Peak Functionals of Driven Dissipative Damage Dynamics: Proofs

Complete proofs of results on the peak M = supt≥0 D(t) of the scalar damage dynamics dD/dt = u(t) − R(D), D(0) = 0, driven by two identical exposure pulses whose starts are Δ apart. Results proved: monotonicity of M(Δ) under linear repair for every unimodal pulse shape, with no log-concavity assumed (Theorem 1); a two-sided comparison (sandwich) bound on M(Δ′)/M(Δ) for repair laws confined to a relative-relaxation band, with a certificate for a given pair of separations (Theorem 2); a sharpened upper bound for repair laws with an inert band at the origin (Proposition 3); the impossibility of any qualitative classification of violating repair laws, given one violating law (Proposition 4); peak monotonicity for every locally Lipschitz repair law with R(0) = 0, and for constant removal, under log-concave inputs (Theorem 5); a general lower bound and a factor-two bound for convex repair, which together give a one-half floor on the peak ratio (Theorems 6a–6b); and a counterexample showing that the floor does not extend to concave repair (Proposition 6c). The proofs support statements made in a companion manuscript on the choice between GUTS survival models (Lee and Joe, in preparation for Environmental Science & Technology). Four deposited Python scripts (NumPy only) reproduce every number quoted in the report; their outputs are included. The report contains no empirical claims.