An Onsager–Prigogine Framework and Terminal Reduction Program for 3D Navier–Stokes Regularity via Auto-Organized Dissipative Structures

We present a fully explicit Onsager–Prigogine framework and terminal reduction programfor the three-dimensional incompressible Navier–Stokes equations on the torus. Themanuscript is intended as an arXiv-ready research proposal for community scrutiny: everystep claimed as a theorem is proved in detail, while the remaining terminal continuationmechanisms are isolated explicitly rather than being folded into an over-strong final claim.The technical core couples: (i) an Onsager-type Gaussian filtering us = esΔu that producesa strictly positive excess-entropy density ws and a filtered relative-entropy balance for Ls;(ii) an Onsager/Prigogine closure in which each shell interface carries a convex quadraticproduction functional with an instantaneous Prigogine minimizer, so that the physical forceslice decomposes into a minimum admissible production, a measurable misalignment excess,and a deterministic baseline; and (iii) a Cheeger-type spectral dichotomy on the induceddyadic shell graph, interpreted on the active shell graph obtained by collapsing massless shellblocks through the exact series law.The rigorous output of the paper is the following. In the coercive regime, the spectralgap forces the shell potential to remain close to the instantaneous Prigogine profile; theresulting oscillation budget feeds a De Giorgi–Nash–Moser argument for ws, yielding parabolicHarnack/Hölder control and a matched low-pass estimate for the filtered velocity at the autoorganizedcutoff. In the bottleneck regime, the no-pocket principle suppresses heterochiraltransport, produces deterministic tail-energy decay above the bottleneck interface, and yieldsa tail-localized Duhamel estimate whose forcing depends only on the high-frequency tail andon the low-high transfer across that interface.What remains, and is stated transparently as part of the proposal status of the manuscript,is the terminal continuation step that turns these two rigorous outputs into a contradiction ata hypothetical blow-up time. Section 7 formulates this final step as two explicit continuationinterfaces: a dynamic coercive interface and a bottleneck-to-Besov-window transfer interface.In this form the paper is mathematically honest, structurally complete up to the statedinterfaces, and aligned with the intended Prigogine-style interpretation of auto-organization.