--- title: "A Candidate Exact Three-Item Independent-Cap Oracle" date: 2026-08-22 version: "0.3.1-candidate" doi: 10.5281/zenodo.22057665 pdf: https://github.com/ipitchford/certified-three-item-jrp-gap/releases/download/v0.3.1-candidate/three-item-jrp-gap-and-logistics-v0.3.1-paper.pdf repository: https://github.com/ipitchford/certified-three-item-jrp-gap archive: https://doi.org/10.5281/zenodo.22057665 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A Candidate Exact Three-Item Independent-Cap Oracle ## Plain-English summary When three products share a supplier order, inbound shipment, receiving slot, or setup event, a common shortcut prices that shared event using only the fastest replenishment stream. This candidate gives an exact warning about that shortcut. For one fully specified rational three-item instance, the relaxed model reports an exact normalized value of approximately 4.033. The true periodic optimum is approximately 4.632. Their ratio is $$ 1.1487291464748205845\ldots. $$ That is a model gap, not a saving. It says the relaxation is 14.8729% optimistic relative to its own denominator on this synthetic instance. No company result, service improvement, or avoided loss has been measured. The mathematical significance is sharper. The unrefereed producer-side Evidence Press predecessor reported a narrow two-item interval with upper endpoint near 1.111889. The new candidate three-item lower bound is 3.3132% above that endpoint. A monotonicity lemma propagates it to every fixed item count of at least three. The two-item interval therefore cannot supply the unrestricted multi-item answer, subject to this candidate’s proof and replay boundary. > **Status:** anonymous, unrefereed successor candidate. Producer-side exact replay passes. No unaffiliated rerun, independently authored implementation, proof-assistant formalisation, specialist review, editorial peer review, operational validation, or field impact is claimed. ## What the certificate proves The exact continuous-time objective combines a common replenishment-event cost, item-specific ordering costs, and deterministic holding costs. The standard relaxation replaces the true density of the union of order epochs by the largest individual frequency. Three items can coordinate in only three structural ways: all periods are commensurable, exactly one pair is commensurable, or all pairs are incommensurable. The oracle solves the boundary regimes exactly, enumerates a finite set of primitive fully coordinated calendars, and proves that every omitted calendar is more expensive than the incumbent. For the strongest rational witness, the optimum uses period multipliers `(3, 3, 2)` and scale `92726/168045`. The certificate enumerates 49,981 primitive triples. The smallest closing cutoff is 21. Exact integer and rational comparisons—not floating-point optimisation—establish the winner and the strict tail margin. The witness comes from an exact five-policy equalisation family. Four neighbouring calendars tie on an explicit rational surface; the selected rational point lies strictly on the `(3, 3, 2)` side. A stationary calculation inside that family guided the selection but is not presented as the exact three-item constant. The lower bound also limits what can be proved using this relaxation alone: no universal inequality comparing exact cost directly with the max-frequency relaxation can use a factor below the certified ratio. Better algorithms may still use stronger lower bounds or different analyses. A simpler witness gives the exact ratio `9586/8355` at `(3, 2, 2)`. It is retained as a transparent audit case. ## Why the exact policy differs The relaxed solution uses frequencies $$ \left(\frac{56015}{92726},\ \frac{153463}{200000},\ 1\right). $$ The exact coordinated solution uses $$ \left(\frac{56015}{92726},\ \frac{56015}{92726},\ \frac{168045}{185452}\right). $$ The exact policy slows the second and third streams so that their events can realize a better joint calendar. Four neighbouring calendars are extremely close in cost. Sixteen additional exact certificates prove two conservative inner boxes under simultaneous relative holding-cost perturbation only: the headline witness has a relative radius of about `9.64e-11`, while the slightly weaker v0.2 witness has a radius of about `1.22e-6`. The five near-tied calendars are frozen as synthetic regression fixtures. A fitted logistics instance must regenerate and verifier-check its own candidate set; the radius contrast is a warning against blind execution of one nominal winner. ## A logistics use that respects the evidence The oracle can serve as a triplet stress test for three SKUs that genuinely share a fixed event. It can answer three useful questions: 1. Is the max-frequency lower bound materially optimistic for this calibrated triplet? 2. Does exact coordination change the accepted deterministic periodic plan? 3. Which instance-specific, verifier-checked near-optimal calendars should be passed to a richer simulation? It can also serve as an exact regression fixture for ERP, APS, optimisation, and replenishment software. A system claiming a lower cost under the identical model has either changed the semantics or failed the benchmark. The oracle is not a complete logistics controller. Minimum order quantities, pack sizes, container fill, nonlinear transport tariffs, stochastic demand, non-zero lead time, service levels, shelf life, supplier calendars, and pointwise dock capacity need an additional operational model. Case and algorithmic research shows why these layers matter; none of it validates this candidate’s field performance. ## Evaluation gates The release supplies a staged protocol: 1. validate that the three products share a meaningful incremental event; 2. reconcile order, receipt, inventory, service, expedite, and cost data; 3. run the exact stress test on every eligible triplet, retaining null findings; 4. add uncertainty and omitted constraints in simulation or a digital twin; 5. run a shadow schedule and review exceptions with planners; 6. pilot only under a preregistered concurrent comparison with service stop rules; 7. scale only if service remains acceptable and total net value is positive. Outcomes include total landed cost, inventory, fill rate, stockouts, expedites, transport utilization, dock peaks, overtime, planner effort, and implementation cost. Before/after company observations without a defensible counterfactual remain associations, not causal effects. ## Assurance boundary The same-producer solver-independent checker recomputes the relaxation, boundary values, primitive-triple set, fixed-pattern costs, tail inequalities, and gap. Version 0.3.1 also rebuilds the active-envelope analysis, complete regression portfolio documents, sixteen robustness vertices, and two holding-box certificates. It rejects inconsistent redundant exact values, floating-point instance literals, incomplete pair partitions, false proof-bound flags, inconsistent tail or boundary data, resource-intensive expression forms, package-escape paths, and structural mutations. The checker was nevertheless written in the same workflow and shares parsing and exact-algebra utilities. Its success is producer-side consistency evidence, not independent reproduction. The general finite-generator and tail proof also remain prose mathematics supported by exact tests rather than a proof assistant. ## What remains open - the exact three-item cap constant; - the exact unrestricted multi-item gap; - a universal three-item upper certificate; - direct arbitrary coupled-resource input in the executable oracle; - unaffiliated rerun, reimplementation, formalisation, and specialist review; - a representative distribution of gaps in real logistics triplets; - preregistered evidence that exact certification changes plans or creates net operational value. ## Release boundary This is an additive successor and does not alter the immutable two-item release. The immutable [GitHub prerelease](https://github.com/ipitchford/certified-three-item-jrp-gap/releases/tag/v0.3.1-candidate) and published [Zenodo version DOI 10.5281/zenodo.22057665](https://doi.org/10.5281/zenodo.22057665) expose six byte-identical assets under unauthenticated public readback. The canonical Evidence Press HTML, machine record, media, graph and protocol surfaces also passed exact post-deploy readback after bounded cache convergence. These checks establish identity and availability, not mathematical correctness, independent assurance, operational validation, savings, or field impact. ## Open directions for follow-up research - Obtain an unaffiliated mathematical audit of the finite primitive-triple generator, global tail theorem, active-envelope elimination and holding-box vertex argument. - Independently rerun the exact DOI archive and reimplement the oracle and checker without importing the released package. - Formalize the generator, totality, tail, monotonicity and vertex lemmas in a proof assistant. - Determine the exact three-item independent-cap constant and construct a universal upper certificate over the parameter space. - Extend the executable oracle to directly accept arbitrary coupled resource rows while preserving exact assurance. - Measure the distribution of relaxation gaps in representative eligible logistics triplets and retain null or ineligible cases. - Run preregistered simulation, shadow and field evaluations with service stop rules, implementation costs and a defensible comparator before any savings or impact claim. ## Research process and reusable methods This is prospective process metadata. It records the intended handoff and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:certified-three-item-jrp-gap - Attempt receipts: ep-attempt:certified-three-item-jrp-gap: published / positive; measured-partial; active human minutes missing; compute minutes missing; rework minutes missing; assurance endpoint measured-partial. Work ledger: https://evidencepress.org/api/work-ledger.json - Intended aims: science, productivity - Artifact roles: research-output, evidence-assessment, method-demonstration, communication - Decision object: certificate — An exact rational three-item gap certificate, fixed-item monotonicity implication, exact holding-box comparison, verifier-checked instance-specific candidate-set builder, and staged decision rule for whether a logistics triplet warrants operational evaluation. Scope: Three-item stationary common-origin periodic policies with rational independent frequency caps; logistics use is diagnostic until omitted constraints and outcomes are evaluated separately. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Exact regime stitching (regime-stitching); Adversarial scientific controls (adversarial-controls); Explicit research-lineage reuse (research-lineage-reuse); Certified decision margins (certified-decision-margin); Staged adoption and impact evaluation (staged-adoption-evaluation); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: discovery, assurance, publication, translation - Semantic bridge: partial — Common and item-specific model coefficients map to incremental replenishment-event, handling, and deterministic holding costs only after domain validation. The exact oracle maps to a triplet benchmark, not directly to executed orders. Remaining risks: Frequency caps may be invalid proxies for pointwise transport or dock capacity.; Stochastic demand, lead times, service, order rounding, minimum quantities, and nonlinear tariffs are outside the certificate.; No representative company dataset or field comparison has been evaluated.. - Human judgement gates: Confirm that the three items share an incremental common event and every coefficient has defensible operational semantics.; Keep exact synthetic lower-bound optimism separate from savings or field performance.; Require simulation and shadow evaluation before any order execution.; Require preregistered service stop rules and a defensible comparator before causal or net-value language.; Confirm Anonymous scholarly attribution, licences, immutable identifiers, and release authority before publication. - Next assurance action: Obtain unaffiliated mathematical review and clean-room replay; for any eligible fitted triplet, regenerate its candidate set and run the preregistered logistics stress-test and shadow protocol. - Claim ceiling: A producer-side unrefereed exact-computation candidate for a bundled three-item lower bound and a proposed logistics evaluation route; not the exact global constant, independent assurance, operational validation, savings, or field impact. - Aim-scoped impact evidence: - productivity: NO_IMPACT_EVIDENCE — Better replenishment plans or lower net logistics cost in Eligible three-SKU clusters with a shared replenishment event; design none; comparator No incumbent-plan comparison has been run.; estimand No plan-change rate, service effect, avoided loss, implementation cost, or net value has been estimated.; no real-world effect evidence asserted - science: NO_IMPACT_EVIDENCE — More reliable mathematical understanding of multi-item joint-replenishment relaxation gaps in Research on continuous-time common-origin fixed-interval joint replenishment with rational independent frequency caps; design none; comparator No matched conventional research or theorem-review workflow was registered.; estimand No effect on discovery time, proof quality, error rate, review effort, method uptake, citation or field outcomes was estimated.; no real-world effect evidence asserted - Parent handoffs: extends-result certified-two-item-jrp; inherited claim: The parent supplies the exact two-item interval, resource-to-cap reduction, and certificate-first model boundary.; inherited ceiling: The parent is an unrefereed producer-side candidate. Reuse is dependency evidence, not independent confirmation. ## Verification status Identity-bearing Anonymous unrefereed successor candidate at PASS_WITH_NOTES after bounded Editor closure. Version 0.3.1 passes final-identity ordinary and optimized 21-test same-producer replay, the 61-file generated ledger, final fresh-extraction replay, and public GitHub Actions replay. The immutable GitHub prerelease and Zenodo version DOI expose six byte-identical assets under public readback. The canonical Evidence Press HTML, paper.json, audio, art, graph, operating ledgers and protocol subtree passed exact post-deploy readback after one bounded custom-domain cache-convergence retry. Unaffiliated rerun, independent reimplementation, formal verification, external specialist review, external editorial peer review, operational validation, and impact evidence remain unassessed. ## Sources and related work - Agent collective and I. Pitchford (2026). Exact two-item resource-constrained joint replenishment and its independent-cap relaxation gap. Evidence Press / Zenodo, v1.2.0-candidate. - Schulz, A. S., and Telha, C. (2024). Integer factorization: Why two-item joint replenishment is hard. Operations Research 72(3), 1192–1202. - Segev, D. (2025). The continuous-time joint replenishment problem: epsilon-optimal policies via pairwise alignment. Management Science 71(5), 4183–4197. - Segev, D. (2026; online 2025). Improved approximation guarantees for joint replenishment in continuous time. Mathematics of Operations Research 51(2), 1683–1712. - Cohen-Hillel, T., and Yedidsion, L. (2018). The periodic joint replenishment problem is strongly NP-hard. Mathematics of Operations Research 43(4), 1269–1289. - Creemers, S., and Stephan, T. (2026). Special Cases of the Joint Replenishment Problem: A Mathematical Analysis. SSRN 6674418. - Suemitsu, I., Miyashita, N., Hosoda, J., Shimazu, Y., Nishikawa, T., and Izui, K. (2024). Integration of sales, inventory, and transportation resource planning by dynamic-demand joint replenishment problem with time-varying costs. Computers & Industrial Engineering 188, 109922. - Bojinov, I., Simchi-Levi, D., and Zhao, J. (2023). Design and analysis of switchback experiments. Management Science 69(7), 3759–3777.