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

==== - Using standard PDE methods (Route A), you can produce a correct, rigorous proof of local existence, conditional regularity results, and global regularity for special cases (small data, symmetries). None of these are new breakthroughs with the present sketch. The Clay problem is precisely the missing global a-priori bound in 3D — the impossibility lies in obtaining it without new ideas. ====
* Using Route B, you may obtain a conditional global regularity result if you can construct and justify the extended algebra with the extra properties listed. That shifts the burden: you must show those axioms are mathematically legitimate. If you can, then your conditional theorem is logically correct — but its acceptance as a Clay solution would hinge on the acceptability of the new foundations.

If you want, I will now do one of the following (pick one):
# Write a complete, formal Picard existence + blowup alternative proof in C([0,T];Hs)C([0,T];H^s)C([0,T];Hs) (with all estimates). This gives the rigorous baseline and makes clear what must be controlled to get global existence.
# Produce a minimal list of precise axioms for an extended limit operator LLL (formal statements like (C1)–(C3)) and check whether any existing generalized-function framework satisfies them.
# Draft a concise conditional theorem: state exactly the extra properties you must assume about LLL and then produce the full rigorous derivation showing global smoothness follows from those assumptions.

Which do you want next?