{
  "schemaVersion": "1.2",
  "slug": "certified-two-item-jrp",
  "title": "Exact Two-Item Resource-Constrained Joint Replenishment and Its Independent-Cap Relaxation Gap",
  "shortTitle": "The exact two-item cap gap in joint replenishment",
  "url": "https://evidencepress.org/releases/certified-two-item-jrp/",
  "oneLine": "An exact certificate system proves finite termination for the stated two-item replenishment model and determines the independent-frequency-cap relaxation gap as the algebraic constant gamma, about 1.11188959394; the unrestricted multi-item gap remains open.",
  "abstract": "Joint replenishment saves a common ordering cost when periodic item schedules coincide, but those coincidences make the exact objective discontinuous in the period ratio. This unrefereed candidate reduces every coprime two-item ratio to a one-dimensional exact scale optimization, retains resource geometry in a Lagrange-dual tail certificate, and proves that unrestricted ratio enumeration terminates after finitely many ratios whenever the common setup cost is positive; no polynomial runtime is claimed. For the standard convex relaxation, a real-coefficient cap-reduction lemma reduces worst-case two-item analysis to itemwise frequency caps. A four-policy envelope, a symbolic critical interval, and a complete 212-cell rational outer certificate prove that the exact real-coefficient two-item independent-cap gap is gamma, where gamma is the relevant root of 262 gamma^4 - 916 gamma^3 + 863 gamma^2 + 150 gamma - 375 = 0 and lies between 1.111889593939396 and 1.111889593940297. Four finite witnesses, exact and coarse two-cap certificates, negative controls, 17 tests, deterministic documents, and producer-side fresh-extraction replay pass. The unrestricted multi-item gap remains open. There is no independent reproduction, formal verification, specialist review, peer review, field validation, complete priority audit, or evidence of operational savings.",
  "datePublished": "2026-08-08",
  "dateModified": "2026-08-09",
  "version": "1.2.0-candidate",
  "doi": "10.5281/zenodo.21855894",
  "doiUrl": "https://doi.org/10.5281/zenodo.21855894",
  "conceptDoi": "10.5281/zenodo.21855566",
  "pdfUrl": "https://github.com/ipitchford/certified-two-item-jrp/releases/download/v1.2.0-candidate/two_cap_integrality_gap.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/certified-two-item-jrp/v1.2.0-candidate/paper/two_cap_integrality_gap.pdf",
  "zenodoUrl": "https://zenodo.org/records/21855894",
  "repoUrl": "https://github.com/ipitchford/certified-two-item-jrp",
  "releaseUrl": "https://github.com/ipitchford/certified-two-item-jrp/releases/tag/v1.2.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/certified-two-item-jrp/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/certified-two-item-jrp/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/certified-two-item-jrp.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/certified-two-item-jrp.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/certified-two-item-jrp.svg",
  "media": [],
  "authors": [
    "Agent collective",
    "Ian Pitchford"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate computer-assisted result. Producer-side solver-independent replay, fresh-extraction regeneration, mutation controls, fail-closed output checks, deterministic document builds, GitHub/Zenodo archive matching, and public file readback pass. The analytic manuscript and supplied certificates have not been independently reconstructed by an unaffiliated party or formalized in a proof assistant. There has been no external specialist audit, editorial peer review, field validation, complete novelty or priority audit, or demonstration that a certified nominal policy yields operational savings. The exact global relaxation gap, multi-item alignment relaxation and rounding, column-sparsity frontier, three-item oracle, robust uncertainty, calendar phases, pointwise capacity, and wider theory remain open."
  },
  "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/21855894",
      "note": "The exact candidate archive and five companion assets are publicly retrievable under a version DOI; all six public downloads matched the GitHub release hashes."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Producer-side fresh-extraction replay, ordinary and optimized release gates, 17 tests, exact two-cap certificates, figures and all three PDFs passed.",
      "evidenceUrl": "https://github.com/ipitchford/certified-two-item-jrp/releases/download/v1.2.0-candidate/certified-two-item-jrp-v1.2.0-candidate.replay-receipt.json"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No unaffiliated party has reported rerunning the archived package."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The solver-independent verifiers were produced within the same research and release workflow."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No theorem has been formalized in a proof assistant."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "No external inventory-theory, approximation-algorithms, or number-theory specialist has audited the paper."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No journal or venue has conducted peer review."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/certified-two-item-jrp/releases/download/v1.2.0-candidate/certified-two-item-jrp-v1.2.0-candidate.replay-receipt.json",
      "note": "Pinned Python 3.14.6 dependencies and the recorded TeX toolchain passed macOS arm64 fresh-extraction replay; no independently recreated container image is supplied."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "Agent collective",
      "OpenAI Codex task"
    ],
    "humanRole": "Problem selection, research direction, mediation, maintenance, licensing and publication by Dr Ian Pitchford.",
    "disclosure": "The original bundle is repository-attributed to an agent collective. An OpenAI Codex task developed the extension proof, revised the software and exposition, ran producer-side checks, and prepared publication. Individual original model runs were not supplied, and no externally auditable Codex run identifier is embedded in the release."
  },
  "problem": {
    "name": "Two-item resource-constrained continuous-time joint replenishment and the gap of relaxation (P)",
    "url": "https://doi.org/10.1287/opre.2022.2390"
  },
  "corrections": [],
  "keywords": [
    "joint replenishment",
    "inventory theory",
    "exact certification",
    "periodic synchronisation",
    "resource constraints",
    "Lagrange duality",
    "integrality gap",
    "frequency caps",
    "Farey mediants",
    "exact integrality gap",
    "computer-assisted proof",
    "reproducible research",
    "unrefereed candidate"
  ],
  "keyResults": [
    "For every fixed coprime period ratio, the resource-constrained two-item objective reduces to an exact one-dimensional convex scale problem with a rational or quadratic-radical optimum.",
    "A nonnegative Lagrange-dual certificate for the constrained desynchronised optimum yields a resource-aware lower bound for every omitted ratio.",
    "When the common ordering cost is positive, a rational synchronised policy lies strictly below the desynchronised optimum, so unbounded ratio enumeration is guaranteed to terminate after finitely many ratios; no polynomial runtime is claimed.",
    "Over nonnegative real resource systems, the supremum gap of relaxation (P) is unchanged when analysis is restricted to item-specific frequency caps; no rational-data continuity identity is claimed.",
    "For nonnegative real coefficients and two independent frequency caps, the exact supremum gap is gamma, the relevant root of 262 gamma^4 - 916 gamma^3 + 863 gamma^2 + 150 gamma - 375 = 0, with 1.111889593939396 < gamma < 1.111889593940297; the canonical family attains it.",
    "A four-policy envelope, symbolic critical interval, and complete 212-cell exact rational outer cover give the matching upper certificate; an independently structured 2,829-cell certificate supplies a coarser producer-side cross-check.",
    "A finite two-item alignment-configuration hull closes all four bundled regression certificates, but no multi-item separation or rounding theorem is supplied."
  ],
  "reviews": [],
  "evidencePackage": "Two 8-page candidate theorem papers and a 3-page practitioner brief in PDF and TeX; exact Python solver and fail-closed CLI; four finite rational/radical witnesses; resource-dual and tail certificates; exact quartic isolation and elimination checks; a four-policy coefficient envelope; complete 212-cell exact and 2,829-cell coarse two-cap certificates; a finite alignment-configuration regression certificate; schema and re-hashed semantic-mutation controls; 17 tests; deterministic figures and PDFs; an exact 92-entry SHA-256 manifest; producer adversarial reviews; and producer-side fresh-extraction replay of the immutable ZIP. These checks do not constitute independent reproduction or external theorem verification.",
  "openProblems": [
    "Determine the unrestricted multi-item integrality gap and whether it equals the now-certified two-item independent-cap constant gamma.",
    "Construct a multi-item alignment-aware relaxation with polynomial separation and a rounding theorem while preserving the exact two-item regression suite.",
    "Classify the column-sparsity k=1, k=2, and fixed-k approximation or hardness frontier.",
    "Build and independently reproduce a complete exact three-item oracle, including its regime and tail proof objects.",
    "Develop robust correlated-uncertainty regions rather than one-at-a-time sensitivity analysis.",
    "Extend the model to calendar phases, fixed delivery dates, and pointwise capacity without erasing synchronisation semantics.",
    "Run a preregistered field pilot with reconciled marginal costs, operational feasibility gates, guardrails, and stop rules.",
    "Obtain independent mathematical reconstruction, specialist review, proof-assistant formalization, and a broader novelty and priority audit."
  ],
  "relatedWorks": [
    {
      "citation": "Wildeman, R. E., Frenk, J. B. G., & Dekker, R. (1997). An efficient optimal solution method for the joint replenishment problem. European Journal of Operational Research, 99, 433-444.",
      "url": "https://repub.eur.nl/pub/15306/AnEfficientOPtimalSolution_1997.pdf"
    },
    {
      "citation": "Porras, E., & Dekker, R. (2006). An efficient optimal solution method for the joint replenishment problem with minimum order quantities. European Journal of Operational Research, 174, 1595-1615.",
      "url": "https://doi.org/10.1016/j.ejor.2005.02.056"
    },
    {
      "citation": "Porras, E., & Dekker, R. (2008). A solution method for the joint replenishment problem with correction factor. International Journal of Production Economics, 113, 834-851.",
      "url": "https://doi.org/10.1016/j.ijpe.2007.11.008"
    },
    {
      "citation": "Schulz, A. S., & Telha, C. (2024). Integer factorization: Why two-item joint replenishment is hard. Operations Research, 72, 1192-1202.",
      "url": "https://doi.org/10.1287/opre.2022.2390"
    },
    {
      "citation": "Segev, D. (2025). The continuous-time joint replenishment problem: epsilon-optimal policies via pairwise alignment. Management Science, 71, 4183-4197.",
      "url": "https://doi.org/10.1287/mnsc.2023.00705"
    },
    {
      "citation": "Segev, D. (2026). Improved approximation guarantees for joint replenishment in continuous time. Mathematics of Operations Research, 51, 1683-1712.",
      "url": "https://doi.org/10.1287/moor.2024.0635"
    },
    {
      "citation": "Segev, D. (2026). Resource-constrained joint replenishment via power-of-m policies. arXiv:2603.18720v1.",
      "url": "https://arxiv.org/abs/2603.18720"
    },
    {
      "citation": "Creemers, S., & Stephan, T. (2026). Special cases of the joint replenishment problem: A mathematical analysis. SSRN 6674418.",
      "url": "https://doi.org/10.2139/ssrn.6674418"
    }
  ]
}