Log-Distance Integrability for Dissipative Flows: BV Selections on Fractal Attractors

We establish a BV chain rule for log-Lipschitz nearest-point projections onto compact attractors of dissipative dynamical systems. The main result provides conditions under which the time derivative of the log-distance functional d(u(t), A) remains integrable along trajectories, yielding BV regularity of the projected trajectory. Two sufficient conditions are identified: (Variant A) exponential distance decay control, and (Variant B) polynomial approach rate. The framework connects to the EDP-convergence route for Navier-Stokes selection regularity developed in the companion paper. Published in English and German versions. Part of the Functional Stability Theory (FST) programme. Changes in Version 1.2 (March 2026) Critical: Fixed thanks placement (was outside author group, causing color code text on title page)