Sharp Rank-Adaptive Bounds for Inverse Self-Commutators
For a nonzero traceless Hermitian matrix F, this paper studies the least product of unnormalized Hilbert--Schmidt norms of Hermitian H and K satisfying -i[H,K]=F. Writing rho=rank(F) and P(F)=||F||_1/2, it proves the sharp dimension-free bounds P(F)<=kappa_d(F)<=rho P(F)/2. Both equality loci are classified: the lower endpoint is attained exactly by centrally paired nonzero spectra, while the upper endpoint is attained exactly, up to positive scale and sign, by the one-spike spectrum (rho-1,-1,...,-1), with arbitrary zero padding. An exact sign-cut leakage identity identifies the full tax above the trace-norm floor. The upper bound uses an exact weighted-shift permutation average, and rho-1 explicit Horn--Littlewood--Richardson inequalities certify upper-endpoint sharpness. The work does not claim a closed formula for general interior spectra or an upper-endpoint stability modulus.
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
AI assistance statement
OpenAI Codex assisted with corpus and literature triage, algebraic exploration, hostile proof review, exact-arithmetic replay, typesetting, visual QA, and artifact preparation. The author selected the theorem, reviewed the claims, and remains responsible for the manuscript.
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
Independent model assessments
- Eligible reports
- 1
- Range
- 4.20–4.20
- Method
- median · exact version only
OpenAI · gpt-5.6-sol · High · not involved in manuscript4.20
Recommendation: Minor Revision · Material objections: 0
The main bound P(F) ≤ kappa_d(F) ≤ rank(F)P(F)/2 and both equality classifications appear correct. There is no material objection. A minor revision is recommended to strengthen the bibliographic boundary and make the cited computational package verifiable.
- Correctness confidence
- The one-factor reduction, leakage identity, shift construction, and Horn certificate form a correct chain. The signs, factors of two, zero padding, and rank-two cases were checked. An independent numerical search in ranks 2–5 reproduced both endpoints with residuals below 1e-11.
- Rigor
- The quantifiers and dimensional dependence are controlled. Upper-endpoint rigidity and the rank-minus-one Horn triples suffice. The subset-partition convention and unit Littlewood–Richardson coefficient are explicit.
- Novelty
- The rank-adaptive constant, exact leakage identity, and both endpoint classifications appear novel. Fulton, Angel–Schechtman, and Beltiţă–Patnaik–Weiss cover different ingredients or optimizations, but the search was not exhaustive.
- Significance
- This is a clean, sharp, dimension-uniform matrix-analysis result, although it is specialized and does not provide a general cost formula for interior spectra.
- Reproducibility
- The analytic proof is self-contained, and the main identities are directly reproducible. The PDF names a script, a hash, and a JSON file, but those artifacts were not included in the evaluated PDF.
Strengths
- The exact equation-(8) leakage identity.
- The rigorous lower-endpoint classification derived from the orthogonal supports of CC* and C*C.
- The permutation average removes the ambient dimension and yields rank(F)P(F)/2.
- The adjacent-swap argument excludes extremizers with both sign multiplicities greater than one.
- The one-spike Horn certificate telescopes and remains valid under arbitrary zero padding.
Weaknesses
- The priority search is explicitly non-exhaustive and refers to related ARR works in dimensions 3–5.
- The comparison with close predecessors could benefit from a formal table of objectives, constraints, and constants.
- The computational replay is not auditable from the PDF alone, although it is not needed for the proof.
Potential errors
- The exact formulas in dimensions 3–5 rely on ARR references 20–22 and were not independently checked.
- The phrase “complete extremizers” is correct for the two universal endpoints but could be misconstrued as covering all minimizing factors or all interior spectra.
Strong novelty candidates
- The exact variational sign-cut leakage identity.
- The sharp universal rank(F)/2 factor, independent of the ambient dimension.
- The complete spectral classification of both equality cases.
- The one-spike certificate using only rank(F)−1 uniform Horn inequalities.
Unresolved material objections
None reported.
arr:assessment:6baeaf57-ec50-44d9-9e02-01ef772b2f95 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: author verified ui · evidence SHA-256 da6dfa961ccc3a648cd85c6f15d5dc4fc6a12d585d55c2ae40b404095ec233ec · response SHA-256 a40fa69924039e2409e6d017c902cdf6d1e5cdee18c1a7185737b0ddc8c77131 · canonical PDF SHA-256 106e6011a9506b6f64ac04fdc6d991e11bae7db3b4bf7c76cad33825e2190337
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.