{ "schemaVersion": "1.6", "slug": "schatten-hlawka-endpoint", "title": "An endpoint lower bound for Schatten–Hlawka constants", "shortTitle": "Small pairwise losses, large joint cancellation", "url": "https://evidencepress.org/releases/schatten-hlawka-endpoint/", "oneLine": "A known two-by-two matrix family forces a quantitative growth rate near the trace norm; the general sharp constant remains open.", "abstract": "Using Esa’s known rank-one projector family, this unrefereed candidate proves a reciprocal-logarithmic lower growth bound for dimension-independent Schatten–Hlawka constants as the exponent approaches one. It determines the family optimum to two reciprocal terms, sharpens the location of every maximizing parameter, and compares compact linear scales and fixed cancellation amplitudes. No matching unrestricted upper bound or uniqueness assertion is made.", "datePublished": "2026-11-06", "dateModified": "2026-11-06", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23999268", "doiUrl": "https://doi.org/10.5281/zenodo.23999268", "conceptDoi": "10.5281/zenodo.23999267", "pdfUrl": "https://github.com/ipitchford/schatten-hlawka-endpoint/releases/download/v0.1.0-candidate/schatten-hlawka-endpoint-v0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23999268", "repoUrl": "https://github.com/ipitchford/schatten-hlawka-endpoint", "releaseUrl": "https://github.com/ipitchford/schatten-hlawka-endpoint/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/schatten-hlawka-endpoint/index.md", "bibtexUrl": "https://evidencepress.org/releases/schatten-hlawka-endpoint/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/schatten-hlawka-endpoint.mp3", "imageUrl": "https://evidencepress.org/assets/og/schatten-hlawka-endpoint.png?v=a991a211e4", "coverArtUrl": "https://evidencepress.org/assets/art/schatten-hlawka-endpoint.svg?v=26102e52b2", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/schatten-hlawka-endpoint.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/schatten-hlawka-endpoint.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "AI-assisted unrefereed candidate. Internal checking is not unaffiliated reproduction, external peer review or formal verification." }, "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/23999268", "note": "GitHub prerelease v0.1.0-candidate and Zenodo record 23999268 are public with the PDF, source/evidence ZIP and checksums. Availability does not establish mathematical correctness or priority." }, { "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 of the final archive passed both finite checkers and all 12 intended-error controls in normal and optimized Python modes on 6 November 2026. Exact-instance and numerical diagnostics do not replace the written proofs." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "Separate implementation within this workflow is not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "New theorems are not proof-assistant verified; citing a Lean source does not transfer its assurance." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "Internal AI-assisted review is not 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": "No journal or venue peer-review decision established." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "passed", "note": "Final archive replay passed in the recorded Python 3.13.5, NumPy 2.4.4 and mpmath 1.3.0 environment. This is producer-side environment reproducibility, not an unaffiliated rerun or universal portability claim." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical development and revision", "Internal AI-assisted proof and source checking", "Codex publication preparation" ], "humanRole": "Publication authorization only; no human scholarly contribution inferred.", "reviewStatus": "Internal checking is not external specialist or editorial review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the revised manuscript and refs.bib on 6 November 2026. GitHub prerelease and Zenodo record 23999268 were read back as public; research assurance is separate from website publication and provider availability.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Schatten norm", "Hlawka inequality", "endpoint asymptotics", "matrix inequalities", "uniform remainder" ], "keyResults": [ "A lower endpoint growth bound with normalized liminf at least 2/3.", "Two-term reciprocal optimum throughout Esa’s known projector family, not arbitrary matrix triples.", "Uniform logarithmic correction to every family maximizing parameter, without uniqueness.", "Compact-scale loss and fixed-amplitude leading comparison, with fixed-parameter quantifiers." ], "reviews": [], "evidencePackage": "Written analytic proofs; finite rational projector checks, SVD comparisons, high-precision diagnostics and intended-error semantic controls. Finite checks do not prove universal claims.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Audenaert and Kittaneh (2012). Problems and conjectures in matrix and operator inequalities. arXiv v3.", "url": "https://arxiv.org/abs/1201.5232", "doi": "10.48550/arXiv.1201.5232" }, { "citation": "Audenaert and Kittaneh (2017). Problems and conjectures in matrix and operator inequalities. Banach Center Publications 112, 15–31.", "url": "https://doi.org/10.4064/bc112-0-2", "doi": "10.4064/bc112-0-2" }, { "citation": "Esa (2026). hlawka-schatten: Blueprint, trace-endpoint construction, and proof source. Pinned commit 182959114052.", "url": "https://github.com/savarin/hlawka-schatten/tree/1829591140522193a81ae721330e4d775c81123e", "doi": null }, { "citation": "Open Problems in Numerical Linear Algebra contributors (2026). MI-25: Schatten Hlawka constants.", "url": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-25/README.md", "doi": null }, { "citation": "Colbrook (2026). MI-25 trace-norm endpoint solution.", "url": "https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-25/solution.tex", "doi": null }, { "citation": "Lee (2026). Constructions and reductions on the Hornich–Hlawka inequality and its generalizations. arXiv v1.", "url": "https://arxiv.org/abs/2610.23785", "doi": "10.48550/arXiv.2610.23785" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:schatten-hlawka-endpoint", "attemptIds": [ "ep-attempt:schatten-hlawka-endpoint-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output" ], "lineageId": null, "accelerationPrimitives": [ "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "Endpoint lower growth bound from a known projector family, with uniform family-optimum refinements.", "scope": "Schatten exponent approaching one; no matching unrestricted upper bound or full sharp constant." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Exact matrix singular values reduce the inherited family to one scalar deficit ratio; analytic uniform remainders and localization support endpoint claims.", "remainingRisks": [ "Internal written-proof checking is not formal verification.", "Floating diagnostics are not rigorous enclosures.", "Bounded prior-art search does not establish absolute priority.", "Fixed-amplitude and family-optimum statements do not cover arbitrary triples or moving amplitudes." ] }, "humanJudgmentGates": [ "Publication authority is not scientific endorsement.", "External specialist validation remains open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "priorityAssessment" ], "nextAction": "Unaffiliated source-led review of uniform remainders, all-maximizer localization and inherited-construction attribution.", "claimCeiling": "Written proof and scoped producer checks; no external validation, formal verification or absolute originality guarantee." }, "impactClaims": [ { "id": "schatten-hlawka-endpoint-no-impact", "aim": "science", "outcome": "Potential reuse of a quantitative obstruction in Schatten matrix inequalities.", "setting": "Matrix inequalities and endpoint asymptotics", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched research-workflow comparison.", "estimand": "No acceleration or downstream impact effect estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "outcomePolicyUrl": "https://evidencepress.org/api/v2/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. 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