A Thermodynamic Obstruction to Finite-Time Blow-Up in the 3D Navier-Stokes Equations

We propose a thermodynamic obstruction to finite-time blow-up in the 3D Navier–Stokes equations. Using a free-energy functional on the global attractor, we derive a conditional regularity criterion: if a BV-regular selection on the attractor exists (Assumption G) and a dissipation-dominance condition holds (Condition D), then finite-time blow-up is excluded. We introduce an EDP-convergence route to selection regularity via rescaled trajectory functionals, Simon's BV compactness lemma, and balanced-viscosity formulations. Numerical validation on the Lorenz attractor confirms that the dissipation-sublevel sets are precompact (DS3 holds). CHANGELOG Changes in Version 2.6 (June 2026) Major: The post-v2.5 strict research review is incorporated: the conditional theorem remains intact, but limsup, G-NS/TLL, EDP, Condition-D-window and H1-route wording are tightened. Major: Pattern-A, mean-field game and Talagrand-transfer language is kept as diagnostic/programmatic context rather than a proof-level shortcut to unconditional Navier-Stokes regularity. Minor: English and German claim hygiene were synchronized after the review pass; the combined bilingual PDF was rebuilt from the updated EN and DE files. DE/EN: English, German and combined bilingual PDFs remain the intended upload set; the German PDF text trace was spot-checked for real umlauts. Changes in Version 2.5 (June 2026) Major: The local post-v2.4 proof/paper synchronization is carried forward: the L2 budget logic, Condition D status, pressure/residual wording and H1 route are stated with stricter guardrails. Major: The bibliography and self-citations are synchronized in English and German after the local citation cleanup; active self-references use Concept DOIs where appropriate. Minor: A Conditional-Input-Ledger table was added after the main theorem to separate the attractor regularity package, Assumption G/G', Condition D and the H1 lift by status and proof function. DE/EN: English, German and combined bilingual PDFs are maintained synchronously; the German text trace was checked for real umlauts. Changes in Version 2.4 (May 2026) Minor: German PDF frontmatter and bibliography labels were localized by adding the correct LaTeX language settings: the German version now renders `Zusammenfassung`, `Inhaltsverzeichnis` and `Literatur` instead of English default labels. Minor: PDF metadata and the combined bilingual PDF were rebuilt from the localized EN/DE sources; no mathematical claim changes were introduced relative to v2.3. Changes in Version 2.3 (May 2026) Major: Math-Sync/Strict-Review corrections are carried forward from the local candidate: attractor regularity is explicit, AGC2 is a separate tracking assumption, and the H1 route is expressed via H1-G, palinstrophy divergence and Bihari dominance. Minor: Source metadata were corrected against Crossref/Springer/Elsevier/Zenodo records: Bae--Cannone 2016, Miranville--Zelik 2008, Grujic--Xu 2024, De Rosa--Drivas--Inversi 2024, Sturm 2025 and internal FST Concept-DOIs. Changes in Version 2.2 (May 2026) Major: Post-Zookeeper/RFEP normal-form context and the local v2.2 candidate state are synchronized for a new Zenodo version while preserving the conditional status of the Navier-Stokes regularity claim. Major: The upload file set is expanded from English and German PDFs to English, German, and combined bilingual PDFs, following the current research-pipeline convention. Minor: Design-check rebuild from 2026-05-07 incorporated: table-of-contents page breaks, overfull boxes, hyperref bookmark warnings, tagged equation anchors, wide table/formula wrapping, and PDF metadata were cleaned up. Minor: One incorrect DOI link was corrected to the verified Journal of Computational Physics DOI 10.1016/j.jcp.2024.112910. Changes in Version 2.1 (March 2026) Major: Assumption G decomposition into three independent components G₁ (attractor existence, proven), G₂ (projection regularity, open geometric property), G₃ (Gronwall integrability, standard) Major: Circularity concern localised to G₂ alone Changes in Version 2.0 (March 2026) Critical: Fixed \thanks placement (was outside \author group)