A dimension-independent capacity bound for amplified determinants of Hermitian matrix tuples
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
For a matrix tuple, we maximize the geometric mean of the squared singular values of its linear pencil over auxiliary matrices with unit normalized Hilbert-Schmidt energy, allowing arbitrary auxiliary size. We compare this regularized optimum with an optimization of Gurvits capacity over coefficient covariance matrices. The capacity optimization is an upper bound for every tuple. If the coefficients are Hermitian and their real span contains a positive definite matrix, it also bounds the optimum from below up to the universal factor e. The lower bound combines the established Mai-Speicher free determinant formula with finite-dimensional regularization. An imaginary shift is represented inside the original coefficient tuple, and its energy cost is computed exactly. Real covariance suffices for the capacity optimum, giving normal auxiliary witnesses without size doubling. A separate dilation transfers general bounded moment approximants to feasible witnesses. Only convergence of a bounded, regularized logarithm is needed; convergence of unregularized Gaussian log determinants is not asserted. The result gives finite witnesses approaching the bound, without a quantitative dimension estimate. We state separately the unresolved extension to arbitrary matrix tuples and the question of an effective construction.
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Cite this paper
Lluis Eriksson (2026). A dimension-independent capacity bound for amplified determinants of Hermitian matrix tuples. AIRR.SCIENCE, ARR-2026-4E4SJ328XF98BSCJ, v1. https://airr.science/papers/ARR-2026-4E4SJ328XF98BSCJ/versions/v1/ Research preprint; ARR-2026-4E4SJ328XF98BSCJ v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Version history
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Pass
- Reproducibility
- Pass
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Substantial OpenAI Codex assistance with research, mathematical development, writing and computation. One GPT-6 Astra xhigh review cycle of the exact manuscript gives ordinary quality 9.0/10 and AIRR research importance 5.1/10. The model contributed the real-covariance simplification during development; shared/reused context was not isolated or blind. No human review, proof-assistant verification or statistical independence is claimed. The initial and final checks are stages of one review cycle, not separate votes. No new Opus report exists because the service was quota-blocked.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 1
- Range
- 5.10–5.10
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/1 benchmark matches. This is an evidence description, not a probability of correctness.
Only one distinct model; One provider; shared blind spots possible; Fresh blinded review context not documented for every model
OpenAI · gpt-6-astra · Xhigh · involved in manuscript · review context: shared5.10
Recommendation: Accept · Material objections: 0
The final six-page PDF and source support C/e <= F_infty <= C for Hermitian tuples with a positive definite real-span direction. No material error was found. The real-covariance simplification and same-size normal witnesses are valid; the original dilation remains a valid more general transfer device. Quality 9.0/10; AIRR importance 5.1/10, separately judged. The result is a useful technical bridge, not a new capacity identity or a resolution of unregularized Gaussian log-determinant convergence. Novelty is plausible only in this narrow formulation and was not exhaustively certified. Context is shared, and this reviewer suggested part of the final simplification, so independence is involved_in_manuscript. Faithful later transcription on 2026-10-16T18:00:34.860Z of the existing local report (text SHA256 372600ce854cfc12fbd17259d86d9fb06049940c29c3990821a96b2cc830be65); no new review, score, assessment date or second vote. Favorable working-paper recommendation, not an editorial Accepted decision.
- Correctness confidence
- Reconstructed covariance orientation, energy, exponents, adjoint symmetrization, both regularizations, congruence and order of limits. Own exact and numerical checks pass. This remains AI review conditional on cited theorems.
- Rigor
- The proof is complete at research-note level under explicit Hermitian and positive-span assumptions. It avoids unjustified unregularized determinant convergence and makes no quantitative witness-size claim.
- Novelty
- The free determinant identity and e factor are Mai–Speicher; Gaussian approximation and regularization are classical. The candidate contribution is their in-tuple, energy-accounted finite-witness transfer and covariance optimization.
- Significance
- A dimension-independent comparison for a substantial but restricted class. No general complex-tuple theorem, optimal constant, effective auxiliary-size bound, efficient algorithm or operational consequence is established.
- Reproducibility
- Short inspectable proof, exact artifact hashes, retained source/PDF snapshots, own seeded checker with negative controls and machine-readable results; no author checker was read or credited as an independent check.
Strengths
- The comparison factor is independent of d, s and tuple coefficients within the stated class, while all energy costs are explicitly charged.
- The upper bound handles arbitrary complex tuples and complex positive semidefinite covariance matrices, including singular covariance.
- The regularization remains inside the original coefficient tuple; the proof takes the large-matrix limit before removing the shift.
- Real covariance suffices by adjoint capacity invariance and concavity, so the main construction uses normal witnesses without doubling auxiliary size.
- The manuscript clearly attributes Mai–Speicher, states the restrictions, and avoids asserting effective size, optimality or a general complex-tuple extension.
Weaknesses
- Most of the deep analytic content and the numerical constant e are imported from Mai–Speicher; the new bridge is a short consequence after the right regularization is recognized.
- The real-span positive-definite hypothesis excludes many naturally traceless or arbitrary noncommutative pencils.
- There is no quantitative bound on auxiliary size or conditioning dependence, no covariance optimization algorithm and no proved operational payoff.
- The examples verify normalization and scope but do not demonstrate a sharp nontrivial C/F_infty gap or establish optimality of e.
- The literature audit is directed rather than exhaustive; related regularization and shifted-determinant literature deserves concise positioning in a future revision.
Potential errors
None reported.
Strong novelty candidates
- Subject to an independent priority search: the explicit finite-witness transfer inside a Hermitian coefficient tuple with a positive direction, preserving normalized energy and yielding the universal C/e lower bound.
- The real-covariance / normal-witness formulation is a useful elementary refinement, derived from prior capacity duality and concavity; it should not be presented as a separate deep invariant or an independent review contribution.
Unresolved material objections
None reported.
arr:assessment:e0b4eaf8-960b-4f9b-93a1-f576d90c8a7c · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: xhigh · basis: platform runtime metadata · evidence SHA-256 d374654c617136f7efa53145e1d15e7d6aab1928de4653ee104875050907411e · response SHA-256 57a453a4e7ed92643f0f60ec7d87a3cf123b75036576b49477829c6df594fc79 · canonical PDF SHA-256 20906b3764ddb86ac14ccbbeab6ae49019521cf7589b3c56c553ff9023827963
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Author-authorized exact-version Working paper. Author is AIRR operator. Advisory AI assessment is retained with its provenance and limitations. This is not Accepted status or independent human peer review.