The K41 Spectrum as the Unique Variational Minimiser of a Scale-Resolved Free Energy

Abstract

This preprint is Paper A in the turbulence split of the Functional Stability Theory programme. It isolates the unconditional variational result: the Kolmogorov K41 energy spectrum is the unique global minimiser of a scale-resolved free-energy functional under joint minimisation over energy profiles and admissible local flux closures subject to constant scale-to-scale flux. Version 1.2 is a maintenance and proof/paper-synchronisation release. It corrects the shell-Hessian scaling language, weakens the local UCU3 constant to a ratio-dependent positive constant, and states the bridge to companion Paper B as conditional on uniform energy injection. The claim level remains the v1.1 reference-normalized K41 selection theorem inside the K41-normalized admissible class and common feasible set. The companion Paper B, Anomalous Dissipation and the Turbulent Cascade: A Variational Free-Energy Derivation of Kolmogorov Scaling, is published under the existing Turbulence/Skeleton concept DOI 10.5281/zenodo.19056813. Paper B treats the conditional Downhill Flux / NLw / ND route to anomalous dissipation. The two papers share the same turbulence programme but have separate DOI lines: Paper A is the reference-normalized K41 variational theorem; Paper B is the conditional cascade/anomalous-dissipation route. The FST Hub / master overview is maintained under concept DOI 10.5281/zenodo.20130499. Public sources and computational artefacts are mirrored in the GitHub repository research-line/functional-stability-theory/fst-physics/k41-variational-minimiser. CHANGELOG Changes in Version 1.4 (literature update and native German UTF-8 text layer, August 2026) Literature update (July 2026 research-landscape checks): nine recent references added and contextualized in both language versions, including Buckmaster–Vicol (2019), Isett (2018), Tanogami (2023, 2025), Sanchis-Agudo & Vinuesa (2025), Mishra (2026), López (2026), Said (2025) and Skalidis (2026); the discussion of the state of research was extended accordingly. All newly cited works were archived in full text during the checks. German edition: full native UTF-8 umlaut text layer and bilingual style audit; terminology parity between the English and German manuscripts verified. No changes to the mathematical content, claims or proofs; the claim discipline of v1.3 is unchanged. Changes in Version 1.3 (June 2026) Minor: The active citation apparatus was rechecked in English and German; all 13 active citation keys and bibliography entries remain synchronized, with Crossref/DataCite metadata confirmed. Policy: AI/LLM acknowledgement forms were removed from the title footnote and acknowledgements section. The neutral AI Disclosure remains. Data and structure: A claim/evidence-status ledger and a numerical-diagnostics gate table were added to make the static theorem, local Hessian stiffness, synthetic diagnostics, K62 bridge and DNS/Paper-B boundary conditions visibly separated. DE/EN: English, German and combined PDFs were rebuilt after the citation, disclosure and data-structure maintenance passes. Claim hygiene: The dimensional-prefactor, minimax/penalty and K62 fluctuation passages were tightened after a strict adversarial review. The K41 theorem is now stated more explicitly as a relative result inside the reduced K41-normalised inertial shell ansatz, while the penalty and intermittency passages are marked as formal modelling bridges. Changes in Version 1.2 (May 2026) Critical: The Hessian stiffness discussion was corrected from k_j^(5/3) to the shell-energy scaling 1/E_j^* ~ k_j^(2/3); k^(-5/3) remains the spectrum-density scaling, not the Hessian stiffness. Major: The local UCU3 constant in the German version was weakened from 1/2 to a ratio-dependent c_R > 0, and the Paper-B bridge is now conditional on uniform energy injection. DE/EN: English and German versions were synchronized after the style and math checks; the combined bilingual PDF was rebuilt. Changes in Version 1.1 (May 2026) Major: Strict-review guardrail integrated: the main claim is now explicitly restricted to reference-normalized K41 selection within the K41-normalized admissible class and common feasible set. Major: Overclaiming routes were removed or marked out of scope: no fixed-closure enforcement, no intrinsic derivation of K41 without the normalization class, and no Navier--Stokes dynamics or intermittency derivation. DE/EN: English and German versions were synchronized after the claim-level revision. Minor: Frontmatter, PDF metadata, hyperref setup, and the combined bilingual PDF were rebuilt during the design pass. Version 1.0 (May 2026) Initial release: Separate Paper A DOI line for the unconditional K41 variational result, linked to the conditional companion Paper B and the FST Hub. Files K41_Variational_Minimiser_en.pdf -- English version. K41_Variational_Minimiser_de.pdf -- German version. K41_Variational_Minimiser_kombi.pdf -- combined English/German PDF. Status Advanced preprint. Subject to revision before journal submission.