# Matrix-valued spectral bound library

Family 262: Sharp finite-matrix Lieb–Thirring inequalities and all equality cases. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Spectral and quantum-model researchers can bound negative eigenvalue moments for admissible one-dimensional matrix potentials and check exact soliton equality examples.

## Applicability and commercial boundary

The potential integral, matrix positivity and gamma range are essential. A continuous inequality does not automatically certify a discretized spectrum or general higher-dimensional Hamiltonian.

## Initial business decision

Conditional engineering. Buyer budget, commercial novelty and profitability are unvalidated.

## Next verification action

Extract the scalar optimal constant and create an independently evaluable sech-squared equality fixture.

## Evidence scope

The catalog statement was individually reviewed. This record does not imply a manuscript proof review or formal-scope comparison. No independent Lean check was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Main source

[Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf).