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

=== 1. Nature of the premises. The three premises you asked me to accept are extremely strong — they go well beyond classical distribution theory and even beyond many nonlinear generalized function frameworks (for example, classical tempered distributions cannot be multiplied arbitrarily). What you have assumed is an algebraic and analytic enlargement of the function space where limits, derivatives, integrals and multiplication are all defined and compatible with classical identities. Such an enlargement, if constructed concretely and shown consistent with classical axioms, would bypass classical obstructions to solving nonlinear PDEs. But neither existence nor consistency of such a structure is a trivial matter. In particular, one must show that the extended operations agree with classical operations where both are defined and do not introduce contradictions. ===
# Relation to known generalized-function approaches. There are known frameworks (Colombeau algebras, ultrafunctions, nonstandard analysis, etc.) that allow multiplication of distributions and extended operations. However, each such framework comes with caveats: either they do not preserve all classical algebraic identities in the way you require, or the embedding of classical spaces is only partial, or they require new equivalence relations that make statements about “classical smoothness” more subtle. Your premises assert a particularly strong version of extension that makes the Picard/Duhamel approach run globally; this is why the proof above is clean and short: the premises remove the classical technical obstructions. But to answer the Millennium Prize question in the standard mathematical community one must either (a) provide such a construction of the extended algebra and prove it is consistent and that the extended solution restricts to a classical smooth solution in the standard sense, or (b) prove the same result within standard functional analysis (without assuming these sweeping extension axioms).
# What remains to be done to make this an accepted Millennium solution? The conditional proof above shows that if the premises are true and if they are shown to be consistent with the standard axioms of mathematics and to agree with classical operations on classical functions, then the Navier–Stokes regularity problem is resolved positively. But the real hard work — and the part that the Millennium Prize requires — is to either: - Provide a rigorous, widely accepted construction of the extended linear limit and extended differentiation/integration/multiplication with all the properties you assumed (and prove it does not conflict with other established mathematics), or - Remove the premises and give a proof that uses only classical, accepted functional-analytic methods.
# Honest claim about novelty. I did not claim that the above axioms are true in current mathematics; I only showed the logical consequences of accepting them. Producing a rigorous, accepted construction of those axioms would be a major mathematical achievement beyond the scope of this derivation and is precisely the nontrivial core.