{ "schemaVersion": "1.6", "slug": "periodic-regulation-budget-monotonicity", "title": "More budget never removes effort: the measure-order form of Gavish and Katriel's budget-monotonicity conjecture for periodic regulation", "shortTitle": "More budget never removes effort", "url": "https://evidencepress.org/releases/periodic-regulation-budget-monotonicity/", "oneLine": "In Gavish and Katriel's model of seasonal control, raising the yearly effort budget never makes it optimal to do less at any time of year; the paper proves this and shows exactly where it stops holding.", "abstract": "Many control problems share one shape: a stock (larvae, pests, a pool of susceptible people, a harvested population or a pollutant) is refilled by a seasonal inflow, decays naturally and is reduced by effort that removes a fixed fraction of what is present, continuously or in instantaneous pulses. With a fixed yearly effort budget the planner minimises the seasonally weighted average stock. Gavish and Katriel (Applied Mathematics & Optimization, 2026) conjectured that a larger budget never makes it optimal to do less at any time. This candidate proves the measure-order (weak) form of that conjecture: in their model, under their uniqueness assumption, the optimal effort assigned to every set of times is nondecreasing in the budget. Supports are nested, every pulse grows, and an optimal plan can be built incrementally. The strict inequality displayed in the source is false in general. The order survives caps on pulses and treatment rates; it can fail under seasonal prices or seasonal efficacy, and with finitely many dates under a nonlinear state cost (exact counterexamples, a sufficient price condition and an elasticity criterion are given). Under a dose budget with finitely many dates the problem becomes linear. The related vaccination conjecture (Conjecture 8 of Gavish and Katriel, Bulletin of Mathematical Biology, 2026) follows in measure form and pointwise except at a precisely described set of points. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-21", "dateModified": "2026-10-21", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23538781", "doiUrl": "https://doi.org/10.5281/zenodo.23538781", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/periodic-regulation-budget-monotonicity/releases/download/v1.0.0-candidate/periodic-regulation-budget-monotonicity-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23538781", "repoUrl": "https://github.com/ipitchford/periodic-regulation-budget-monotonicity", "releaseUrl": "https://github.com/ipitchford/periodic-regulation-budget-monotonicity/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/periodic-regulation-budget-monotonicity/index.md", "bibtexUrl": "https://evidencepress.org/releases/periodic-regulation-budget-monotonicity/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/periodic-regulation-budget-monotonicity.mp3", "imageUrl": "https://evidencepress.org/assets/og/periodic-regulation-budget-monotonicity.png?v=df91078e7e", "coverArtUrl": "https://evidencepress.org/assets/art/periodic-regulation-budget-monotonicity.svg?v=1832409820", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/periodic-regulation-budget-monotonicity.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/periodic-regulation-budget-monotonicity.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. The main theorem and extensions have complete written proofs with the stated scope; finite-date results are labelled as such; internal AI-assisted and adversarial review only. External specialist review, independent reproduction and formal verification have not been 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/23538781", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish 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": "verify_all.py regenerates every check output byte for byte. A separate clean replay reproduced all outputs byte-identically, and the frozen archive was checked against that replay and replayed before release." }, { "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 has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "AI reviewers recomputed key values (the counterexample derivatives, the worked example and random in-band price models) with separate code inside the producer workflow; this is implementation diversity, not external reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation; the measure-theoretic statements rest on written proofs." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review 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 external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Checks need Python with NumPy, SciPy, SymPy and Matplotlib; tested versions are recorded but not locked in a container." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, verification re-review and adversarial review (including OpenAI gpt-5.6-sol)", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the approved revised manuscript and its exact check outputs. Versioned GitHub and Zenodo research archives are checked against the frozen release assets on 21 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "periodic optimal control", "impulsive control", "measure-valued controls", "monotone comparative statics", "budget monotonicity", "pulse vaccination", "pest and vector control", "harvesting" ], "keyResults": [ "Budget monotonicity (Theorem 1.1, proof): for every pair of budgets M1 ≤ M2 the optimal effort measures satisfy μ_M1(E) ≤ μ_M2(E) for every Borel set E of times, under Gavish and Katriel's uniqueness assumption, including pulses and nonsmooth data. Supports are nested, pulses grow and optimal levels fall at every time.", "The strict inequality displayed in the source conjecture is false in general; the weak form is the one the source's prose and first formulation assert.", "Limits (proofs and exact certificates): monotonicity survives caps on pulses and treatment rates; it can fail with seasonal prices or seasonal efficacy, including among all effort measures with one continuous price, and with finitely many dates under a quadratic state cost; a price-band sufficient condition holds for finitely many dates.", "Dose budgets with finitely many dates make the problem linear: with a unique best date, the optimum below an eradication threshold is one campaign there (proof).", "The vaccination density conjecture (Gavish and Katriel, Conjecture 8) holds in measure form and pointwise except where the density formula is undefined or uses a discontinuous derivative representative (proof)." ], "reviews": [], "evidencePackage": "Written proofs of every theorem; exact rational and symbolic certificates for the identities, worked examples and counterexamples (241 exact instances of the core sign identities, the price and quadratic-cost counterexamples, dose-budget identities), replayed byte for byte by verify_all.py; labelled floating-point explorations for caps and prices.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Gavish and Katriel (2026), Optimal regulation in a time-periodic environment: insights from a simple model. Applied Mathematics & Optimization 93, Article 22 (preprint arXiv:2502.06267).", "url": "https://doi.org/10.1007/s00245-025-10374-z", "doi": "10.1007/s00245-025-10374-z" }, { "citation": "Gavish and Katriel (2026), Optimal vaccination for contagious diseases with seasonal transmission. Bulletin of Mathematical Biology 88, Article 170.", "url": "https://doi.org/10.1007/s11538-026-01741-0", "doi": "10.1007/s11538-026-01741-0" }, { "citation": "Topkis (1978), Minimizing a submodular function on a lattice. Operations Research 26, 305–321.", "url": "https://doi.org/10.1287/opre.26.2.305", "doi": "10.1287/opre.26.2.305" }, { "citation": "Milgrom and Shannon (1994), Monotone comparative statics. Econometrica 62, 157–180.", "url": "https://doi.org/10.2307/2951479", "doi": "10.2307/2951479" }, { "citation": "Quah (2007), The comparative statics of constrained optimization problems. Econometrica 75, 401–431.", "url": "https://doi.org/10.1111/j.1468-0262.2006.00752.x", "doi": "10.1111/j.1468-0262.2006.00752.x" }, { "citation": "Agur, Cojocaru, Mazor, Anderson and Danon (1993), Pulse mass measles vaccination across age cohorts. PNAS 90, 11698–11702.", "url": "https://doi.org/10.1073/pnas.90.24.11698", "doi": "10.1073/pnas.90.24.11698" }, { "citation": "Shulgin, Stone and Agur (1998), Pulse vaccination strategy in the SIR epidemic model. Bulletin of Mathematical Biology 60, 1123–1148.", "url": "https://doi.org/10.1016/S0092-8240(98)90005-2", "doi": "10.1016/S0092-8240(98)90005-2" }, { "citation": "Bayen, Rapaport and Tani (2020), Optimal periodic control for scalar dynamics under integral constraint on the input. Mathematical Control and Related Fields 10, 547–571.", "url": "https://doi.org/10.3934/mcrf.2020010", "doi": "10.3934/mcrf.2020010" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:periodic-regulation-budget-monotonicity", "attemptIds": [ "ep-attempt:periodic-regulation-budget-monotonicity-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "A proof that optimal periodic effort measures are ordered in the budget (the measure-order form of the Gavish–Katriel conjecture), with exact boundaries: prices, efficacy, nonlinear costs and dose budgets, and the vaccination Conjecture 8.", "scope": "Scalar linear periodic model S' = c − (δ + η)S with exponential pulses, linear weighted state cost and a budget of log-kill units, under the source's uniqueness assumption; price-band, elasticity and dose results only for finitely many treatment dates. Not an empirical claim about any intervention." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Conjecture 1 and Conjecture 8 are quoted verbatim from arXiv:2502.06267v2 and the open-access version of record of the vaccination paper; the weak measure-order reading is justified and the strict display shown false.", "remainingRisks": [ "The regulation conjecture was compared with arXiv v2; the version of record's wording was not compared.", "Pointwise Conjecture 8 has a stated residual exception set at discontinuities of the derivative of the transmission rate.", "The bounded literature check is not exhaustive priority certification." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "independentRerun", "specialistReview", "formalVerification", "priorityAssessment" ], "nextAction": "Have a specialist in periodic or impulsive optimal control check the finite-date comparison (Theorem 4.7) and the grid-restriction passage to all measures (Section 5).", "claimCeiling": "Measure-order budget monotonicity in the stated scalar linear model under the source's uniqueness assumption, with the stated finite-date extensions. 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