A pure-tensor counterexample to a literal analytic-rank tensorization inequality

Summary

An archived AIM problem-list page displays a tensorization inequality for analytic rank or partition rank. Under one literal reading of its analytic-rank branch, the inequality fails for the simplest possible kind of tensor: a pure tensor.

The result depends on the wording. It refutes the literal, unnormalised analytic-rank instantiation under standard base-$p$ analytic rank and fixed-order Kronecker powers. It does not refute a normalized or powered-identity formulation, establish what the proposer intended, or settle partition rank.

The exact statement: the literal comparator

The archived page prints a bound of the form

where $I$ is the identity tensor on the same space, $c<1$, and $\mathbf r$ is analytic rank or partition rank. For the analytic-rank branch, the candidate uses

and groups corresponding tensor legs in each Kronecker power. The archived display does not specify that logarithm base, grouping, normalization or the relationship between its two rank branches. Those conventions are assumptions of the conditional theorem, not recovered historical facts.

The technical mechanism: the pure-tensor witness

Work over $\mathbb F_2$, at order three and side length two. In fixed coordinate bases, take

For a nonzero pure order-$d$ tensor over $\mathbb F_p$, character orthogonality gives

Here that bias is $3/4$. Writing $a=\log_2(4/3)$,

The hypothesis $\operatorname{arank}(A)<\operatorname{arank}(I)$ therefore holds strictly.

Why every positive power eventually fails

Under the fixed-order Kronecker product, a pure tensor remains pure. Its active diagonal support has size one at every power, so

for every positive integer $m$. The proposed scalar right-hand side is $(2ac)^m$. The exact rational comparison

implies $2a<1$. Thus $2ac<1$ for every fixed $0<c<1$, so $(2ac)^m$ tends to zero and is eventually smaller than the constant left-hand side $a$.

This is an all-power proof, not an extrapolation from finite computation.

The powered-identity diagnostic

The same diagonal calculation shows exactly where normalization matters. If $D_{k,n,d}$ has $k$ active diagonal coordinates inside side length $n$, then

The one-dimensional diagonal scale cancels. This exact ratio is a useful diagnostic for corrected tensorization statements, but it is only a diagonal comparison. The candidate does not prove a general normalized tensorization theorem or claim that this is the uniquely intended historical formulation.

What this does not show

A tempting partition-rank route uses the order-four determinant, whose partition rank is below that of the identity. It does not close the problem: the needed order-four tight-support entropy equality is unavailable. The package preserves this stopped route instead of presenting it as evidence.

Evidence, assurance and limitations

The package contains the five-page DOI-bearing manuscript, a self-contained proof, exact Python replay in normal and optimized modes, a same-producer JavaScript reimplementation, and direct coefficient-level enumeration. The enumerator checks 64 assignments at $m=1$ and 4,096 at $m=2$, reproducing biases $3/4$, $9/16$ and $81/256$. Seven deliberately corrupted assertions are all rejected.

One supplied Major Revision review was actioned through a full response matrix. A producer-coordinated five-role internal editorial gate then returned PASS_WITH_NOTES with no new P0 or P1 scientific issue. The roles test different questions, but they were produced within the same workflow and are not cognitively or organizationally independent.

Public Linux CI runs the replay on four Python/Node combinations and separately rebuilds the PDF. GitHub and Zenodo expose the same PDF, 51-file ZIP and manifest; fresh downloads match their local SHA-256 values. These facts establish availability, integrity and producer-side replay. They do not establish independent reproduction, independent reimplementation, formal verification, external specialist review, editorial peer review, novelty, priority or field acceptance.

Relationship to earlier work

Lovett developed analytic rank in the standard bias-based framework and recorded the diagonal scaling used here. Bhrushundi and collaborators gave bias lower bounds for multilinear forms, with the order-three pure-tensor constant $3/4$ appearing naturally. Later work on geometric rank and subrank makes normalization factors explicit when comparing tensor parameters.

Partition rank has a different identity normalization. Naslund introduced the parameter in the relevant combinatorial setting, and Lampert and Moshkovitz proved that the order-four determinant has partition rank three. That fact is why the stopped determinant route is interesting, but it does not supply the missing asymptotic equality.

The candidate contribution, if novel, is narrower: applying known diagonal bias and support formulas to the literal archived comparator and isolating the normalization failure. No first or priority claim is made.

Who should care, and why

Why the distinction matters

Tensorization statements amplify small one-shot gaps into exponential ones. That amplification is meaningful only when numerator and comparator scale in compatible ways. Here a rank-one tensor remains rank one on the relevant diagonal scale, while an unpowered scalar identity rank is repeatedly multiplied. The resulting decay is an artifact of the comparator, not a deep high-power phenomenon.

The example therefore acts as a compact design test: before attempting an asymptotic tensorization proof, verify the statement on pure and diagonal tensors and compare against the powered identity. That test cannot solve the normalized or partition-rank problems, but it can prevent effort being spent on a literally false formulation.

How to reproduce the recorded checks

Use tag v0.2.0-candidate or the Zenodo version DOI, not the moving main branch. From a fresh extraction with Python 3.12 or later and Node.js 20 or later, run:

sh run_all.sh

The script runs Python normally and with optimization, runs the JavaScript reimplementation, performs direct coefficient checks at $m=1,2$, and requires all seven negative controls to reject their corrupted claims. The expected terminal marker is:

ALL REPLAY AND NEGATIVE-CONTROL GATES PASS

Successful execution confirms the shipped finite predicates. The all-$m$ result rests on the written proof, and replay does not establish novelty, independence or peer review.

The most valuable next projects

1. Historical source reconstruction. Determine the intended normalization, logarithm and tensor grouping from contemporary notes or the proposer.
2. Normalized analytic rank. Formulate and test a powered-identity or scale-normalized tensorization statement beyond diagonal tensors.
3. Partition rank. Find a valid asymptotic mechanism or a genuine counterexample for the separately normalized partition-rank branch.
4. Independent reconstruction. Rebuild the proof and finite checks in a materially separate stack without importing the producer implementation.
5. Formalization and specialist review. Formalize the diagonal proof and obtain authenticated analytic-rank and historical-priority assessment.

What is in the evidence package

The immutable release contains the canonical PDF and LaTeX, aligned Markdown, proof and claim maps, source capture, Python and JavaScript replay, coefficient enumeration, negative controls, citation and novelty audits, supplied-review response, five internal role reports, provenance, licences, environment and compute receipts, research metrics, release notes and a complete SHA-256 manifest. The version DOI is the citation target; any correction should be a versioned successor, not a silent edit.

@misc{puretensoranalyticrankcounterexample2026,
  title        = {A pure-tensor counterexample to a literal analytic-rank tensorization inequality},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22181327},
  url          = {https://doi.org/10.5281/zenodo.22181327},
  version      = {0.2.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/pure-tensor-analytic-rank-counterexample/}
}