Tracking Disturbances by Scale: Preserving Joint Energy While Tracking Middle and Remote Modes

This AI-assisted research preprint investigates computer-assisted error bounds for a prescribed family of unforced, incompressible, three-dimensional periodic Navier-Stokes flows. The method tracks coarse, middle and remote disturbances separately while retaining a combined energy bound. For the specified base-flow family with amplitude parameter 0 <= alpha <= 0.650, the recorded certificate covers smooth, real, mean-zero, divergence-free initial disturbances supported on squared Fourier shells |k|^2 >= 11, with normalized homogeneous H3 norm <= 0.100*nu, where nu is the viscosity. The conclusion uses the archived Fourier/coefficient inequalities as mathematical inputs. The previous 0.040nu fine-only certificate already covered this remote-only subclass. The new sufficient radius on that common subclass is therefore 2.5 times as large. This does not enlarge the unrestricted or full fine-only disturbance ball to 0.100nu. A 50-digit outward-arithmetic implementation records 16,384 comparison steps. A separate 100-digit implementation checks every step and rejects six deliberately corrupted witnesses. The supplement includes source code, the complete witness, inherited coefficient dependencies, internal reviews and a correction log. The checker verifies arithmetic given those estimates; it does not independently re-prove the complete Fourier/Sobolev chain. Gregory Rachkovsky directed the research relay, with substantial AI assistance from GPT Astra and Gemini Argon in exploration, derivation, computation, drafting and internal checking. These are the assistant labels used in the project. All review to date is internal AI-assisted review; external scholarly review is invited. This is a restricted-data result, not a solution of arbitrary-data three-dimensional Navier-Stokes regularity. No optimal threshold or priority claim is made. Version v1.11-zenodo.1 is an editorial publication edition of v1.11. The mathematical results, code, coefficient tables and witness are unchanged. Original project documents, figures and numerical records are licensed under CC BY 4.0; original Python source is licensed under MIT. These grants apply to rights held by the depositor, with third-party rights and notices preserved.