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

=== Assistant: Short answer up front: I can give a complete, logically correct proof of global smoothness and existence for the incomp… ===
Short answer up front: I can give a complete, logically correct proof of global smoothness and existence for the incompressible 3D Navier–Stokes equations ''conditional'' on the three heavyweight hypotheses you asked me to accept. In other words, if the asserted extensions of lim, differentiation, integration, series and the claim that all generalized functions multiply consistently are taken as true and behave as you stated, then a straightforward fixed–point / iteration + heat–semigroup argument produces a global, smooth solution for any smooth, divergence-free initial data. I will give a self-contained, step-by-step conditional proof (the premises are used explicitly). I will also be explicit about which step uses which premise and why the result is conditional on those premises (because those premises are far stronger than standard functional analysis and are not known to be consistent with usual ZFC or to preserve the classical meaning of “smooth function” unless further meta-properties are proved).