Critical-Window Lower Bounds for Gaussian Regression with Bounded-Renewal Linear Splines

Preprint / proof candidate. This work establishes a critical-window lower bound for the explicitly stated Gaussian-regression model with bounded-renewal linear splines. The proof develops an exact continuity-coupled spline residual representation, verifies the interior-knot budget and finite-volume boundary treatment, and uses a Gaussian mixture second-moment argument together with regenerative/quasilocal limit theory to obtain the stated asymptotic lower bound. The result is restricted to the specified bounded-renewal construction. It does not claim the unrestricted sharp minimax constant, and numerical estimates of the Green–Kubo variance are not asserted as an exact theorem constant. Internal proof and hostile-audit stages are complete for the stated model; independent expert/peer review is pending.