{
  "schemaVersion": "1.6",
  "slug": "pure-tensor-analytic-rank-counterexample",
  "title": "A pure-tensor counterexample to a literal analytic-rank tensorization inequality",
  "shortTitle": "Pure-tensor analytic-rank obstruction",
  "url": "https://evidencepress.org/releases/pure-tensor-analytic-rank-counterexample/",
  "oneLine": "A side-two pure trilinear tensor refutes one literal, unnormalised analytic-rank tensorization comparator, while the powered-identity diagonal ratio remains exactly (k/n)^m.",
  "abstract": "An archived AIM problem-list page displays a tensorization inequality for analytic rank or partition rank. Under standard base-p analytic rank and fixed-order Kronecker powers, its literal, unnormalised analytic-rank instantiation is false. Over F2, a side-two pure trilinear tensor has analytic rank a=log_2(4/3), the identity has rank 2a, and every Kronecker power of the pure tensor still has rank a. Since 2a<1, the proposed scalar bound (2ac)^m tends to zero for every fixed 0<c<1. On diagonal tensors, comparison with the powered identity instead gives the exact ratio (k/n)^m. The diagonal formula and normalization principle are known; the possible contribution is only their application to this literal comparator. The release does not refute a normalized or powered-identity formulation, determine the source's intended meaning, or settle partition rank. Exact derivation, finite producer replay, seven negative controls, review repair, public CI and byte-matched GitHub and Zenodo assets support the candidate; independent reproduction, formal verification, external specialist review, editorial peer review, novelty and priority remain unestablished.",
  "datePublished": "2026-08-30",
  "dateModified": "2026-08-30",
  "version": "0.2.0-candidate",
  "doi": "10.5281/zenodo.22181327",
  "doiUrl": "https://doi.org/10.5281/zenodo.22181327",
  "conceptDoi": "10.5281/zenodo.22181326",
  "pdfUrl": "https://github.com/ipitchford/pure-tensor-analytic-rank-counterexample/releases/download/v0.2.0-candidate/paper.pdf",
  "altPdfUrl": "https://zenodo.org/records/22181327/files/paper.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22181327",
  "repoUrl": "https://github.com/ipitchford/pure-tensor-analytic-rank-counterexample",
  "releaseUrl": "https://github.com/ipitchford/pure-tensor-analytic-rank-counterexample/releases/tag/v0.2.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/pure-tensor-analytic-rank-counterexample/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/pure-tensor-analytic-rank-counterexample/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/pure-tensor-analytic-rank-counterexample.mp3?v=404fb9633f",
  "imageUrl": "https://evidencepress.org/assets/og/pure-tensor-analytic-rank-counterexample.png?v=930491615c",
  "coverArtUrl": "https://evidencepress.org/assets/art/pure-tensor-analytic-rank-counterexample.svg?v=6ca3369a3e",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/pure-tensor-analytic-rank-counterexample.mp3",
      "name": "Audio briefing — pure-tensor analytic-rank obstruction",
      "description": "Plain-English synthetic-voice explanation of the literal unnormalised comparator, exact pure-tensor witness, powered-identity diagnostic and assurance limits; a communication aid, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/pure-tensor-analytic-rank-counterexample.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed conditional counterexample candidate at PASS_WITH_NOTES after actioning a supplied Major Revision report and completing a producer-coordinated five-role internal editorial gate. Reviewer identity, credentials and unaffiliated status were not authenticated. The diagonal bias formula and normalization principle are known; no first-discovery or priority claim is made."
  },
  "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/22181327",
      "note": "The tagged GitHub prerelease and Zenodo record expose the DOI-bearing PDF, full ZIP and manifest; all three assets were independently downloaded from both services and matched local SHA-256 values."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Public Linux CI runs normal and optimized exact replay across Python 3.12 and 3.13 with Node.js 20 and 22, rejects seven negative controls, and independently rebuilds the PDF.",
      "evidenceUrl": "https://github.com/ipitchford/pure-tensor-analytic-rank-counterexample/actions/runs/33336067178"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No authenticated unaffiliated clean-environment rerun is reported."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Python and JavaScript were produced within one coordinated workflow and are not independent implementations in the assurance sense."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No proof assistant formalizes character orthogonality, diagonal bias multiplication, Kronecker support growth or the eventual inequality."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The supplied review and five internal role reports are not authenticated unaffiliated analytic-rank 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 comparable external venue has conducted editorial 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/pure-tensor-analytic-rank-counterexample/blob/v0.2.0-candidate/ENVIRONMENT.txt",
      "note": "Commands, dependency versions, source capture, manifests and Linux CI are public, but no bit-reproducible system image or authenticated unaffiliated reconstruction is supplied."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Python 3 standard library",
      "Node.js standard library"
    ],
    "humanRole": "Research direction, candidate selection, review provision, claim-boundary decisions, publication authorization and institutional responsibility by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.",
    "disclosure": "AI systems contributed proof reconstruction, exact computation, manuscript drafting, source and novelty auditing, review remediation, editing, replay and release engineering within one producer-coordinated workflow. Same-producer cross-language agreement, the supplied review and five-role internal gate do not establish unaffiliated reconstruction, specialist review or editorial peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "AIM Problem 1.3: Polynomial method via tensorizing",
    "url": "https://web.archive.org/web/20191214012625id_/http://aimpl.org/addcombapp/1/"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "reader-first-reference",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Authored prospectively against the DOI-bearing tagged candidate, supplied-review response, internal editorial gate, public hosted replay and exact GitHub and Zenodo asset readback on 30 August 2026; the first guarded Evidence Press deployment, exact canonical release readback and publication-preservation gate subsequently passed without changing the claim ceiling.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "analytic rank",
    "tensor bias",
    "Kronecker power",
    "diagonal tensor",
    "partition rank",
    "tensor normalization",
    "counterexample",
    "unrefereed candidate"
  ],
  "keyResults": [
    "Under base-p analytic rank and fixed-order Kronecker powers, the literal unnormalised analytic-rank instantiation of the archived comparator is false.",
    "Over F2 at order three and side length two, a pure tensor has analytic rank a=log_2(4/3), the identity has rank 2a, and every positive Kronecker power of the pure tensor has rank a.",
    "The exact inequality 16/9<2 gives 2a<1, so the proposed scalar right-hand side (2ac)^m tends to zero for every fixed 0<c<1.",
    "For a k-supported diagonal tensor in side length n, comparison with the powered identity gives the exact ratio (k/n)^m."
  ],
  "reviews": [],
  "evidencePackage": "A five-page DOI-bearing manuscript and aligned Markdown; a self-contained exact proof; a checksum-bound archived source extract; dependency-free Python normal and optimized replay; same-producer JavaScript reimplementation; direct coefficient evaluation over 64 assignments at m=1 and 4,096 assignments at m=2; seven semantic negative controls; claim, convention, source, citation and bounded novelty audits; a supplied Major Revision review and full response matrix; one producer-coordinated five-role internal editorial gate; a 51-file manifested ZIP; a five-job public GitHub Actions replay and PDF rebuild; and exact GitHub and Zenodo asset readback. These establish public availability, integrity, producer replay and the stated conditional theorem—not historical intention, partition-rank resolution, independent reproduction, formal verification, authenticated external specialist review, editorial peer review, novelty, priority or impact.",
  "openProblems": [
    "Determine the historically intended logarithm base, tensor grouping, normalization and branch logic of the archived AIM statement.",
    "Prove or refute an appropriately normalized or powered-identity analytic-rank tensorization inequality beyond the diagonal family.",
    "Settle the partition-rank formulation; the determinant route recorded in the package does not currently close it.",
    "Independently reconstruct the source-to-theorem bridge and the all-power proof in a materially separate stack.",
    "Formalize the pure and diagonal bias formulas and obtain an authenticated specialist and priority assessment."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "American Institute of Mathematics. Polynomial method via tensorizing, Problem 1.3 in Additive combinatorics and its applications. Archived 14 December 2019.",
      "url": "https://web.archive.org/web/20191214012625id_/http://aimpl.org/addcombapp/1/",
      "doi": null
    },
    {
      "citation": "Lovett, S. (2019). The analytic rank of tensors and its applications. Discrete Analysis 2019:7.",
      "url": "https://doi.org/10.19086/da.8654",
      "doi": "10.19086/da.8654"
    },
    {
      "citation": "Bhrushundi, A., Harsha, P., Hatami, P., Kopparty, S., & Kumar, M. (2020). On multilinear forms: bias, correlation, and tensor rank. APPROX/RANDOM 2020.",
      "url": "https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.29",
      "doi": "10.4230/LIPIcs.APPROX/RANDOM.2020.29"
    },
    {
      "citation": "Derksen, H., Makam, V., & Zuiddam, J. (2024). Subrank and optimal reduction of scalar multiplications to generic tensors. Journal of the London Mathematical Society 110:e12963.",
      "url": "https://doi.org/10.1112/jlms.12963",
      "doi": "10.1112/jlms.12963"
    },
    {
      "citation": "Lampert, A., & Moshkovitz, G. (2026). Slice rank and partition rank of the determinant. ITCS 2026.",
      "url": "https://doi.org/10.4230/LIPIcs.ITCS.2026.90",
      "doi": "10.4230/LIPIcs.ITCS.2026.90"
    }
  ],
  "claimAssurance": null,
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      "scope": "Base-p analytic rank, fixed-order Kronecker powers and the printed scalar identity-rank comparator; not a reconstruction of historical intention, a refutation of normalized or powered-identity formulations, a partition-rank theorem, or an external-assurance or priority finding."
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      "remainingRisks": [
        "The archived display does not establish the proposer's intended normalization or grouping.",
        "Producer-written proof and programs may share a convention error despite seven negative controls.",
        "The bounded novelty search cannot establish historical priority, and partition rank remains unresolved."
      ]
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      "Keep the result conditional on the literal unnormalised analytic-rank instantiation.",
      "Do not transfer the pure-tensor mechanism to partition rank.",
      "Treat the known diagonal formula and normalization principle as prior art and make no priority claim.",
      "Inspect the all-m written proof rather than treating finite enumeration as extrapolation.",
      "Keep producer replay, internal editorial review, unaffiliated reconstruction, formal verification and peer review distinct."
    ],
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        "inheritedClaim": "The archived AIM page prints a tensorization comparator using the scalar r(Id)^m and names analytic rank or partition rank as alternatives.",
        "inheritedAssuranceCeiling": "The archived page fixes the displayed wording and attribution but does not establish the intended logarithm base, normalization, grouping, quantifier logic or this candidate's correctness and priority."
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        "status": "NO_IMPACT_EVIDENCE",
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        "estimand": "No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation or field outcomes was estimated.",
        "evidenceRefs": [],
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