Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

=== Conditional theorem. Assume (P1)–(P3) above. Then for every smooth, divergence-free initial datum u0u_0u0 (Schwartz, say) there is a unique global-in-time solution u(x,t)u(x,t)u(x,t) of the incompressible Navier–Stokes equations with u∈C∞(R3×(0,∞))u\in C^\infty(\mathbb{R}^3\times(0,\infty))u∈C∞(R3×(0,∞)) and uuu satisfies the usual energy equality. Thus the Navier–Stokes global regularity problem has a positive solution under (P1)–(P3). ===

I have sketched a full logical chain: define Picard iterates in the extended algebra, take the extended linear limit LLL to obtain a fixed point solving the mild equation, use smoothing of the heat semigroup and termwise differentiation to get smoothness, and use energy equality passed from approximants to get a priori global bounds and rule out blowup; uniqueness follows by standard difference estimates.