Domain 02 · Navier–Stokes Critical Integrability¶

Campaign ID: NS-CI-001
Mathematical status: open problem
Primary setting: three-dimensional incompressible Navier–Stokes on R^3
Governance: ADR-0003; ADR-0013

Canonical challenge¶

For smooth rapidly decreasing divergence-free initial data and a Leray–Hopf weak solution, determine whether the critical quantity

integral from 0 to T of ||u(t)||_L6^4 dt

is finite for every finite T.

This is the critical Ladyzhenskaya–Prodi–Serrin pair (q,p)=(4,6). Any result for a narrower data class or a different spatial domain remains a restricted result until an explicit extension theorem is proved.

Programme posture¶

WP00 normalized the problem and source boundary. WP01 and WP02 catalogued false routes and conditional regularity facts. Later work isolated restricted targets and tested equation-specific mechanisms. No route is promoted merely because it produces numerically plausible dissipation, excursion persistence, or generic decorrelation.

NS-CI-WP06 is a separate non-blocking computability and reduction lane. It audits whether a genuine reduction from halting to a precisely represented event of the true viscous equation could exist. Current evidence concerns related Euler flows, modified equations, finite-dimensional ODEs, and a bounded software interface fixture. None of those artifacts supports Turing completeness, undecidable blow-up, singularity, noncomputability of the critical integral, or formal independence for the true equation.

Canonical artifacts¶

Claim boundary¶

The campaign has not proved regularity, global smoothness, the universal critical-integral estimate, a true-equation computational simulation, undecidability, or formal independence. Restricted estimates, negative results, route terminations, literature audits, and bounded interface fixtures are retained only as controlled changes in the search space.