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Abstract

We sharpen the relative tail-payment estimate for a self-adjoint diagonal operator perturbed by a bounded self-adjoint operator with bounded first commutator. A smooth spectral cutoff replaces the growing-band argument and gives complete relative tail payment at the threshold A ≥ max{16P, (1152K²/δ)^(1/3)}, where P bounds the perturbation, K bounds its first commutator, and δ is the positive finite-head margin. The estimate retains the original finite head and the complete receiving tail. A shifted version yields indexed lower enclosures for every low Ritz eigenvalue, including an indefinite finite head. A fixed rank-two perturbation of a compact-resolvent diagonal operator shows that the cubic truncation power is optimal in this first-commutator class. Jacobi examples illustrate the resulting enclosures with explicit commutator budgets. This supplement develops the relative tail-payment method of the arithmetic matrix criteria predecessor.