An internally audited global-continuation certificate for one unforced periodic Navier-Stokes datum
Description
This preliminary research record documents an internally audited global-continuation certificate for one explicit smooth, mean-zero, divergence-free datum for the unforced, viscosity-one Navier–Stokes equations on the three-dimensional torus. It presents the initial datum, Fourier norm normalization, terminal bound, analytic small-data continuation argument, scope of the stored composed audit, and selected SHA-256 digests. Eight selected file digests were computed from available bytes during preparation; the intact slab-18 digest is explicitly identified as recorded in preserved evidence rather than freshly rehashed. The complete raw coefficient chain and verifier are omitted, so this is not a standalone reproducible proof. The global conclusion depends on the stated PDE background and finite-certificate premises. No arbitrary-data regularity theorem, external peer review, or established novelty or priority is claimed. The report discloses assistance from ChatGPT (OpenAI) and Claude (Anthropic).
Other
Related-work note — 16 October 2026
A posteriori analysis of approximate Navier–Stokes solutions has established precedents, including Morosi–Pizzocchero [1] and Chernyshenko–Constantin–Robinson–Titi [2]. Applications using Reynolds expansions for selected periodic initial data are discussed by Morosi–Pernici–Pizzocchero [3]. Silva [4] presents a closely related Fourier–Wiener framework for computer-assisted global regularity, including neighbourhoods of cyclic-shear, ABC and TG3 flows.
The present record concerns one specified initial datum and reports a composed internal audit. Its stated assumptions and limitations on independent reproducibility remain unchanged.
References
[1] C. Morosi and L. Pizzocchero, “On approximate solutions of the incompressible Euler and Navier–Stokes equations,” Nonlinear Analysis 75 (2012), 2209–2235.
"https://doi.org/10.1016/j.na.2011.10.020"
[2] S. I. Chernyshenko, P. Constantin, J. C. Robinson and E. S. Titi, “A Posteriori Regularity of the Three-dimensional Navier-Stokes Equations from Numerical Computations,” arXiv:math/0607181 (2006).
"https://arxiv.org/abs/math/0607181"
[3] C. Morosi, M. Pernici and L. Pizzocchero, “Large order Reynolds expansions for the Navier-Stokes equations,” Applied Mathematics Letters 49 (2015), 58–66; arXiv:1402.0487v4.
"https://arxiv.org/abs/1402.0487v4"
[4] Jose Luis Lima de Jesus Silva, “Computer-assisted global regularity across nonlinear families of three-dimensional periodic Navier–Stokes flows,” arXiv:2609.16157v1 (2026).
"https://arxiv.org/abs/2609.16157v1"