prophet-limits", "releaseUrl": null, "markdownUrl": "https://evidencepress.org/releases/iid-multiunit-prophet-limits/index.md", "bibtexUrl": "https://evidencepress.org/releases/iid-multiunit-prophet-limits/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/iid-multiunit-prophet-limits.mp3", "imageUrl": "https://evidencepress.org/assets/og/iid-multiunit-prophet-limits.png?v=abfdb2ffb3", "coverArtUrl": "https://evidencepress.org/assets/art/iid-multiunit-prophet-limits.svg?v=ef03e27a25", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/iid-multiunit-prophet-limits.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/iid-multiunit-prophet-limits.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate with integrated written proofs, completed producer replay and verified public archive assets. No external peer review, unaffiliated reproduction, formal verification, exhaustive novelty finding or priority clearance is established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "note": "Public Zenodo record https://zenodo.org/records/23404235 and GitHub release https://github.com/ipitchford/iid-multiunit-prophet-limits/releases/tag/v0.1.0-candidate provide the ZIP, paper.pdf and SHA256SUMS. All three assets were fully downloaded from both providers and matched the frozen SHA-256 values. This establishes archive availability, not external scientific endorsement or canonical site delivery." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "The clean frozen replay passed 10 runs: mixture, upper-bound, nonlinear-system and deterministic-policy certifiers plus the regression suite, each in normal and optimized (-O) Python modes. Producer-operated Ubuntu/Python 3.11 CI also passed. These checks cover encoded certificates and tested controls, not the universal analytic 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 established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Coordinated producer checking is not independent reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "Written proofs and ball arithmetic are not proof-assistant verification." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Internal AI scrutiny is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No journal peer review established." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Witness inputs, checking code, pinned python-flint environment and replay instructions are publicly archived with verified byte identity. Frozen local replay and producer-operated Linux CI passed; no unaffiliated rerun or independent reimplementation is established." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "OpenAI Codex coordinated mathematical development, checking and drafting" ], "humanRole": "Publication authority only; no human research contribution inferred from account ownership.", "reviewStatus": "Internal AI-assisted proof and source checking; not external human peer review or independent reproduction." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared on 15 October 2026 from the integrated manuscript, targeted re-review, producer replay and dated prior-art refresh. Inspected source SHA-256 f1cd1e3a4450588acaae58a436eb5f24e4a2b2d62df5ffda74e96d103085d0a3; final PDF SHA-256 872b5afab8c7bed53995dfe2578a9790a08412389b37ccb5e011e0c6eb8d5aa7; archive SHA-256 aaeccd251a733ccd3d710608f0e5869125ee3ed7ad2a9f2126f70bca79d508a7. Zenodo record 23404235 is published. Full-download parity passed for the ZIP, paper.pdf and SHA256SUMS on Zenodo and GitHub release v0.1.0-candidate at commit 30789207c579d1a873e507fc7b22820056815093. Linux CI run 38799210335 passed. These are archive and producer-check observations, not a canonical site-delivery receipt.", "grandfatheredAtSchemaVersion": null, "keywords": [ "iid prophet inequalities", "online selection", "multiple stopping", "continuum minimax", "quantile policies", "certified bounds" ], "keyResults": [ "For each fixed positive integer k, the worst-case optimal-online/top-k-prophet ratio has a limit as the number of observations grows.", "The limit equals a continuum profile infimum and a compact cumulative-coverage game maximum; this does not assert attainment by a bounded policy or worst-case profile.", "For two selections, the supplied certified witnesses give [0.83695, 0.8372914], and hence the rounded interval [0.83695, 0.83730].", "The specified earlier two-unit nonlinear-system value is strictly below this asymptotic constant; the all-horizon infimum question remains separate.", "An occupancy-weighted threshold construction dominates finite policy mixtures; this is attributed to established occupancy/flow principles, not claimed as a new purification principle." ], "reviews": [], "evidencePackage": "Integrated written analytic proofs and explicit two-unit witnesses with Arb ball-arithmetic certification, supported by internal AI-assisted checking and a bounded primary-source comparison. Targeted integrated-manuscript re-review found no remaining mandatory mathematical revision. Frozen replay passed all 10 runs: four certifiers and the regression suite, each in normal and optimized Python modes. Producer-operated Linux CI also passed. Full downloads of all three release assets from GitHub and Zenodo matched the frozen SHA-256 values.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Jiang, Ma and Zhang, Tightness without counterexamples: finite-horizon occupancy/quantile linear programming and numerical two-unit bounds.", "url": "https://arxiv.org/abs/2205.00588v5", "doi": "10.48550/arXiv.2205.00588" }, { "citation": "Brustle, Perez-Salazar and Verdugo, Splitting guarantees for prophet inequalities via nonlinear systems: asymptotic lower-bound construction.", "url": "https://arxiv.org/abs/2406.17767v2", "doi": "10.48550/arXiv.2406.17767" }, { "citation": "Molina, Gast, Loiseau and Perchet, Prophet inequalities: competing with the top l items is easy: multi-selection lower-bound systems.", "url": "https://arxiv.org/abs/2408.07616v2", "doi": "10.48550/arXiv.2408.07616" }, { "citation": "Loiseau, Molina, Perchet, Perez-Salazar and Verdugo, Kernel methods for refined prophet inequalities: adjacent single-threshold minimax methodology.", "url": "https://arxiv.org/abs/2608.08662", "doi": "10.48550/arXiv.2608.08662" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:iid-multiunit-prophet-limits", "attemptIds": [ "ep-attempt:iid-multiunit-prophet-limits-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "adversarial-controls", "assurance-vector" ], "decisionObject": { "type": "bound", "description": "A continuum characterisation for fixed selection budgets and an explicit certified two-unit interval.", "scope": "Known iid nonnegative law with finite positive mean; optimal adaptive at-most-k selection versus the expected top-k sum; n tends to infinity with k fixed." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Tail-quantile layer-cake identities connect cumulative policy coverage to expected rewards; analytic finite-horizon transfer connects the continuum game to the worst-case iid ratio.", "remainingRisks": [ "Written proof and mathematical encoding require semantic scrutiny.", "Arb certificates certify supplied witnesses, not all statements in the manuscript.", "No inference to unknown distributions, correlated arrivals, growing budgets or all-horizon infima." ] }, "humanJudgmentGates": [ "The user authorises publication; no scientific endorsement is inferred.", "Development-independent proof, novelty and priority scrutiny remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "availability", "internalReplay", "semanticValidation", "specialistReview", "noveltyAssessment" ], "nextAction": "Seek development-independent scrutiny of the written proof and certificate-to-theorem connection; retain the archived producer checks, bounded prior-art comparison and external-assurance limits.", "claimCeiling": "Unrefereed mathematical candidate with internal producer checks; no formal verification, external peer review, priority clearance or measured impact." }, "impactClaims": [ { "id": "iid-multiunit-prophet-no-impact", "aim": "science", "outcome": "Inspectable limiting characterisation and bounded numerical certificates", "setting": "Coordinated AI-assisted mathematical research", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None registered.", "estimand": "No causal research-speed or real-world decision improvement 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:iid-multiunit-prophet-limits-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "research-through-publication", "scopeBoundary": "Only remaining extension-proof validation, manuscript integration, exact replay, archive, media and site delivery from this registration. Earlier research and current initial integration are left-censored; no complete discovery or intake-to-publication clock is reconstructed.", "forecast": { "frozenAt": "2026-10-15T19:41:09Z", "procedureClass": "remaining-extension-validation-and-integrated-publication", "targetOutcome": "Validated, self-contained scoped candidate and complete exact public delivery, or an explicit blocker without overstating the result.", "expectedActiveMinutes": 180, "plausibleLowMinutes": 120, "plausibleHighMinutes": 240, "expectedUnattendedWaitMinutes": 40, "referenceClass": { "label": "Judgmental remaining-task decomposition", "sampleSize": 0, "basis": "Uncalibrated uncertain planning prior, not measured history, proof confidence or a hard cap. Schema-required probabilities below are subjective planning estimates, not empirical frequencies." }, "fermiComponents": [ { "component": "Extension proof validation, manuscript integration and exact replay", "count": 1, "lowMinutesPerUnit": 70, "centralMinutesPerUnit": 110, "highMinutesPerUnit": 150, "basis": "All-fixed-k extension remains conditional on proof validation; errors can require substantial revision." }, { "component": "Frozen archives, media, site authoring, guarded delivery and readback", "count": 1, "lowMinutesPerUnit": 50, "centralMinutesPerUnit": 70, "highMinutesPerUnit": 90, "basis": "Mandatory publication gates remain; provider and CI waits are separately uncertain." } ], "probabilityPositiveSignal": 0.6, "probabilityTargetClosure": 0.6, "probabilityHorizonMinutes": 360, "assumptions": [ "The proposed all-fixed-k proof survives validation; otherwise retain the narrower supported scope or record a blocker.", "Required providers remain available and no rights or exact-byte failure blocks publication.", "No ongoing research job is stopped or reprioritised for this intake." ], "reforecastTriggers": [ "A load-bearing proof gap changes the mathematical target.", "Replay, provenance, archive parity or delivery gates fail.", "Provider waits materially exceed the planning assumption." ], "stopRule": "Forecast is uncertain and not a hard cap. Preserve ongoing research; continue mandatory authorised gates after an overrun. Do not publish unsupported claims, claim inf_n equality or exact constants, or weaken a gate to meet the forecast." }, "outcome": { "status": "unavailable", "policyVersion": "2.0", "recordedAt": "2026-10-15T20:30:51Z", "reason": "Separate active, waiting, blocked, rework and aggregate runtime telemetry was not instrumented for this remaining research-through-publication scope. Earlier research and initial integration remain left-censored. Observed milestone clocks and confirmed machine delivery are retained separately; no retrospective effort estimate or calibration update is made.", "missingFields": [ { "field": "calendarElapsedMinutes", "reason": "Opening and terminal timestamps are observed and retained; their wall-clock difference is derivable, not active effort. No numeric metrics series is emitted in this unavailable outcome." }, { "field": "activeAgentMinutes", "reason": "The scoped activeAgentMinutes counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "unattendedWaitMinutes", "reason": "The scoped unattendedWaitMinutes counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "blockedMinutes", "reason": "The scoped blockedMinutes counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "reworkMinutes", "reason": "The scoped reworkMinutes counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "agentRuns", "reason": "The scoped agentRuns counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "maxParallelAgents", "reason": "The scoped maxParallelAgents counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "modelTurns", "reason": "The scoped modelTurns counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "researchCycles", "reason": "The scoped researchCycles counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "falsificationGatesRun", "reason": "The scoped falsificationGatesRun counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "candidateArchitecturesTested", "reason": "The scoped candidateArchitecturesTested counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "candidateArchitecturesRejected", "reason": "The scoped candidateArchitecturesRejected counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "substantiveReviewRounds", "reason": "The scoped substantiveReviewRounds counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "p0Findings", "reason": "The scoped p0Findings counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "p1Findings", "reason": "The scoped p1Findings counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "prepublicationClaimCorrections", "reason": "The scoped prepublicationClaimCorrections counter was not instrumented; no aggregate is inferred from messages, elapsed time or left-censored research and review." }, { "field": "resultState", "reason": "Candidate publication is observed in statusHistory and canonical readback; the aggregate metrics outcome is unavailable because required telemetry is incomplete, not because publication is unknown." }, { "field": "positiveSignalObserved", "reason": "Publication evidence is retained separately; no calibrated positive-signal metric is emitted with the incomplete telemetry series." }, { "field": "targetReached", "reason": "The publication target is recorded as achieved in statusHistory; no calibrated target-closure metric is emitted with the incomplete telemetry series." }, { "field": "forecastErrorMinutes", "reason": "Active effort was not measured, so forecast error cannot be calculated from wall-clock duration." }, { "field": "forecastRatio", "reason": "Active effort was not measured; wall-clock duration is not a substitute for the forecast denominator's active-time basis." }, { "field": "withinForecastInterval", "reason": "Active effort was not measured, so membership in the active-time forecast interval is unknown." }, { "field": "activeHumanMinutes", "reason": "Scoped human effort was not instrumented; unknown is not zero." }, { "field": "computeMinutes", "reason": "Scoped substantive computation time was not aggregated. 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