Geometric Depletion and Diffusive-Scale No-Concentration: An Option A Programme for 3D Navier-Stokes Regularity

This preprint presents a geometric framework for addressing Option A (global regularity) of the three-dimensional incompressible Navier–Stokes equations. The approach operates at the diffusive scale and combines three mechanisms: a forced coherence–incoherence dichotomy for the vorticity direction, geometric dissipation from q²|∇ξ|², and cancellation of near-field vortex stretching. THIS IS A RESEARCH PROGRAMME, NOT A FINISHED PROOF. The argument reduces global regularity to three scale-sharp lemmas and a diffusive-time no-concentration proposition. The document explicitly identifies the technical bottlenecks requiring rigorous verification: • Lemma 3: Principal-value and commutator estimates for near-field velocity depletion• Lemma 2: Quantitative epsilon-delta compatibility in coherent stretching absorption• Dyadic closure: Explicit summability of transport flux after near-/far-field splitting The contribution of the work is the isolation of a minimal set of geometric statements at the diffusive scale whose verification would close Option A. The framework combines the vorticity-direction programme of Constantin–Fefferman (1993) with the modern scale-of-sparseness / dynamic dissipation scale framework of Bradshaw–Farhat–Grujić (2019). The novelty is organisational and programmatic: A forced coherence–incoherence dichotomy at each dyadic vorticity level Explicit near-field Biot–Savart cancellation via principal-value commutator estimates A time-averaged diffusive-scale no-concentration proposition replacing pointwise sparseness conditions Counterexamples to any of these statements are equally valuable as rigorous proofs. This work addresses Option A (regularity).For a separate Option B (blow-up) approach using physical arguments, see:https://doi.org/10.5281/zenodo.18300240 Call for verification We invite experts to: (a) provide full principal-value/commutator estimates proving Lemma 3(b) quantify epsilon(delta) in Lemma 2(c) supply or refute explicit dyadic summation in the closure argument Author contributions Lewis Thorpe-Aiken: Conceptualisation, programme architecture, diffusive-scale dichotomy and S2/S2′ formulation, bottleneck identification, strategy integration, direction and editorial decisions. Claude (Anthropic AI) & ChatGPT: Mathematical formalisation and LaTeX typesetting under Lewis’s guidance; stress-testing and critique of key lemmas; drafting of verification framework and iterative refinement for preprint release. Contact: lewis@consciousnesspartnership.com