{ "schemaVersion": "1.6", "slug": "linear-nonbacktracking-jordan-growth", "title": "Linear Jordan-block growth in non-backtracking graphs", "shortTitle": "Jordan blocks grow linearly", "url": "https://evidencepress.org/releases/linear-nonbacktracking-jordan-growth/", "oneLine": "A constructive proof candidate shows that non-backtracking Jordan blocks can grow linearly with graph order, even under one fixed maximum-degree bound.", "abstract": "For the counting non-backtracking matrix of a finite simple undirected leafless n-vertex graph, this anonymous unrefereed theorem candidate proves that the extremal complex Jordan-block size f(n) is Theta(n). Explicit bounds are max(1,floor((n-1)/131072)-3)<=f(n)<=2n for n>=3. Linear growth also holds at maximum degree at most 393217, with lower bound max(1,floor(n/12714016)). A banded determinant valuation, fixed-dimensional rational encoding and signed-fiber graph realization provide the arbitrary-parameter proof. Exact finite computations are supporting controls, not the proof of asymptotic growth. AIM Problem 1.3(1) is addressed; part (2), optimal constants, exact extremal values and historical priority are not claimed.", "datePublished": "2026-09-05", "dateModified": "2026-09-05", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.22375005", "doiUrl": "https://doi.org/10.5281/zenodo.22375005", "conceptDoi": "10.5281/zenodo.22375004", "pdfUrl": "https://github.com/ipitchford/linear-nonbacktracking-jordan-growth/releases/download/v0.1.0-candidate/linear-nonbacktracking-jordan-growth-0.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/22375005/files/linear-nonbacktracking-jordan-growth-0.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22375005", "repoUrl": "https://github.com/ipitchford/linear-nonbacktracking-jordan-growth", "releaseUrl": "https://github.com/ipitchford/linear-nonbacktracking-jordan-growth/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/linear-nonbacktracking-jordan-growth/index.md", "bibtexUrl": "https://evidencepress.org/releases/linear-nonbacktracking-jordan-growth/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/linear-nonbacktracking-jordan-growth.mp3?v=655c52b7dd", "imageUrl": "https://evidencepress.org/assets/og/linear-nonbacktracking-jordan-growth.png?v=05fa5ceaab", "coverArtUrl": "https://evidencepress.org/assets/art/linear-nonbacktracking-jordan-growth.svg?v=e7c3664a5f", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/linear-nonbacktracking-jordan-growth.mp3", "name": "Audio briefing — linear Jordan-block growth", "description": "Plain-English synthetic narration; communication, not additional evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/linear-nonbacktracking-jordan-growth.txt" }, { "type": "video", "url": "https://youtu.be/VciR9OZUfvE", "name": "Defeating Diagonalizability The Constructive Proof of Linear Jordan Growth", "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." } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous AI-assisted unrefereed theorem candidate. Five producer-coordinated editorial roles completed one round; the sole required Minor exposition repair is closed and the final internal decision is Accept/PASS_WITH_NOTES. No external validation, formal verification, exhaustive novelty or historical priority is asserted." }, "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/22375005", "note": "All five assets downloaded from both GitHub and published Zenodo match local SHA-256 values. DOI issuance and byte identity establish availability, not correctness." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Producer fresh-extraction normal and optimized finite checks pass. The arbitrary-parameter proof is written, not formally certified by these tests. Hosted Linux Python 3.11 and 3.13 normal/optimized checks passed.", "evidenceUrl": "https://github.com/ipitchford/linear-nonbacktracking-jordan-growth/actions/runs/33983850624" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "The supplied separate-host report is not authenticated unaffiliated evidence." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Supplied bounded reconstruction and coordinated roles do not establish independent theorem reconstruction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No end-to-end proof-assistant verification." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "No authenticated 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": "Producer-coordinated editorial approval is not journal 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/linear-nonbacktracking-jordan-growth/blob/v0.1.0-candidate/ENVIRONMENT.txt", "note": "Pinned arithmetic libraries and exact inputs; no hermetic OS or byte-reproducible PDF toolchain." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "OpenAI Codex under human direction", "Python, SymPy and python-flint exact controls", "Pandoc and pdfLaTeX" ], "humanRole": "Research direction, candidate selection, supplied review, scope and full-publication authorization; scholarly creator remains Anonymous.", "disclosure": "AI assistance contributed proof development, code, exposition, review repair and producer-coordinated editorial review. Supplied review observations are not authenticated unaffiliated validation." }, "attributionBoundary": null, "problem": { "name": "AIM Problem 1.3(1): non-backtracking Jordan-block growth", "url": "https://www.unsolvedmath.com" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Research-package assurance assessed at 2026-09-05T18:27:02.545Z against immutable payload 49e763e511a4cbff260b87c7f53e6cecb2d3afaa, internal editorial closure, hosted CI and ten matching public downloads. Canonical site readback is a separate pending event.", "grandfatheredAtSchemaVersion": null, "keywords": [ "non-backtracking graphs", "Jordan blocks", "spectral graph theory", "quadratic pencils", "unrefereed candidate" ], "keyResults": [ "For n>=3, max(1,floor((n-1)/131072)-3)<=f(n)<=2n, hence f(n)=Theta(n).", "At maximum degree at most 393217, block sizes still grow linearly: f(n)>=max(1,floor(n/12714016)).", "Constructed long blocks occur at the fixed conjugate eigenvalues plus or minus 512i.", "The graph family is simple, connected and leafless; zero-valued padding reaches every larger order." ], "reviews": [], "evidencePackage": "Eight-page arbitrary-parameter proof, exact construction code, historical r-through-20 pencil controls, fresh normal and optimized r-through-12 replay, three fully enumerated small graph embeddings, published baselines, padding, 50,104 bounded determinant permutations and ten scientific negative-control instances. Complete manifests, source note, review response, licences and separate historical and publication receipts are supplied.", "openProblems": [ "Obtain unaffiliated reconstruction of the valuation, encoding and graph lift.", "Improve constants and determine exact extremal sizes or the smallest useful degree bound.", "Complete broader prior-art and priority assessment.", "Investigate formal verification of the arbitrary-parameter argument; AIM part (2) remains separate." ], "mathObjects": [], "relatedWorks": [ { "citation": "AIM Problem Lists. Spectra of hypergraphs, Problem 1.3(1). Dated source and notation bridge in the package.", "url": "https://aimpl.org/spectralhypergraph/1/", "doi": null }, { "citation": "Glover, C., and Kempton, M. (2021). Some spectral properties of the non-backtracking matrix of a graph. LAA 618, 37–57.", "url": "https://doi.org/10.1016/j.laa.2021.01.022", "doi": "10.1016/j.laa.2021.01.022" }, { "citation": "Heysse, K., Lorenzen, K., and Reinhart, C. (2025). Defective eigenvalues of the non-backtracking matrix. ELA 41, 511–528.", "url": "https://doi.org/10.13001/ela.2025.8835", "doi": "10.13001/ela.2025.8835" }, { "citation": "Takata, K., Mock, A., Notomi, M., and Shinya, A. (2026). Higher-order exceptional points unveiled by nilpotence and mathematical induction. Communications Physics 9, 274.", "url": "https://doi.org/10.1038/s42005-026-02649-w", "doi": "10.1038/s42005-026-02649-w" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:nonbacktracking-jordan-growth", "attemptIds": [ "ep-attempt:nonbacktracking-jordan-growth-assurance-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "method-demonstration", "communication" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "Two-sided linear Jordan-block growth bounds with explicit graph generation and proof bridges.", "scope": "Counting non-backtracking operators of finite simple undirected leafless graphs, n>=3." }, "bottleneckTargeted": [ "assurance", "publication", "translation" ], "semanticBridge": { "state": "explicit", "description": "Banded valuation forces a long pencil chain; uniform rational encoding preserves it; signed fibers realize the adjacency and degree actions; incidence transfer and zero-valued padding preserve the chain in a simple graph.", "remainingRisks": [ "The universal proof is not formally verified or externally reconstructed.", "Finite checks do not establish exact all-parameter block sizes.", "Priority and real-world relevance are not established." ] }, "humanJudgmentGates": [ "Audit the source counting convention and all quantifiers.", "Reconstruct valuation, rational encoding and graph realization.", "Preserve third-party rights and prior attribution.", "Keep internal review and publication separate from external assurance." ], "parentLinks": [ { "relation": "reuses-method", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://doi.org/10.1016/j.laa.2021.01.022", "inheritedClaim": "Reduced non-backtracking matrix and quadratic-pencil framework.", "inheritedAssuranceCeiling": "Standard antecedent framework, not independent confirmation of the present growth construction." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Seek unaffiliated mathematical reconstruction and wider source reconciliation before stronger assurance claims.", "claimCeiling": "Unrefereed constructive theorem candidate for part (1), with producer controls; no part (2), exact extremum, optimal constants, priority, external validation or impact established." }, "impactClaims": [ { "id": "linear-jordan-no-impact", "aim": "science", "outcome": "More inspectable spectral graph constructions", "setting": "Producer-coordinated research and publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched conventional-workflow comparator.", "estimand": "No causal acceleration, reliability, uptake or scientific-impact 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:nonbacktracking-jordan-growth-assurance-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Prospective manuscript and telemetry repairs, internal editorial gate, immutable archiving and canonical publication. Intake and the earlier completed research are excluded; historical research receipts remain separate and are not reconstructed.", "forecast": { "frozenAt": "2026-09-05T18:00:07Z", "procedureClass": "reviewed-theorem-to-full-candidate", "targetOutcome": "Publish the review-repaired anonymous unrefereed Jordan-growth candidate with exact replay, internal editorial disposition, immutable GitHub and Zenodo assets and canonical Evidence Press readback, or preserve the exact blocking gate.", "expectedActiveMinutes": 150, "plausibleLowMinutes": 120, "plausibleHighMinutes": 210, "expectedUnattendedWaitMinutes": 30, "referenceClass": { "label": "Full-candidate route prior", "sampleSize": 0, "basis": "Procedural benchmark, not a matched empirical estimate." }, "fermiComponents": [ { "component": "Review repairs and source audit", "count": 1, "lowMinutesPerUnit": 25, "centralMinutesPerUnit": 30, "highMinutesPerUnit": 45, "basis": "Minor revisions without a supplied mathematical gap." }, { "component": "Deterministic package and five-role editorial gate", "count": 1, "lowMinutesPerUnit": 35, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 60, "basis": "One frozen submission, existing exact controls." }, { "component": "Immutable archive, reader-first page and media", "count": 1, "lowMinutesPerUnit": 35, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 60, "basis": "Existing generators and authenticated GitHub and Zenodo." }, { "component": "Two seal, CI, deployment and readback cycles", "count": 1, "lowMinutesPerUnit": 25, "centralMinutesPerUnit": 30, "highMinutesPerUnit": 45, "basis": "Normal new-slug workflow." } ], "probabilityPositiveSignal": 0.95, "probabilityTargetClosure": 0.85, "probabilityHorizonMinutes": 240, "assumptions": [ "No load-bearing defect emerges", "Required publishing services remain available", "No conflicting rights or prior-art evidence emerges" ], "reforecastTriggers": [ "Material mathematical repair", "Unavailable required service", "Overlapping site changes" ], "stopRule": "Fail closed on unresolved integrity or authority gates. Consolidate at 75 minutes; pause at the workflow's 240-minute hard stop rather than weakening a gate." }, "outcome": { "measuredAt": "2026-09-05T18:40:44Z", "calendarElapsedMinutes": 41, "activeAgentMinutes": 1, "activeHumanMinutes": null, "computeMinutes": null, "unattendedWaitMinutes": 0, "blockedMinutes": 0, "reworkMinutes": 0, "agentRuns": 6, "maxParallelAgents": 4, "modelTurns": 6, "deduplicatedModelTokens": null, "uncachedInputTokens": null, "researchCycles": { "positive": 0, "negative": 0, "inconclusive": 0 }, "falsificationGatesRun": 6, "candidateArchitecturesTested": 0, "candidateArchitecturesRejected": 0, "substantiveReviewRounds": 1, "p0Findings": 0, "p1Findings": 0, "prepublicationClaimCorrections": 0, "resultState": "target-closed", "resultSummary": "Supplied review actioned without mathematical claim changes; five internal editorial roles and exact replay passed. Six existing scientific negative-control classes were rerun, not six new research cycles. Immutable GitHub and Zenodo bytes agree and the first canonical release is live. Earlier discovery is excluded. Final paired replay used 18.6973869 wall seconds and 16.338528 CPU seconds; this is partial process telemetry, not total compute. Active-agent time is a deliberately incomplete lower bound: one whole minute of directly supervised coordinator work in the clock-tool-bounded 18:38:52Z to 18:40:44Z recovery/readback window (112 seconds, rounded down). Earlier generation, other agents, compaction and uninstrumented waits are not reconstructed. Zero unattended and blocked minutes refer only to that observed window, not the complete attempt. Forecast error and ratio therefore compare an incomplete lower bound, not total effort; they must not be used to infer acceleration or update a class prior.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -149, "forecastRatio": 0.006666666666666667, "withinForecastInterval": false, "varianceReason": "Active-agent time is a deliberately incomplete lower bound: one whole minute of directly supervised coordinator work in the clock-tool-bounded 18:38:52Z to 18:40:44Z recovery/readback window (112 seconds, rounded down). Earlier generation, other agents, compaction and uninstrumented waits are not reconstructed. Zero unattended and blocked minutes refer only to that observed window, not the complete attempt. Forecast error and ratio therefore compare an incomplete lower bound, not total effort; they must not be used to infer acceleration or update a class prior.", "missingFields": [ { "field": "activeHumanMinutes", "reason": "No task-local human-work timer." }, { "field": "computeMinutes", "reason": "Paired replay process receipts are retained, but complete substantive-computation elapsed coverage is unavailable; no reconstructed total." }, { "field": "deduplicatedModelTokens", "reason": "No supported fork-aware task-local counter supplied; rollout token events were not summed." }, { "field": "uncachedInputTokens", "reason": "No supported task-local uncached-input counter supplied." } ] } }, "missingnessReason": null } ] } }