{ "schemaVersion": "1.6", "slug": "triangulated-graph-product-picard", "title": "The Picard group of a triangulated product of graphs", "shortTitle": "The Picard group of a graph product", "url": "https://evidencepress.org/releases/triangulated-graph-product-picard/", "oneLine": "An integral cut–cycle argument classifies ridge and full tropical Picard groups, with an explicit multigraph convention and commuting diagonal flips.", "abstract": "This unrefereed candidate gives a uniform integral classification for every diagonal triangulation of a product of finite connected loopless unit-edge graphs. Under branch-local Cartier conventions the ridge group is Z^(2+gh) plus the two finite critical groups; the full polyhedral group has the two Jacobian tori and the same free rank. It proves Lazar's product formula for simple factors and establishes the stated multigraph extension. Integer lifting, primitivity, factor injection, commuting flips and primitive cycle classes are explicit. Classical foundations and an inherited catalogue reduction are credited. Finite diagnostics are not the universal proof, and historical priority remains unestablished.", "datePublished": "2026-09-08", "dateModified": "2026-09-08", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.22656789", "doiUrl": "https://doi.org/10.5281/zenodo.22656789", "conceptDoi": "10.5281/zenodo.22656788", "pdfUrl": "https://github.com/ipitchford/triangulated-graph-product-picard/releases/download/v0.1.0-candidate/SOLUTION.pdf", "altPdfUrl": "https://zenodo.org/records/22656789/files/SOLUTION.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22656789", "repoUrl": "https://github.com/ipitchford/triangulated-graph-product-picard", "releaseUrl": "https://github.com/ipitchford/triangulated-graph-product-picard/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/triangulated-graph-product-picard/index.md", "bibtexUrl": "https://evidencepress.org/releases/triangulated-graph-product-picard/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/triangulated-graph-product-picard.mp3?v=e2b26453e5", "imageUrl": "https://evidencepress.org/assets/og/triangulated-graph-product-picard.png?v=a3161649b7", "coverArtUrl": "https://evidencepress.org/assets/art/triangulated-graph-product-picard.svg?v=adc8d6c712", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/triangulated-graph-product-picard.mp3", "name": "Plain-English audio briefing", "description": "AI-generated communication summary, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/triangulated-graph-product-picard.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed uniform written proof candidate under explicit branch-local conventions. Producer exact diagnostics and internal AI review do not establish unaffiliated whole-proof acceptance, formal verification, historical priority or impact." }, "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/22656789", "note": "Exact public GitHub and Zenodo assets; availability does not establish correctness." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Exact finite diagnostics and fresh-extraction release checks; optimized verifier refuses disabled assertions.", "evidenceUrl": "https://github.com/ipitchford/triangulated-graph-product-picard/blob/v0.1.0-candidate/verification.json" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "Producer checks and internal AI review do not establish this dimension; supplied private review is not authenticated unaffiliated assurance." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Producer checks and internal AI review do not establish this dimension; supplied private review is not authenticated unaffiliated assurance." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "Producer checks and internal AI review do not establish this dimension; supplied private review is not authenticated unaffiliated assurance." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Producer checks and internal AI review do not establish this dimension; supplied private review is not authenticated unaffiliated assurance." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "Producer checks and internal AI review do not establish this dimension; supplied private review is not authenticated unaffiliated assurance." }, { "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/triangulated-graph-product-picard/blob/v0.1.0-candidate/requirements.txt", "note": "Pinned SymPy and recorded replay; not a hermetic environment or independent reconstruction." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "OpenAI Codex under human direction", "Python and SymPy", "Pandoc and pdfLaTeX" ], "humanRole": "Problem selection, supplied review, research direction and publication authority; scholarly creator Anonymous.", "disclosure": "AI-assisted proof development, diagnostics, bounded source screening, internal editorial review and publishing. No unaffiliated assurance inferred." }, "attributionBoundary": null, "problem": { "name": "AIM chip-firing Problem 31; Lazar product formula", "url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Internal mathematical assurance assessment is separate from canonical publication readback.", "grandfatheredAtSchemaVersion": null, "keywords": [ "tropical Picard group", "graph products", "chip-firing", "integral lattices", "critical group" ], "keyResults": [ "Uniform integral ridge Picard classification with no extra mixed torsion.", "Full polyhedral Picard group and its Jacobian tori.", "Commuting integral diagonal flips.", "Primitive cycle-product normalization." ], "reviews": [], "evidencePackage": "Uniform written proof, exact finite producer diagnostics, file manifest, recovered catalogue antecedent and five internal AI editorial reports. https://github.com/ipitchford/triangulated-graph-product-picard/blob/v0.1.0-candidate/EDITORIAL_DECISION.md", "openProblems": [ "Unaffiliated mathematical review and independent implementation.", "Specialist historical-priority audit.", "General-purpose integral normal-form software.", "Extensions to arbitrary lengths and excluded graph classes." ], "mathObjects": [], "relatedWorks": [ { "citation": "Lazar (2017), A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group, Conjecture 1.2 and Proposition 4.5.", "url": "https://doi.org/10.37236/5533", "doi": "10.37236/5533" }, { "citation": "Cartwright (2015), Combinatorial tropical surfaces, Section 3 and Example 3.2.", "url": "https://arxiv.org/abs/1506.02023", "doi": "10.48550/arXiv.1506.02023" }, { "citation": "Cartwright (2020; online 2019), Tropical complexes.", "url": "https://doi.org/10.1007/s00229-019-01148-3", "doi": "10.1007/s00229-019-01148-3" }, { "citation": "AIM chip-firing workshop (2013), Problem 31.", "url": "https://aimath.org/pastworkshops/chipfiringproblems.pdf", "doi": null }, { "citation": "UnsolvedMath, catalogue record AIM-COMBINATORICS-0121, imported attempt 1; snapshot archived with explicit provenance limits.", "url": "https://huggingface.co/datasets/ulamai/UnsolvedMath", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:triangulated-graph-product-picard", "attemptIds": [ "ep-attempt:triangulated-graph-product-picard-assurance-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "communication" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "reusable-method", "description": "Integral compatibility lattice and ridge/full Picard classification.", "scope": "Connected loopless unit-edge graph products, branch-local Cartier convention, all diagonal choices." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Local branch lifts and endpoint incidence equations identify the integer Cartier lattice; primitive principal diagonals control torsion; the cited sheaf comparison supplies the full-group bridge.", "remainingRisks": [ "Uniform written proof awaits unaffiliated scrutiny.", "Finite diagnostics cannot establish universal correctness.", "The global-star convention on parallel edges is a different object.", "Equivalent earlier results are not excluded." ] }, "humanJudgmentGates": [ "Audit integral local-to-global sufficiency and sheaf comparison.", "Assess antecedents and historical priority.", "Preserve branch conventions, attribution and rights." ], "parentLinks": [ { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://doi.org/10.37236/5533", "inheritedClaim": "Product construction, conjectural formula and factor/tree antecedents.", "inheritedAssuranceCeiling": "Earlier theorem does not validate this general closure or altered multigraph convention." }, { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://arxiv.org/abs/1506.02023", "inheritedClaim": "Ridge/full sheaf comparison and exponential sequences.", "inheritedAssuranceCeiling": "Classical framework, not external review of the present specialization." }, { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://huggingface.co/datasets/ulamai/UnsolvedMath", "inheritedClaim": "AIM-COMBINATORICS-0121 attempt 1 diagonal exact-sequence and primitivity reduction; hash-bound snapshot in package.", "inheritedAssuranceCeiling": "Unreviewed catalogue input with explicit upstream version limits and corrected claims." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Unaffiliated mathematical audit, independent implementation and primary antecedent comparison.", "claimCeiling": "Unrefereed uniform written proof candidate under explicit branch-local conventions. Producer exact diagnostics and internal AI review do not establish unaffiliated whole-proof acceptance, formal verification, historical priority or impact." }, "impactClaims": [ { "id": "triangulated-graph-product-picard-no-impact", "aim": "science", "outcome": "Inspectable integral product classification candidate", "setting": "Producer-coordinated publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched comparison", "estimand": "No acceleration or downstream effect estimated", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/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:triangulated-graph-product-picard-assurance-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Remaining review repairs, internal assurance and first canonical publication only. Completed discovery, supplied review, initial source reading and extraction predate this registration and are excluded; no historical research clock is reconstructed. Final ledger reseal and closeout are measured separately.", "forecast": { "frozenAt": "2026-09-08T08:24:24.581Z", "procedureClass": "compact-proof-full-candidate", "targetOutcome": "Review repairs, deterministic baseline, five-role internal acceptance, immutable GitHub and Zenodo assets, and first canonical Evidence Press publication.", "expectedActiveMinutes": 150, "plausibleLowMinutes": 120, "plausibleHighMinutes": 210, "expectedUnattendedWaitMinutes": 30, "referenceClass": { "label": "Full-candidate skill prior", "sampleSize": 0, "basis": "Procedural benchmark, not measured matched evidence." }, "fermiComponents": [ { "component": "Review repairs and sources", "count": 1, "lowMinutesPerUnit": 25, "centralMinutesPerUnit": 30, "highMinutesPerUnit": 40, "basis": "Minor attribution and metadata revisions." }, { "component": "Package baseline and internal review", "count": 1, "lowMinutesPerUnit": 35, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 65, "basis": "Compact proof with fast exact replay." }, { "component": "Public identity and media", "count": 1, "lowMinutesPerUnit": 30, "centralMinutesPerUnit": 35, "highMinutesPerUnit": 50, "basis": "Existing publishing tools." }, { "component": "Site seals and canonical readback", "count": 1, "lowMinutesPerUnit": 30, "centralMinutesPerUnit": 40, "highMinutesPerUnit": 55, "basis": "Mandatory new-slug two-deployment cycle." } ], "probabilityPositiveSignal": 0.95, "probabilityTargetClosure": 0.85, "probabilityHorizonMinutes": 240, "assumptions": [ "No fatal proof or rights defect.", "Authorized publishing providers remain available." ], "reforecastTriggers": [ "Scientific defect.", "Provider failure or concurrent source conflict." ], "stopRule": "Fail closed on integrity or authority failure; time thresholds are telemetry and do not terminate authorized publication." }, "outcome": { "measuredAt": "2026-09-08T08:50:44.632Z", "calendarElapsedMinutes": 26, "activeAgentMinutes": 2, "activeHumanMinutes": null, "computeMinutes": null, "unattendedWaitMinutes": 3, "blockedMinutes": 0, "reworkMinutes": 0, "agentRuns": 6, "maxParallelAgents": 4, "modelTurns": 107, "deduplicatedModelTokens": 12125855, "uncachedInputTokens": 417577, "researchCycles": { "positive": 0, "negative": 0, "inconclusive": 0 }, "falsificationGatesRun": 1, "candidateArchitecturesTested": 0, "candidateArchitecturesRejected": 0, "substantiveReviewRounds": 1, "p0Findings": 0, "p1Findings": 0, "prepublicationClaimCorrections": 3, "resultState": "target-closed", "resultSummary": "Review-repaired candidate, exact complete archive, internal editorial acceptance, public GitHub/Zenodo byte readback, standard media and first guarded canonical publication completed. Final ledger reseal follows outside this measurement cut. No external mathematical assurance or discovery-speed claim. The finite diagnostic and negative-control suite is one adversarial gate, not a new research cycle.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -148, "forecastRatio": 0.013333333333333334, "withinForecastInterval": false, "varianceReason": "Active time is the union of instrumented Reasoning and AgentMessage intervals; unattended time includes instrumented compaction only. Both are observable lower bounds, not complete labour/wait totals. Tool-call generation and uninstrumented provider waits are excluded. Deduplicated response IDs remove fork copies. Cached inputs remain part of raw tokens. No historical discovery clock is inferred. Offline gaps and unnecessary administrative pauses are not classified as external blocking. Rework is an uninstrumented zero lower bound, not absence of repairs: the simple-factor versus branch-local scope, the full Picard quotient, and inherited catalogue provenance were clarified in response to the supplied review, alongside minor exposition fixes. No new mathematical proof claim was introduced. Incomplete active-time coverage cannot support a speed-gain claim; discovery is excluded.", "missingFields": [ { "field": "activeHumanMinutes", "reason": "No contemporaneous human active-time instrument." }, { "field": "computeMinutes", "reason": "No complete scoped substantive-computation wall clock; ordinary CI/build time is not research compute." } ] } }, "missingnessReason": null } ] } }