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. DRAFT version. Subject to revisions. The Navier–Stokes transfer remains conditional and should be read as a selection-regularity framework, not as an unconditional 3D Navier–Stokes regularity theorem. Two sufficient routes to LDI are identified: bounded variation of the log-distance and power-law sublevel-time control. The v1.4 revision adds guardrails for strong versus trajectory-attractor formulations, no-positive-measure contact, nonconvex regularised selectors, and skeleton compactness. CHANGELOG Changes in Version 1.4 (May 2026) Critical: Guardrails added for the 3D Navier–Stokes attractor status: the framework now distinguishes strong single-valued semiflows from weak/trajectory-attractor formulations. Major: LDI Variant B clarified as power-law sublevel-time control; no-positive-measure contact is no longer treated as an automatic Leray–Hopf consequence. Critical: The LDI tail-integral equivalence was corrected to include the speed weight ||u'(t)|| log(R/d(t)); the unweighted int log(1/d(t)) dt condition is no longer used as the LDI formula. Major: The time-averaged selection and damped-to-classical transfer are recast through pre-registered selectors and a skeleton compactness criterion. Changes in Version 1.3 (March 2026) Major: PROOF OF LIFE — TLL+LDI numerically verified on the Lorenz attractor (all 5 tests passed: C_TLL=0.19, LDI finite, BV chain rule ratio=0.52, STC α=1.11, exponential tracking λ=0.064) Major: First numerical verification of the TLL+LDI framework on a non-trivial chaotic attractor Changes in Version 1.2 (March 2026) Critical: Fixed thanks placement