A universal capacity bound for amplified determinants and polynomial witnesses in the Hermitian case
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
For an arbitrary finite tuple of complex square matrices, we maximize the geometric mean of the squared singular values of its linear pencil over auxiliary matrices with unit normalized Hilbert-Schmidt energy and unrestricted size. We prove that this optimum lies between C/e and C, where C is the maximum, over coefficient covariances, of the dimension-normalized operator capacity. No Hermitian or positive-span assumption is imposed on this comparison. An invertible finite evaluation supplies a perturbation inside the original matrix space, subharmonicity selects its phase, and quantitative resolvent comparison, algebraic spectral regularity and Han’s least-singular-value theorem control the entire logarithmic tail. This transfers the Mai–Speicher free determinant formula to feasible finite witnesses. Separately, for Hermitian coefficients whose real span contains the identity, we give an explicit polynomial auxiliary-size bound and a Gaussian sampling guarantee. We do not assert an optimal constant or a general polynomial-time witness algorithm.
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Cite this paper
Lluis Eriksson (2026). A universal capacity bound for amplified determinants and polynomial witnesses in the Hermitian case. AIRR.SCIENCE, ARR-2026-1DGGW5KX2M9R4VJT, v1. https://airr.science/papers/ARR-2026-1DGGW5KX2M9R4VJT/versions/v1/ Research preprint; ARR-2026-1DGGW5KX2M9R4VJT 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 Not peer reviewed by AIRR.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. GPT-6 Astra xhigh assessed this exact eight-page PDF: ordinary quality 9.2/10 and AIRR research importance 6.0/10. The reviewing context contributed to the affine transfer and phase-selection argument; context was shared and reused, not isolated or blind. This is one involved AI judgment, not an independent second vote, human review or formal verification. The complete report was saved before the reviewer service reached its usage limit. No new Opus report exists.
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
- 6.00–6.00
- 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: shared6.00
Recommendation: Accept · Material objections: 0
Favorable scientific recommendation for circulation as a working paper; no unresolved material objection was found in the exact eight-page PDF. The universal capacity sandwich and the separate Hermitian polynomial-size and sampling theorems withstand this audit. Ordinary quality score: 9.2/10. AIRR Millennium research-importance score: 6.0/10, not a conversion of the quality score or a probability of correctness. This is a shared-context OpenAI GPT-6 Astra AI assessment: this same reviewing context contributed to the affine transfer and phase-selection argument and exchanged technical feedback with the authoring context. It is involved_in_manuscript, not blind, isolated, human, formal, or an independent second vote. The judgment depends on the cited external theorems, especially the recent Han preprint, and does not certify priority, editorial acceptance or exceptional historical impact.
- Correctness confidence
- No material error found after reconstruction of both proofs and covariance, tail, normalization and concentration checks. Confidence is substantial but conditional on the cited external results, notably Han Theorem 1.5; this involved AI review is not independent or formal verification.
- Rigor
- The affine proof supplies both mesoscopic spectral-mass control and a least-singular-value cutoff, fixes the phase before N tends to infinity, and pays the actual perturbation energy. Polynomial size and sampling constants are correctly derived. Minor explicitness improvements remain.
- Novelty
- The unrestricted finite energy-constrained transfer and explicit polynomial witness guarantee are plausible new consequences relative to the primary sources inspected. The free determinant identity, operator capacity, subharmonicity, resolvent comparison and concentration are established tools; directed search does not establish worldwide priority.
- Significance
- Removes two substantive restrictions from the prior bridge and replaces extremely large constructive witnesses by a polynomial auxiliary-size guarantee in the positive-span class. General polynomial size, bit complexity, optimality of e and consequences beyond this optimization problem remain open.
- Reproducibility
- Exact source and PDF hashes, a frozen source snapshot, two own diagnostic programs and machine-readable results are preserved. These diagnostics inspect finite identities and constants; they do not reproduce the external infinite-dimensional proofs or provide independent model support.
Strengths
- Universal comparison C/e <= F_infty <= C for arbitrary complex square tuples, including spaces without invertible scalar combinations or positive real span.
- A finite invertible evaluation supplies an admissible perturbation; phase subharmonicity and normalization retain the target capacity constant without asserting convergence of the unperturbed model.
- Affine determinant convergence explicitly controls the whole negative logarithmic tail using algebraicity, a shrinking-scale resolvent estimate and Han’s least-singular-value bound.
- The Hermitian result gives every integer m >= max(2,ceil(4 d (a beta)^4 theta^-3)), normal auxiliary matrices, and a separately quantified Gaussian sampling guarantee.
Weaknesses
- The principal infinite-dimensional inputs are external; Han’s recent 50-page preprint was checked for exact applicability and read selectively, not re-proved in full.
- No general polynomial auxiliary-size bound, efficient optimization of Q, finite-bit sampling analysis or sharpness of e is established.
- Literature search was directed and limited. The exact combined statement was not found, but priority is unconfirmed and shifted free determinant theory has close prior work, including Kargin.
- Reviewer involvement and reused shared context limit evidential independence. Numerical checks cannot rule out errors in the external theorems.
Potential errors
None reported.
Strong novelty candidates
- The energy-normalized finite-dimensional realization of the Mai–Speicher capacity value to a universal factor e for arbitrary complex tuples, via an invertible in-space affine perturbation. Priority remains unconfirmed.
- The covariance-dependent polynomial auxiliary-size estimate and explicit concentration guarantee for the Hermitian positive-span class. These are new consequences relative to sources checked, not new Gaussian comparison or concentration theorems.
Unresolved material objections
None reported.
arr:assessment:f430a110-baf4-401d-b0b4-c60208d078f8 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: xhigh · basis: platform runtime metadata · evidence SHA-256 8a24a5ef6be06d981b1f9c7e61b005d9ff75fd60964a84af1677d9027c8d7a81 · response SHA-256 a299c6860fbb66818beed10ad09f89b4c4c0d30c5d14705095c0e846eaf7e8fd · canonical PDF SHA-256 04fe7efd3a75ac14259349610ca1602a6d5d5487cb5982b0bf2b259c83e9ed0b
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.