Widom asymptotics for spatial compressions of Schrödinger spectral projections

We establish a Widom trace formula for spatial compressions of exact spectral projections of semiclassical Schrödinger operators. The potential is smooth near a compact classically allowed region, and the observation domain lies strictly inside that region. For every continuous spectral function with positive power decay at both endpoints, we obtain a trace formula at scale $h^{1-n}\log h^{-1}$ whose coefficient separates into a boundary integral and a universal spectral functional. This yields the exact von Neumann and Rényi entanglement asymptotics for every fixed positive order, and proves their universal ratios to the particle-number variance. The proof combines commutator estimates in every fixed positive Schatten class with a local trace formula obtained from a boundary-adapted normal form and a Hardy model. In dimensions $n\ge 2$, a positive partition and the known variance asymptotic control the grazing contribution; in one dimension all boundary Fermi points are transverse. A harmonic-oscillator counterexample shows that continuity and weighted integrability alone are insufficient for the trace formula.