{ "schemaVersion": "1.6", "slug": "seven-cycle-fourth-power-upper112", "title": "A tighter bound for four seven-cycles", "shortTitle": "Four seven-cycles: at most 112", "url": "https://evidencepress.org/releases/seven-cycle-fourth-power-upper112/", "oneLine": "An exact finite exclusion narrows the independence-number interval to 108–112; the precise answer remains open.", "abstract": "This computer-assisted candidate excludes every 113-word independent set in the fourth strong power of the seven-cycle. A complete cubic catalogue, exhaustive band-allocation checks and integer completion certificates give 108 ≤ α(C₇^⊠4) ≤ 112, improving the historical upper endpoint 115 recorded in 2019. Auxiliary results classify maximum-square boundary traces, exclude the zero-saturation-excess balanced 112 case, determine the square-factor gadget maximum 106 and certify feasibility of a specified two-block moment relaxation at 109–112. These scoped results do not determine the exact independence number or Shannon capacity.", "datePublished": "2026-10-05", "dateModified": "2026-10-05", "version": "v0.1.0-candidate", "doi": "10.5281/zenodo.23160075", "doiUrl": "https://doi.org/10.5281/zenodo.23160075", "conceptDoi": "10.5281/zenodo.23160074", "pdfUrl": "https://github.com/ipitchford/seven-cycle-fourth-power-upper112/releases/download/v0.1.0-candidate/paper.pdf", "altPdfUrl": "https://zenodo.org/records/23160075/files/paper.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23160075", "repoUrl": "https://github.com/ipitchford/seven-cycle-fourth-power-upper112", "releaseUrl": "https://github.com/ipitchford/seven-cycle-fourth-power-upper112/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/seven-cycle-fourth-power-upper112/index.md", "bibtexUrl": "https://evidencepress.org/releases/seven-cycle-fourth-power-upper112/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/seven-cycle-fourth-power-upper112.mp3?v=39a74e252c", "imageUrl": "https://evidencepress.org/assets/og/seven-cycle-fourth-power-upper112.png?v=bea84d5e12", "coverArtUrl": "https://evidencepress.org/assets/art/seven-cycle-fourth-power-upper112.svg?v=34b136be59", "media": [ { "type": "video", "url": "https://youtu.be/nD60xdkyXJ4", "name": "A Tighter Bound For Four Seven-Cycles", "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence." }, { "type": "audio", "url": "https://evidencepress.org/assets/audio/seven-cycle-fourth-power-upper112.mp3?v=39a74e252c", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/seven-cycle-fourth-power-upper112.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed computer-assisted mathematical candidate. Producer-side exact replay and internal semantic checks are distinct from unaffiliated reproduction, formal verification and external peer review." }, "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/23160075", "note": "Immutable GitHub prerelease and published Zenodo DOI 10.5281/zenodo.23160075 verified against frozen assets. Full downloads matched for the manuscript formats and checksum list; provider digests and sizes matched for the evidence bundle." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Fresh complete main replay passed all 1,016 classes in both A and N, with prerequisite checks and actual successful exits. All scoped auxiliaries passed separately. Frozen package checks, normal/optimized runs and semantic negative controls passed. Producer-side verification, not unaffiliated reproduction." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "Not established by this producer-side workflow." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Not established by this producer-side workflow." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "Not established by this producer-side workflow." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Not established by this producer-side workflow." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "Not established by this producer-side workflow." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Portable Python and C++ source and exact certificates; no unaffiliated replay established." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "Codex mathematical analysis, code and exact replay", "Codex review response and publication preparation" ], "humanRole": "The user authorized the research and publication and supplied an AI-assisted review; no human scholarly contribution is inferred.", "disclosure": "Anonymous is the scholarly creator; Evidence Press is the publisher. Internal checks and the supplied AI-assisted review are not authenticated external specialist review." }, "attributionBoundary": null, "problem": { "name": "What is the independence number of the fourth strong power of the seven-cycle?", "url": "https://arxiv.org/html/1808.07438v2" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Derived from the revised manuscript and dated claim-to-source audit. Scientific replay, archive publication and canonical website readback are recorded separately.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Shannon capacity", "strong graph product", "seven-cycle", "independence number", "computer-assisted proof", "exact certificates" ], "keyResults": [ "108 ≤ α(C₇^⊠4) ≤ 112; the exact value remains open.", "Consequently α(C₇^⊠5) ≤ 392.", "All 1,003,520 oriented maximum-square traces classified by their largest cubic extension.", "At size 112 with no 33-band, zero saturation excess is excluded.", "Maximum 106 within two square-factor Gao gadgets, including nonmaximum seeds.", "The specified two-block invariant degree-three relaxation is feasible at 109–112." ], "reviews": [], "evidencePackage": "Complete cubic census; exhaustive adjacent and nonadjacent band allocations; additions-only RUP checks and exact integer completion inequalities. Acceptance receipts identify the complete replay scope.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Vesel and Žerovnik (2002). Improved lower bound on the Shannon capacity of C7. Original 108 construction; publisher abstract checked, full text unavailable in this revision.", "url": "https://doi.org/10.1016/S0020-0190(01)00229-0", "doi": "10.1016/S0020-0190(01)00229-0" }, { "citation": "Polak and Schrijver (2019). New lower bound on the Shannon capacity of C7 from circular graphs. Table 1 records the historical fourth-power bounds 108–115.", "url": "https://doi.org/10.1016/j.ipl.2018.11.006", "doi": "10.1016/j.ipl.2018.11.006" }, { "citation": "Itty, Rosin, Carstensen and Reichman (2026), v2. Improved lower bounds for the Shannon capacity of odd cycles.", "url": "https://arxiv.org/abs/2607.21517v2", "doi": "10.48550/arXiv.2607.21517" }, { "citation": "Gao (2026), v1. A recursive construction improving the lower bound on the Shannon capacity of C7. The gadget construction examined here.", "url": "https://arxiv.org/abs/2607.27869v1", "doi": "10.48550/arXiv.2607.27869" }, { "citation": "Buys, Polak and Zuiddam (2026), v1. Lean-verified lower bounds for the Shannon capacity of odd cycles. Valid-tuple reformulation and higher-dimensional iteration.", "url": "https://arxiv.org/abs/2607.29681v1", "doi": "10.48550/arXiv.2607.29681" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:seven-cycle-fourth-power-upper112", "attemptIds": [ "ep-attempt:seven-cycle-fourth-power-upper112-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "Exact finite exclusion of every 113-word independent set.", "scope": "Fourth strong power of C7; lower witness 108, upper bound 112, exact optimum unknown." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Band normalization, exact trace relations, physical allocation coverage and integer completion bounds connect graph codes to the checked finite instances.", "remainingRisks": [ "Written reductions and checkers are not formally verified.", "Complete census and encoding semantics remain part of the trusted mathematical argument.", "Bounded source search does not establish priority." ] }, "humanJudgmentGates": [ "Publication authorized by the user.", "Specialists can assess the semantic reductions, exhaustive coverage and literature comparison." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "specialistReview", "formalVerification", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Unaffiliated full replay and audit of the graph-to-encoding reductions.", "claimCeiling": "Producer-checked unrefereed candidate; exact independence number remains open." }, "impactClaims": [ { "id": "seven-cycle-fourth-power-upper112-no-impact", "aim": "science", "outcome": "An inspectable finite upper-bound certificate and scoped auxiliary results", "setting": "AI-assisted mathematical research and publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched-assurance comparison is registered.", "estimand": "No discovery-speed, cost, adoption or impact effect is 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:seven-cycle-fourth-power-upper112-publication", "status": "result-ready", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Remaining assurance and publication after this registration. Earlier discovery and preparation remain outside the measured interval.", "forecast": { "frozenAt": "2026-10-05T10:59:53.405Z", "procedureClass": "remaining-full-candidate-publication", "targetOutcome": "Reviewed upper112 candidate, full acceptance replay, identical public evidence assets, standard media and canonical Evidence Press readback.", "expectedActiveMinutes": 150, "plausibleLowMinutes": 120, "plausibleHighMinutes": 210, "expectedUnattendedWaitMinutes": 120, "referenceClass": { "label": "Full Evidence Press candidate route", "sampleSize": 0, "basis": "Prospective judgment forecast following the current route planning prior; not a measured speed claim." }, "fermiComponents": [ { "component": "Final exact-version package and assurance gates", "count": 1, "lowMinutesPerUnit": 35, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 65, "basis": "Existing certificates; full main and saturation acceptance replays requested by review." }, { "component": "Archive providers, page, required media and guarded site cycles", "count": 1, "lowMinutesPerUnit": 85, "centralMinutesPerUnit": 105, "highMinutesPerUnit": 145, "basis": "New-slug full-candidate workflow with configured standard tools." } ], "probabilityPositiveSignal": 0.9, "probabilityTargetClosure": 0.85, "probabilityHorizonMinutes": 300, "assumptions": [ "The candidate proofs survive final exact-version checks.", "Configured publication providers remain available and existing site records are preserved." ], "reforecastTriggers": [ "A load-bearing proof, attribution or licensing defect changes the payload.", "Provider failure, archive mismatch or site preservation defect delays delivery." ], "stopRule": "Timing is planning telemetry. Continue authorized work unless an unresolved integrity, rights or provider-access defect blocks publication; never promote an unchecked claim to meet the forecast." }, "outcome": null }, "missingnessReason": null } ] } }