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

Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2<\gamma <3/2$ and arbitrary finite-matrix potentials $W\ge 0$ with $\int \mathrm{tr}(W^{\gamma +1/2})<\infty$: 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 ${\mathrm{sech}}^2$ solitons with independent scales and centers, and zero channels.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.