IDMate: Finite-temperature error bounds for window-resolved self-consistent-field screening

Abstract

We formulate a finite-temperature residual test in IDMate that bounds window-resolved electronic errors without a spectral-gap assumption. For a fixed Hamiltonian and exact electron number, strong convexity of the matrix Fermi entropy bounds the density-matrix distance and free-energy error within a selected window. The bound remains finite at spectral crossings and extends to weighted k points with a shared chemical potential. We also derive a distance correction for particle-number mismatch. Across $3{,}586$ stress trials in $70$ seeded perturbation ladders, $1{,}942$ proposals satisfy the screen with no observed violation of the $0.05$ window-distance criterion plus its numerical allowance. This criterion differs from the uncorrected exact-trace bound, which six of ten historical in-loop candidates exceed at the numerical-error scale. An accept-or-recover loop replaces ten reference-map evaluations while meeting terminal comparison criteria in three configurations that include oracle-subspace controls. Additional candidates built only from preceding-iteration orbitals yield two acceptances and one abstention. The silicon candidate has a window distance of $1.891\times10^{-13}$ but a normalized real-space density error of $3.754\%$. Analytic examples separate errors from complement occupations and interblock coupling. Window-level accuracy therefore does not imply full-state accuracy; the screen tests compressed proposal quality, independently of nonlinear SCF convergence or net acceleration.