Six Birds for Navier-Stokes: A Mechanized No-Zeno Scaffold

Energy estimates for 3D incompressible Navier–Stokes control L² norms but do not prevent highly localized gradient peaks, which is why global regularity is typically reduced to "peak-channel" criteria (e.g. Beale–Kato–Majda). We develop a multiscale decision scaffold, inspired by the Six-Primitives/Six-Birds framework, that separates (i) system-agnostic inequality logic, (ii) Navier–Stokes-facing hinge assumptions, and (iii) numerical diagnostics.

Operationally, we (A) mechanize in Lean a No-Zeno engine: refinement events are encoded by crossing times, and Zeno cascades are ruled out under work/capacity/divergence hypotheses; (B) mechanize a route-stability ⇒ polynomial-capacity lemma; and (C) mechanize a discrete no-go theorem: if the feasible interface inputs at a scale admit highly localized tests, then any uniform ICAP-style capacity certificate forces a pointwise positive-gain bound—the discrete analogue of a peak/strain control in the PDE interpretation. Motivated by this obstruction, we propose an SBT-legal variant in which feasibility excludes strong localization, and we operationalize legality as a reproducible certificate based on per-shell anti-localization and channel diagnostics. In numerical demos, typical smooth random seeds pass the certificate with maxⱼ κⱼ = O(1), while intentionally localized shell wavepackets fail immediately (e.g. maxⱼ κⱼ ~ 10²).

These results clarify exactly where a Clay-style proof would require a genuinely new Navier–Stokes inequality (a provable channelization/delocalization-to-anti-localization hinge) and provide a reproducible workflow for testing such hypotheses. We do not claim a proof of Clay regularity, nor do we prove that the legality certificate is preserved under the classical Navier–Stokes flow; PDE-facing implications are stated explicitly as OPEN assumptions.