OpenAI · 262-01 · 2026-10-05 · Mathematical physics

Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026

Family 262. Sharp one-dimensional Lieb–Thirring inequalities.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2 < γ < 3/2$ and arbitrary finite-matrix potentials $W \geq 0$ with $\int tr (W^{γ + 1/2}) < \infty$ : the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar $sech^{2}$ solitons with independent scales and centers, and zero channels.

We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and $1/2 < γ < 3/2$ . For measurable Hermitian positive semidefinite potentials W with $\int_{R} tr (W^{γ + 1/2}) < \infty$ , equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.

Preprint source and Lean scope

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A Lean scope note is stored for family 262, but it does not name this preprint. It names 262-03. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.