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

==== 1. Decide which route you prefer. (A) Classical functional route (standard, rigorous, but will only give conditional results unless you beat Clay’s core difficulty). (B) Extended algebra route (foundational; requires proving extra axioms). ====
# If you pick Route A (recommended): - Fix X=HsX=H^sX=Hs with s>3/2s>3/2s>3/2. Write a full Picard existence theorem in C([0,T];Hs)C([0,T];H^s)C([0,T];Hs) with constants. I can produce a complete, formal write-up with all constants and estimates. - State and prove the blowup alternative: if maximal time T∗<∞T^''<\inftyT∗<∞, then lim⁡t↑T∗∥u(t)∥Hs=∞\lim_{t\uparrow T^''}\|u(t)\|_{H^s}=\inftylimt↑T∗∥u(t)∥Hs=∞. - Then either attempt to produce a novel a-priori estimate to prevent blowup (this is hard and likely where the original obstruction lies) or prove conditional theorems (Serrin, BKM) clearly and sharply.
# If you pick Route B: - Write precise axioms for LLL (not informal P1–P3) including (C1)–(C3) above. - Either (i) show a concrete construction of LLL satisfying them, or (ii) demonstrate consistency relative to standard mathematics (this is a foundational project). - Then convert your conditional proof into a formal theorem with the axioms explicitly stated.
# Whichever route you pick, be explicit about admissibility. Add an energy/admissibility requirement to the solution concept (energy inequality or equality), then prove uniqueness in that admissible class. This prevents pathological extra solutions from disqualifying your result.