We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and $1/2\lt \gamma\lt 3/2$. For measurable Hermitian positive semidefinite potentials W with $\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty$, 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.

We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent $1/2\lt \gamma\lt 3/2$. The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying $\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty$. The matrices may have arbitrary rank and need not commute at different points.

We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every $\frac12\lt \gamma\lt \frac32$, the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in $L^{\gamma+1/2}(\mathbb R)$, with all negative eigenvalues included.