{ "schemaVersion": "1.6", "slug": "davies-levitin-lens-conjecture", "title": "A proof of the Davies–Levitin lens conjecture for the indefinite tridiagonal pencil", "shortTitle": "The Davies–Levitin lens, for every n", "url": "https://evidencepress.org/releases/davies-levitin-lens-conjecture/", "oneLine": "A candidate proof that every complex eigenvalue of the Davies–Levitin indefinite tridiagonal pencil lies in the lens where its two Gershgorin disks overlap, for every matrix size.", "abstract": "This unrefereed candidate proves Conjecture 5.3 of Davies and Levitin (2014), which implies the closed-disk form of AIM 2015 Problem 11.1. Let H(c) be the 2n×2n symmetric tridiagonal matrix with diagonal c and off-diagonal entries 1, and let S = diag(1,…,1,−1,…,−1) with n entries of each sign. For every n ≥ 1 and real c, every non-real eigenvalue λ of the pencil H(c) − λS satisfies |λ − c| < 2 and |λ + c| < 2, so the non-real spectrum lies in the intersection, not merely the union, of the two Gershgorin disks. The conjecture was previously known for n ≤ 3 and for eigenvalues equidistant from c and −c. A familiar Weyl-function mechanism, applied to the known Chebyshev form of the characteristic equation, reduces it to a new comparison lemma for an explicit sum of Lorentzians centred at the zeros of U(n−1), evaluated at two points of equal height. The lemma is proved by explicit estimates with finitely many Arb-certified constants for n ≥ 8, elementary arguments for n ≤ 2 and, for 3 ≤ n ≤ 7, an explicit estimate combined with a short interval certificate with rational endpoints and Arb ball arithmetic. Certified examples show that the lens fails for some other spectral measures, for a balanced pencil with one weakened coupling, and for unbalanced pencils. The work was produced by an AI research agent and has not been checked by a human specialist.", "datePublished": "2026-09-25", "dateModified": "2026-09-25", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.22962499", "doiUrl": "https://doi.org/10.5281/zenodo.22962499", "conceptDoi": "10.5281/zenodo.22962498", "pdfUrl": "https://github.com/ipitchford/davies-levitin-lens-conjecture/releases/download/v0.1.0-candidate/davies-levitin-lens-conjecture-v0.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/22962499/files/davies-levitin-lens-conjecture-v0.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22962499", "repoUrl": "https://github.com/ipitchford/davies-levitin-lens-conjecture", "releaseUrl": "https://github.com/ipitchford/davies-levitin-lens-conjecture/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/davies-levitin-lens-conjecture/index.md", "bibtexUrl": "https://evidencepress.org/releases/davies-levitin-lens-conjecture/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/davies-levitin-lens-conjecture.mp3?v=042ef3b349", "imageUrl": "https://evidencepress.org/assets/og/davies-levitin-lens-conjecture.png?v=1cc53414ce", "coverArtUrl": "https://evidencepress.org/assets/art/davies-levitin-lens-conjecture.svg?v=9cba2c343b", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/davies-levitin-lens-conjecture.mp3", "name": "Audio briefing", "description": "AI-generated communication, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/davies-levitin-lens-conjecture.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed candidate produced by an AI research agent. The theorem rests on a written proof and two short interval-arithmetic certificates. There is no formal verification, independent reproduction or specialist or journal review; asymptotic sharpness of the lens is numerical only; novelty is established only within a bounded literature search." }, "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/22962499", "note": "Public GitHub prerelease v0.1.0-candidate and published Zenodo record (3 files each); every file matched the SHA-256 sums by server checksum and by independent download." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "Fresh-extraction replay: manifest; certificates C1 and C2, regression tests and five negative controls in normal and optimised Python; certificate C3 for n ≤ 60; numerical checks.", "evidenceUrl": "https://github.com/ipitchford/davies-levitin-lens-conjecture/blob/v0.1.0-candidate/REPLAY_RECEIPT.md" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "Producer-coordinated internal model reviews do not establish this external assurance dimension." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "An independent model re-certified the n ≤ 60 inequality in mpmath iv from the definitions, inside one producer workflow: implementation diversity, not independent reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "Neither the written proof nor the certificates are formalised in a proof assistant." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Producer-coordinated internal model reviews do not establish this external assurance dimension." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "Producer-coordinated internal model reviews do not establish this external assurance dimension." }, { "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/davies-levitin-lens-conjecture/actions", "note": "Linux CI (Python 3.12, python-flint 0.9.0) runs the replay; versions are recorded. Typesetting is not pinned." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "Claude (Anthropic) research agent under user direction", "Python with python-flint (FLINT/Arb ball arithmetic), mpmath, NumPy and SymPy", "pdfTeX and pandoc" ], "humanRole": "The user supplied publication authority and copies of two cited sources; scholarly creator Anonymous. No human mathematical contribution is claimed.", "disclosure": "AI-produced research, computation and drafting. A blind model referee, an independent model re-certification, two refute passes by a different AI model, one commissioned AI-based external review, a producer-coordinated five-role AI editorial gate (Minor Revision) and a two-reviewer AI confirmation round (Accept with minor edits). None of these is specialist or journal peer review." }, "attributionBoundary": null, "problem": { "name": "Davies–Levitin lens conjecture (Conjecture 5.3; AIM 2015 Problem 11.1)", "url": "https://doi.org/10.1016/j.laa.2014.01.025" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Research, the response to a commissioned external AI review, the internal editorial gate with its confirmation round, and publication are recorded separately.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Davies–Levitin conjecture", "Conjecture 5.3 of Davies and Levitin", "indefinite linear pencil", "non-self-adjoint matrices", "non-real eigenvalues", "Krein space", "indefinite Jacobi matrix", "Chebyshev polynomials", "Herglotz functions", "interval arithmetic" ], "keyResults": [ "For every n ≥ 1 and real c, every non-real eigenvalue λ of the 2n×2n pencil H(c) − λS satisfies |λ − c| < 2 and |λ + c| < 2; hence |Re λ| < 2 − |c| (Theorem 1.1).", "A familiar Weyl-function mechanism, applied to the known Chebyshev form, reduces the conjecture to one new comparison lemma, Ψ(x) ≥ Ψ(y) at equal heights, for an explicit Lorentzian sum Ψ over the zeros of U(n−1) (Lemma 3.1); the lemma and its proof are the original contribution.", "The comparison lemma is proved by explicit estimates with finitely many Arb-certified constants for n ≥ 8, elementary arguments for n ≤ 2, and a short fail-closed Arb interval certificate for 3 ≤ n ≤ 7 (minimum margin 0.107).", "Every non-real eigenvalue off the imaginary axis satisfies the hypothesis of Davies–Levitin's asymptotic Theorem 5.4, and their localisation (50) holds for every n in strict form.", "Certified examples show the lens fails for some other spectral measures, for a balanced pencil with one weakened coupling, and for unbalanced pencils m ≠ n (Remark 8.4)." ], "reviews": [ { "schemaVersion": "1.0", "reviewId": "davies-levitin-lens-internal-editorial-v1", "reviewClass": "editorial-assessment", "status": "published", "subject": { "canonicalUrl": "https://evidencepress.org/releases/davies-levitin-lens-conjecture/", "doi": "10.5281/zenodo.22962499", "version": "0.1.0-candidate", "releaseTag": "v0.1.0-candidate", "title": "A proof of the Davies–Levitin lens conjecture for the indefinite tridiagonal pencil" }, "reviewer": { "name": "Five internal model editorial roles and two confirmation reviewers", "independence": "producer-coordinated" }, "assessment": { "title": "Internal five-role editorial assessment with confirmation round", "date": "2026-09-25", "recommendation": "Minor Revision" }, "assuranceImpact": { "changesVerificationStatus": false, "changesPeerReviewedFlag": false, "changesIndependentReproductionFlag": false, "changesFormalVerificationFlag": false, "explanation": "Producer-coordinated model review does not establish external peer review, reproduction or formal proof." }, "publication": { "publicUrl": "https://github.com/ipitchford/davies-levitin-lens-conjecture/blob/v0.1.0-candidate/review/EDITORIAL_DECISION.md", "publicSummary": "Five roles reviewed one frozen target (all Minor Revision; no error found in the proof). All 51 items were answered in one repair batch; 50 were made and one journal-version note was deferred. Material repairs added certified counterexamples for unbalanced pencils and a weakened coupling, reframed the reduction as a known mechanism, and hardened the essential certificates. Two confirmation reviewers accepted the repairs with minor edits, which were applied." } } ], "evidencePackage": "Written proof (11 pages); two essential Arb certificates (numerical constants; the inequality for 3 ≤ n ≤ 7) with per-constant and coverage mutation controls; an additional certificate for n ≤ 60 with an independent mpmath re-certification; certified root isolation of the theorem (20,744 non-real eigenvalues, none outside the lens) and of the counterexamples; regression tests for two repaired defects; model-based and internal editorial records.", "openProblems": [ "Obtain specialist checking of the written proof and an independent rerun of certificates C1 and C2.", "Replace the interval certificate for 3 ≤ n ≤ 7 by a short elementary inequality.", "Prove the observed 2/3 slack in the comparison lemma and the asymptotic sharpness of the lens.", "Find the correct localisation for unbalanced pencils m ≠ n and other Jacobi blocks, where the lens itself fails (Remark 8.4)." ], "mathObjects": [], "relatedWorks": [ { "citation": "Davies and Levitin (2014): spectra of a class of non-self-adjoint matrices; Conjecture 5.3 and Theorems 2.3, 3.3 and 5.4.", "url": "https://doi.org/10.1016/j.laa.2014.01.025", "doi": "10.1016/j.laa.2014.01.025" }, { "citation": "Öztürk (2023): the conjecture for n ≤ 3 and for eigenvalues equidistant from c and −c; the Chebyshev statement (Remark 1.2).", "url": "https://doi.org/10.1002/mma.8766", "doi": "10.1002/mma.8766" }, { "citation": "Levitin and Öztürk (2018): an associated two-parameter eigenvalue problem.", "url": "https://doi.org/10.1007/978-3-319-75996-8_19", "doi": "10.1007/978-3-319-75996-8_19" }, { "citation": "American Institute of Mathematics (2015): open problems from the workshop Mathematical aspects of physics with non-self-adjoint operators, Problem 11.1.", "url": "https://aimath.org/pastworkshops/nonselfadjointproblems.pdf", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:davies-levitin-lens-conjecture", "attemptIds": [ "ep-attempt:davies-levitin-lens-conjecture-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "communication" ], "lineageId": null, "parentLinks": [], "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "For every n ≥ 1 and real c, every non-real eigenvalue of H(c) − λS lies in the lens |λ − c| < 2, |λ + c| < 2.", "scope": "The balanced 2n×2n pencil with constant diagonal c and unit couplings; the lens fails for unbalanced pencils, weakened couplings and other spectral measures (Remark 8.4)." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The written reduction (Section 3) and Lemmas 4.1 and 5.1–5.4 turn the theorem into the one-variable inequality of Proposition 6.2; Table 1 lists the argument used for each range; certificate C1 checks its numerical constants and certificate C2 covers 3 ≤ n ≤ 7.", "remainingRisks": [ "The written proof has not been checked by an independent specialist.", "The certificates trust python-flint 0.9.0 (FLINT/Arb) and exact rational conversion.", "Asymptotic sharpness and the 2/3 slack are numerical observations." ] }, "humanJudgmentGates": [ "Assess the written reduction and the x-side and y-side lemmas.", "Preserve the candidate status and the bounded novelty wording.", "Publication is authorised; external review is a separate dimension." ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Specialist checking of the written proof and an independent rerun of certificates C1 and C2; external review is not a publication prerequisite.", "claimCeiling": "Unrefereed candidate. A written proof with two essential interval certificates; formal verification, external review and unconditional priority are not established." }, "impactClaims": [ { "id": "davies-levitin-lens-conjecture-no-impact", "aim": "science", "outcome": "Inspectable proof of a stated conjecture", "setting": "Producer-coordinated mathematical publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None.", "estimand": "No acceleration or 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:davies-levitin-lens-conjecture-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "Prospective publication-scope measurement from this registration: action of the commissioned external AI review (already applied at registration), deterministic readiness baseline, the mandatory five-role internal editorial gate with one repair batch, research repository and immutable GitHub/Zenodo identity, Evidence Press page, media and canonical readback. The procedural forecast below was frozen in the local publication state at 2026-09-25T15:49:44Z and is copied unchanged. Problem selection, discovery, the proof, certificates, earlier model reviews, the LaTeX conversion and the external-review repairs before registration are left-censored and are not reconstructed. No acceleration comparison.", "forecast": { "frozenAt": "2026-09-25T16:11:46Z", "procedureClass": "mathematical-candidate-publication-small-archive", "targetOutcome": "Internal editorial acceptance, immutable public research assets (GitHub release and published Zenodo DOI 10.5281/zenodo.22962499 with byte-identical assets) and canonical Evidence Press availability.", "expectedActiveMinutes": 180, "plausibleLowMinutes": 120, "plausibleHighMinutes": 300, "expectedUnattendedWaitMinutes": 45, "referenceClass": { "label": "Reviewed mathematical candidate publication with a sub-megabyte archive", "sampleSize": 1, "basis": "Seymour 2δ+3 release (190 active minutes including a 3 GB archive); this package is under 1 MB with a fast replay but needed LaTeX conversion. Procedural prior only; no acceleration comparison. The totals were first frozen in the local publication state at 2026-09-25T15:49:44Z and are recorded here unchanged at ledger registration." }, "fermiComponents": [ { "component": "Deterministic readiness baseline and research-package assembly", "count": 1, "lowMinutesPerUnit": 20, "centralMinutesPerUnit": 35, "highMinutesPerUnit": 55, "basis": "Procedural prior; no comparative acceleration claim." }, { "component": "Five-role internal editorial gate and one repair batch", "count": 1, "lowMinutesPerUnit": 30, "centralMinutesPerUnit": 50, "highMinutesPerUnit": 80, "basis": "Procedural prior; no comparative acceleration claim." }, { "component": "GitHub repository, tag, release and Zenodo publication", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 20, "highMinutesPerUnit": 40, "basis": "Procedural prior; no comparative acceleration claim." }, { "component": "Evidence Press page, explanatory art, OG card, thumbnail and audio", "count": 1, "lowMinutesPerUnit": 35, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 70, "basis": "Procedural prior; no comparative acceleration claim." }, { "component": "Two-commit seals, CI, guarded deployment and two readback cycles", "count": 1, "lowMinutesPerUnit": 20, "centralMinutesPerUnit": 30, "highMinutesPerUnit": 55, "basis": "Procedural prior; no comparative acceleration claim." } ], "probabilityPositiveSignal": 0.9, "probabilityTargetClosure": 0.8, "probabilityHorizonMinutes": 300, "assumptions": [ "No fatal integrity defect surfaces in the editorial gate.", "GitHub, Zenodo, Cloudflare and OpenAI providers remain available." ], "reforecastTriggers": [ "Provider outage, credential expiry or integrity failure.", "Editorial decision of Major Revision or Reject." ], "stopRule": "Timing is telemetry, not a cap; continue unless integrity, rights or provider access blocks publication." }, "outcome": { "measuredAt": "2026-09-25T19:11:30Z", "calendarElapsedMinutes": 180, "activeAgentMinutes": 60, "activeHumanMinutes": null, "computeMinutes": null, "unattendedWaitMinutes": 15, "blockedMinutes": 105, "reworkMinutes": 10, "agentRuns": 7, "maxParallelAgents": 5, "modelTurns": 8, "deduplicatedModelTokens": null, "uncachedInputTokens": null, "researchCycles": { "positive": 0, "negative": 0, "inconclusive": 0 }, "falsificationGatesRun": 10, "candidateArchitecturesTested": 1, "candidateArchitecturesRejected": 0, "substantiveReviewRounds": 1, "p0Findings": 0, "p1Findings": 2, "prepublicationClaimCorrections": 3, "resultState": "target-closed", "resultSummary": "Five-role internal gate: all Minor Revision, no error found in the proof; 51 items, 50 applied and 1 deferred (2 Major: AI-use disclosure; false m != n extension replaced by certified counterexamples). Two-reviewer confirmation round: Accept with minor edits, applied. Deterministic batches passed on the confirmation target and the final archive; GitHub/Zenodo byte parity verified by server checksum and download; canonical page read back after guarded deployment (one registry change-log fix PR). DOI registration at DataCite was still pending at readback.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -120, "forecastRatio": 0.3333, "withinForecastInterval": false, "varianceReason": "Active minutes are an estimate from session timestamps: registration 16:11:46Z to the pause at about 16:45Z, and resumption at about 18:50Z to readback. About 105 minutes were blocked by a model-provider usage limit that stopped both confirmation reviewers; about 15 minutes were reviewer and CI waiting. Part of the five-role gate ran before registration and is left-censored. Rework covers the sanitisation re-run and the registry change-log fix. modelTurns is a lower bound (distinct root and reviewer invocations). Below the forecast interval because the pre-registration LaTeX and review repairs had already been done; no speed or impact comparison.", "missingFields": [ { "field": "activeHumanMinutes", "reason": "Human effort not instrumented." }, { "field": "computeMinutes", "reason": "No separately instrumented total." }, { "field": "deduplicatedModelTokens", "reason": "Fork-aware runtime counter unavailable." }, { "field": "uncachedInputTokens", "reason": "Runtime token/cache accounting unavailable." } ] } }, "missingnessReason": null } ] } }