Endpoint-Controlled Two-Term Asymptotics for Nonlinear Caputo Relaxation: Memory–Constitutive Competition and Hitting Times

We derive two‑term asymptotic expansions for nonlinear Caputo relaxation while isolating the recent‑history concentration that first‑order regular variation does not control. Under a terminal uniform‑integrability condition, the Caputo dissipation retains its finite‑initial‑value term and has an explicit memory correction. Ultimate convexity and a local derivative bound are sufficient for this condition, whereas a plateau–catch‑up construction proves that bare regular variation is insufficient. For a restoring law \(φ(x) = cxβ\{1 + axσ + o(xσ)\}\), the state satisfies \(u(t) ~ K t-α/β\). Its first relative correction has order \(t-ασ/β\) when \(σ < 1\) and order \(t-α/β\) when \(σ ≥ 1\), separating constitutive dominance, resonance, and memory dominance. For the exact‑power law, the coefficient changes sign at \(α = β/(β + 1)\); the same sign determines whether the corrected residual hitting time is earlier or later than its leading estimate. At \(T = 3000\), implicit L1 diagnostics recover the subcritical and supercritical state coefficients within \(0.85%\) and \(4.6%\), respectively, and the resonant coefficient within \(1%\). These computations support the predicted signs and scales but do not replace the analytical endpoint hypothesis.