Lieb-Thirring bounds for Melik-Adamyan canonical Hamiltonians

We study a class of positive matrix Hamiltonians arising from the canonical differential expressions of Melik-Adamyan and appearing in the appendix of Alpay--Gohberg. Let $J$ and $B$ be self-adjoint involutions on $\mathbb{C}^{2n}$ satisfying $JB=-BJ$, and let $\mathcal{H}>0$ satisfy $\mathcal{H} J\mathcal{H}=J$. For $m>0$ we consider $$\mathcal A_{m,\mathcal{H}} = \mathcal{H}^{-1}\left(-iJ\frac d{dt}+mB\right)$$ in the weighted space $L^2_{\mathcal{H}}$. A locally absolutely continuous $J$-unitary gauge $\Theta$ representing $\mathcal{H}$ reduces this expression to the free massive Dirac operator plus the Hermitian coefficient $$P_{m,\Theta}=-i\Theta^*J\Theta'+m(\Theta^*B\Theta-B).$$ Whenever this coefficient belongs to $L^2$, the corresponding self-adjoint realization, including its operator domain, is independent of the chosen representing gauge. Minimizing $\int{\rm Tr}|P_{m,\Theta}|^2$ over the gauge fibre defines an intrinsic energy. A two-sided Birman--Schwinger decoupling, combined with a truncated pseudo-relativistic estimate proved here, gives a $3/2$-moment bound for all eigenvalues in the gap $(-m,m)$ in terms of this energy. The Dirac estimate applies to arbitrary Hermitian matrix coefficients in $L^2$ and requires no sign condition. On the half-line we treat every self-adjoint Lagrangian boundary condition. Two reflection-compatible conditions require no endpoint correction, while an arbitrary condition contributes at most $2nm^{3/2}$. At zero mass, the optimal-gauge energy is computed explicitly in terms of $\mathcal{H}^{-1/2} \mathcal{H}' \mathcal{H}^{-1/2}$. For a scalar hyperbolic-rotation family the massive gauge minimization reduces exactly to a one-dimensional phase functional. We prove existence of a minimizer in the principal phase sector and give an explicit trial phase that strictly and quantitatively improves the positive lift whenever the corresponding first variation is nonzero.