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 routes to LDI are identified: bounded variation of the log-distance and power-law sublevel-time control. The v1.5 release publishes bibliographic source corrections; the v1.4 revision added guardrails for strong versus trajectory-attractor formulations, no-positive-measure contact, nonconvex regularised selectors, and skeleton compactness. CHANGELOG Changes in Version 1.6 (June 2026) Bibliographic sourcefix: The active EN/DE bibliography was rechecked against Crossref/DOI, AIMS, ScienceDirect, RePEc/Wiley, MathNet, Zenodo API and AMS reference data. Duplicate DOI lines for DeBlasiMyjak1993 and Zamfirescu1990 were removed; the canonical Crossref DOIs 10.1007/BF01201454 and 10.1007/BF01188685 remain exactly once. Maintenance bundle: This release carries forward the local v1.6 maintenance work since v1.5: source metadata cleanup, claim-sensitive English wording, German terminology cleanup, neutral AI/KI disclosure, Tide-Clock/Bad-Channel guardrail wording, and the corrected u'-regularity caveat. DE/EN: English, German and combined PDFs are published synchronously; the German text track was checked for genuine umlauts. Changes in Version 1.5 (May 2026) Bibliographic correction: Source metadata were checked against Crossref, DOI/publisher pages, MathNet, RePEc/Wiley and Zenodo. Bae–Cannone 2016, Ilyin–Kalantarov–Zelik 2026, Romanov 1994, the NS-Skeleton Zenodo self-citation and the Lytchak DOI were corrected or completed. 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