Quantum Mechanics for Closure of Dynamical Systems

The paper explicitly proves the positivity of the compression map ΠL on operator algebras and the induced positivity of projected multiplication operators πL, and it establishes large-data consistency of matrix elements and operators via spectral approximation of kernel integral operators and ergodic averages. The candidate solution reproduces these arguments at essentially the same level: (i) a direct quadratic-form proof that ΠL preserves positivity and thus tr(ρ πL f) ≥ 0 for f ≥ 0; (ii) convergence of Nyström eigenpairs and matrix elements ⟨φi,N, ÂN φj,N⟩N → ⟨φi, A φj⟩; and (iii) positivity preservation through the forecast/analysis cycle by composing positive maps and conditioning by effects. Minor differences are in presentation (the model supplies a standard operator-norm argument and multiplicity remarks), but there is no substantive conflict.