A computer-assisted proof of global stability of two-dimensional plane Couette flow beyond the energy method

For two-dimensional plane Couette flow with streamwise period L = 1.659, the energy method proves global stability of the laminar flow only for Reynolds numbers below Re_E ≈ 177.2. Darrow, Carlson and Goluskin (arXiv:2606.18232) introduced a three-parameter family of quartic Lyapunov functionals and showed numerically that one of them decays for all Re ≤ 197.3. They left a rigorous verification for future work. We give a computer-assisted proof. For L = 1659/1000 and every Re ≤ R* ≈ 197.6390650 (an explicit rational), every two-dimensional perturbation of finite energy decays to zero. To our knowledge this is the first rigorous proof of global stability of plane Couette flow above its energy stability threshold. The proof follows their construction, with three changes: the two modes are explicit polynomials rather than exact eigenfunctions, and their residual errors enter the estimates; the spectral bounds on the infinite-dimensional remainder are proved by an integrated-Legendre tail lemma; the enclosures of the mode integrals and the sum-of-squares certificate are verified in exact rational and ball arithmetic (python-flint / FLINT). The certified threshold is slightly above the published value, mostly because the two energy eigenmodes are rotated within their span by about −0.038. Two independently written programs check the certificate. The proof assumes that the computing environment is correct. Files: r7_couette.pdf (the paper). r7_deposit.zip contains: the certificate and the frozen specification; both checkers; the full proofs as reviewed (the main proof with the checker source and a line-by-line code–proof correspondence, and the spectral lemmas), each with a replay script and its blind reviews; the failed first version of the main proof with its reviews; the list of trusted inputs; numerical Navier–Stokes tests (evidence only, not used in the proof); the scripts that constructed the certificate; a one-command re-check (reverify.py, needs python-flint 0.9.0); SHA-256 checksums. Use of AI: this work was carried out with an AI research system directed by the author. It used Claude Opus 5.5 (Anthropic, through Claude Code) and GPT-6 Astra (OpenAI, through the Codex CLI), on 5–7 November 2026. Claude Opus 5.5 prepared the candidate problems and wrote the certificate specification, one checker, the search, the proofs and the manuscript. GPT-6 Astra reviewed the specification and wrote the second checker from the specification text alone. GPT-6 Astra also reviewed the proofs and the manuscript blind. The complete re-check was also run on a separate computer, with the same result. No human has yet checked the written proofs line by line. No AI system is an author; the author takes full responsibility for the content. Details are in Section 10 of the paper.