Research paperAcceptedARR-2026-5QQF95VHTC9GABH8 · v1 · 2026-08-30

Sharp Onset and Unbounded Growth of Norm-Optimal Self-Commutator Rank

Lluis Eriksson

Abstract

For a traceless Hermitian target F, this paper studies the least rank r_*(F) among factors C minimizing one half of the squared Hilbert--Schmidt norm subject to CC^*-C^*C=2F. It proves r_*(F)=max(n_+(F),n_-(F)) for every target through dimension seven, including singular spectra, and proves sharp failure in dimension eight on an explicit two-ray cone whose interior has r_*=5>4. An exact dimension-nine seed and a symbolic hive coarse-graining theorem yield targets G_t in dimension 27t with kappa(G_t)=87t and 17t+1<=r_*(G_t)<=18t, so the additive excess above inertia is unbounded. The proof is computer-assisted through exact rational Horn/polyhedral and hive certificates with independent replay routes. It does not determine the exact amplified rank, classify all optimizers, formally verify the imported Horn/hive theorem, or claim exhaustive priority.

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Definitions and rankings →
7.00 · Exceptional · n=1

Verification record

Frontier-model screening
Pass · 1 models
Source integrity
Pass
Bibliographic integrity
Partial
Reproducibility
Partial
Lean 4
Not applicable

Recorded under ARR-VERIFY-1.0. ARR verification and screening are not peer review.

Version history

The ARR identifier remains stable. Each version has its own immutable release, timestamp and version identifier.

  • v1 · source snapshot available · viewing

Related ARR records

  • ARR-2026-1D2QV1RP1292JREWRelated Work · Provides general rank-adaptive cost bounds and equality classifications; it does not contain the complete threshold through dimension seven, the sharp dimension-eight onset, or unbounded optimal-rank excess.
  • ARR-2026-7NPRNBW4488HG90KRelated Work · Provides the complete one-spike inverse-cost formula and exact kick-synthesis laws; it does not contain the threshold or amplification theorem proved here.
  • ARR-2026-24M24KDPZK8HDBQ9Referenced By · Linked from Sharp inertia ceilings and optimal stability for inverse self-commutators.
  • ARR-2026-12W84G9BVC8BQAEQReferenced By · Linked from Exact Rank Transitions through p=32 and a Half-Integral Optimum at p=53.

AI assistance statement

OpenAI Codex assisted with corpus and literature triage, mathematical exploration, adversarial proof review, exact-arithmetic replay, manuscript revision, typesetting, visual QA, reproducibility hardening, independent model-assessment coordination, and ARR record preparation. The author selected the claims, reviewed the manuscript, authorized publication, and remains responsible for the work.

Frontier-model screening

Status: pass. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.

  • gpt-5.6-solOpenAI · reasoning effort high · pass
Longitudinal frontier-model record

Independent model assessments

Read the scale and limits
7.00 / 10.00Exceptional
Eligible reports
1
Range
7.00–7.00
Method
median · exact version only
OpenAI · gpt-5.6-sol · High · not involved in manuscript7.00

Recommendation: Accept · Material objections: 0

The corrected reproducibility package resolves the sole material packaging objection. Its ZIP and manifest are identified by fixed hashes; all 89 payload files were independently matched to the manifest; the missing helper is now included; and the formerly failing exact WSL/GMP verifier passes all six chambers. Together with the previously successful complete replay and visually clean 11-page canonical PDF, this supports acceptance. The score rises modestly from the prior strict 6.80 because the correction improves reproducibility and completeness without changing the paper's underlying mathematical novelty or significance.

Correctness confidence
Confidence is supported by the successful exact WSL/GMP verification across all six chambers and the previously successful complete default replay.
Rigor
The supplied record indicates a systematic verification workflow, exact artifact checks, and clean presentation, with no newly reported mathematical defect.
Novelty
The packaging correction supplies no new evidence that changes the prior assessment of mathematical novelty, so this dimension remains moderate.
Significance
The correction concerns artifact completeness rather than mathematical importance; significance therefore remains at the previously supported level.
Reproducibility
All 89 payload files match the corrected manifest in size and hash, the omitted helper is present, and both exact verification and full replay have passed.

Strengths

  • Exact verification now passes all six chambers.
  • The corrected package is complete and fully hash-checked against its manifest.
  • The complete default replay passed.
  • The canonical 11-page PDF was visually clean.

Weaknesses

  • The correction does not add evidence warranting a higher assessment of mathematical novelty or significance.
  • The evaluation record supports solid acceptance rather than an exceptional Millennium-anchor rating.

Potential errors

None reported.

Strong novelty candidates

None reported.

Unresolved material objections

None reported.

arr:assessment:5c1785af-7187-46a1-b45e-dc418396e38e · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 53a3c91414d05d6b12d11cdfb48276eee8f2838c62a3efa4662f04b58d547e91 · response SHA-256 3cbac19f857a384d8622e30034bcd79d069472cc6a690dacde4d708df1067993 · canonical PDF SHA-256 17b2361721027649b8ef5b98440032b91b5b04e89e2a9535fcbe7a27954ac1c5

A model assessment is not peer review or a correctness certificate. ARR preserves disagreement, exact-version provenance and later reassessments.

Editorial disclosure

Founder-owned record: the author and ARR founder-editor are the same person. Acceptance records a technically valid deposit after explicit audits; it is not independent peer review or a guarantee that the theorem is correct or novel.