Anomalous Dissipation and the Turbulent Cascade: A Variational Free-Energy Characterisation of Kolmogorov Scaling

DRAFT / PREPRINT. This is an advanced draft and may still be revised before journal submission. This record is Paper B / Skeleton in the turbulence split of the Functional Stability Theory programme. It develops the conditional Downhill Flux / NLw / ND route from K41-type free-energy monotonicity to anomalous dissipation. The unconditional K41 variational-minimiser theorem is isolated as Paper A: 10. 5281/zenodo. 20131305 (latest version 10. 5281/zenodo. 20562341). Thus this record should be read as the conditional cascade/anomalous-dissipation companion to the K41 variational theorem, not as the sole public location of the K41 result. CHANGELOG Changes in Version 1. 7 (June 2026) Major: The bibliography and self-citation layer were synchronised across the English and German versions; missing source anchors were added, stale unused entries were removed, and the AI-use text was kept as a neutral disclosure rather than an acknowledgement. Major: The Dual-DFC1 ledger is now reflected in the paper text: the Sabra validation remark names computedualdfc. py as an existing toy/ledger and negative-control test, and the new ledger table separates positive controls, tolerated backscatter and bad-corridor failures. Major: The June 2026 Shell-DFC waterline smoke test is documented as a guardrail rather than evidence for a stronger theorem. The transition window is blocked by non-monotone weights, and the tail window fails in the actual flux ordering. Minor: The linked K41 companion latest-version DOI was updated to v1. 3, 10. 5281/zenodo. 20562341, while the stable K41 Concept DOI remains 10. 5281/zenodo. 20131305. DE/EN: English, German and combined bilingual PDFs remain available as a three-file set. The German source/PDF use real umlauts. Changes in Version 1. 6 (May 2026) Critical: The shell-energy Hessian scaling is corrected to the energy-scale stiffness 1/Eⱼ^* ~ kⱼ^2/3 rather than the stronger kⱼ^5/3 statement. Major: Theorem 3. 4 is restricted to positive stationary profiles only when a feasible closure exists; it is no longer framed as an unconditional global profile statement. Major: The anomalous-dissipation lower bound is explicitly conditional on uniform energy input, and the DNS falsification test no longer relies on impossible negative KL-Hessian eigenvalues. DE/EN: English, German and combined bilingual PDFs were rebuilt and synchronised from the 2026-05-16 proof/paper-math-hotfix file set. Changes in Version 1. 5 (May 2026) Minor: The RFEP / Spectrum Duality citation in both English and German PDFs was switched from the version DOI 10. 5281/zenodo. 19036191 to the concept DOI 10. 5281/zenodo. 19036190. No theorem, claim or proof content changed relative to v1. 4. Changes in Version 1. 4 (May 2026) Critical: The anomalous-dissipation claim is now explicitly conditional on DFC1∨, DFC2, DFC3, non-degeneracy and uniform energy input; no unconditional Navier-Stokes regularity or anomalous-dissipation theorem is claimed. Major: Paper A / K41 variational minimiser was separated into its own Zenodo concept DOI 10. 5281/zenodo. 20131305; this record is now Paper B / Skeleton for the conditional cascade route. Major: The DFC hierarchy, projection-bridge caveat, Slack/Minimax guardrails, KHM/Eyink/DR context and DNS falsification path were sharpened after strict review. Minor: PDF metadata, German umlauts and upload file set were cleaned; a combined bilingual PDF was added. DE/EN: English, German and combined bilingual PDFs were rebuilt from the 2026-05-14 local post-review file set. Changes in Version 1. 3 (March 2026) No substantive changes (already journal-grade). Recompiled with latest cross-references. Changes in Version 1. 2 (March 2026) Critical: Fixed placement. Changes in Version 1. 1 (March 2026) Major: Numerical verification of Theorem 3. 1: FEK41=0 exactly, 300/300 perturbation tests confirm strict convexity. Major: DFC recast as energetic stability class. Major: Finite-N spectral gap via shell-model Markov structure (GOY/SABRA).