← NS Zone · PROGRAM-NS
3D Incompressible Navier–Stokes Global Regularity · one of the Clay Millennium Prize problems · 67/67 built · global regularity OPEN
The document opens by explicitly listing eight non-claims: it does not claim to have proven global regularity, does not claim to have constructed a finite-time blow-up, does not claim that True ETN (the infinite-dimensional tension field) or the X-integral can, by itself, derive PDE regularity, and does not claim that smoothness on an arbitrary numerical lattice directly implies continuum smoothness. Its core tool is a two-layer language stack: True ETN reads N–S's Fourier/Littlewood–Paley multiscale evolution as an infinite-dimensional tension field; the X-integral is a typed, source-traceable, scale-by-scale-guarded calculus of formation legality, used to determine whether a given set of local nonlinear interactions is entitled to be promoted to a cross-scale concentration or singularity-formation mechanism — formation happens only once it passes the guard, and a conclusion that fails to pass does not mean "the quantity is zero," but "this higher-level claim is not currently entitled to form."
The whole proof architecture compresses into two complementary claims: C1 (Chain Necessity) — a blow-up must generate an X-legal ultraviolet concentration chain; C2 (Finite Obstruction) — any such chain must be genuinely blocked by N–S structure (incompressibility, triad geometry, viscous scale tax) at a finite scale. If both are proved: blow-up ⟹ chain exists, but ¬(chain exists), hence ¬blow-up — that is, global regularity. Neither claim has yet been fully proved.
Honest state of play: C1a (UV-escape necessity) and C1b (causal provenance of the nonlinear replenishment — the nonlinear Duhamel term must make up for the high-frequency tail lost to dissipation) are CLOSED; C1c (persistent triadic genealogy — whether a single traceable causal branch exists) remains OPEN. This round's genuine contribution to C2 is a NO-GO: using an abstract geometric cascade ledger, it explicitly constructs a scalar model that simultaneously satisfies "finite time + finite energy dissipation + Λ∈L¹ + Λ∉L^(5/2) + a nonzero critical toll at every scale," proving that every presently natural scalar/additive budget argument is insufficient to rule out a blow-up-shaped ledger. The frontier therefore formally shifts to C3, Cross-Scale Coupling Rigidity (25 independent sub-investigations), followed by C4 (9 rounds), C5 (13 rounds), and C6 (17 rounds).
Read these three papers first, before any of the C3-C6 rounds below — ETN and the X-integral are a research language, not proven theorems; the genuine theorems are only the parts each document itself marks CLOSED.
25/25 COMPLETE. Runs from the first necessary condition for cross-scale coupling all the way through operator escape, multi-core geometry, pressure concentration, and an algebraic bridge to the Grujić–Xu derivative-chain framework — genuine exclusion of blow-up remains OPEN, and the frontier hands off to C4.
9/9 COMPLETE. C3 split the survivor into independent necessary channels; C4 instead asks whether these channels can legally coexist within a single genuine state-transition chain — ultimately routing every recurrent UV singular event either to a synchronized structure or to one of six finite compensation motifs. C4 closes structurally as a research phase, not as a regularity theorem, and hands off to C5.
13/13 COMPLETE. C4 routed UV singular events to the six-member family of finite compensation motifs; C5 compactifies these motifs round by round, routing pseudo-defects into a finite residual alphabet $\{A,T,G,P,H,F\}$. C5-M formally declares that C5 closes structurally as a research phase — a finite defect graph does not imply global regularity — and hands off to C6 to extract the genuine recurrent sink cycles.
17/17 fully live, and PROGRAM-NS's full series is now complete at 67/67 papers (framework 1 + C1 1 + C2 1 + C3 25 + C4 9 + C5 13 + C6 17). C5 compressed every recurrent survivor state into the six-letter residual alphabet $\{A,T,G,P,H,F\}$ and one certified compatibility graph, flagging three candidate recurrent cycles; C6-A through C6-Q — seventeen rounds — progressively extract sink strongly-connected components, correct the semantics of visibility, and open up multiple carrier channels: spectral, pressure, peak, satellite, eternal, and others. C6-Q proves that TS's nonlocal tail can be localized, that spatial escape can be re-bound by translation, and that recurrent defects can even be extended into an eternal solution — the document's own proposed next round, C6-R, has not yet been written, and whether the series will be extended remains undecided. A finite defect graph does not imply global regularity; global regularity remains fully open.