Update (v2, June 2026): This version adds a diagnostic companion, Two Obstructions to Geometric Incoherent-Absorption and the Critical Transport Sum, alongside the original programme. The companion stress-tests the first cornerstone of the programme, the incoherent-absorption lemma (Lemma 1), and finds that it does not hold in the form the programme requires. Two explicit, self-contained counterexamples defeat both the full-ball and the dyadic-shell readings, and the only reading surviving both coincides with the weighted H¹–BMO machinery the geometric shortcut was meant to bypass. Following the difficulty to the inter-level enstrophy transport, the note shows that sum is scale-critical and argues the coherence mechanism falls in the same algebraic class that Bradshaw–Farhat–Grujić (2019) brought to a reduction, not a closure, of the scaling gap. The conclusion is therefore negative: by this route the programme reproduces at best a BFG-type algebraic reduction and does not close Option A. The two counterexamples and the critical-scaling computation are proven and elementary. The identification of the surviving reading with H¹–BMO duality, and the claim that coherence-depletion and sparseness occupy the same algebraic class, are structural judgements rather than theorems, and are flagged as such throughout. The original programme is retained below for context. A checkable negative result is preferable to an unverified programme, and the companion is released in that spirit. 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 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 The companion document added in this version is itself a response to this call: it supplies explicit counterexamples to the incoherent-absorption step (Lemma 1) and traces the resulting obstruction to a scale-critical inter-level transport sum. In the same spirit, we continue to invite experts to: provide full principal-value/commutator estimates proving Lemma 3 quantify epsilon(delta) in Lemma 2 supply or refute explicit dyadic summation in the closure argument check the two counterexamples and the same-class argument of the companion, on which the negative conclusion rests 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) stress-testing and critique of key lemmas; drafting of verification framework and iterative refinement for preprint release. Contact: lewis@consciousnesspartnership.com
Lewis Thorpe-Aiken (Thu,) studied this question.