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

==== 1. Definition vs existence gap. You defined mild solutions in C([0,T];X)C([0,T];X)C([0,T];X) and produced a fixed point there. But the Clay statement asks: every smooth divergence-free initial datum leads to a global classical (real-valued C∞C^\inftyC∞) solution. Your proof constructs a solution inside a chosen solution class; it does not logically exclude the possibility that there also exist other solutions (distributional/pathological) that fail to be classical. The Clay problem requires that the only physically relevant solution (given usual admissibility) is classical — equivalently, that no finite-time singularity can exist for any admissible evolution. ====
# Passing limits and nonlinearities. Your key step takes Picard iterates u(n)u^{(n)}u(n) (classical for t>0t>0t>0) and applies an extended linear limit LLL. That step is only legitimate for concluding the limit is classical if you either: - prove LLL sends sequences of classical functions to classical functions (extra property), or - prove convergence in a topology that embeds into the classical space (e.g. strong HsH^sHs or uniform convergence plus control of derivatives), so the limit inherits classical regularity. Neither has been shown.
# Nonlinear functionals and energy. You used energy equalities/inequalities and algebraic manipulations across the limit. Those are nonlinear operations; to pass them through the limit you need continuity properties of LLL or quantitative norm convergence. This was not established.

Because of the three points above, the draft is a conditional argument that only yields: “if the extended limit acts nicely (preserves classicality and commutes with the nonlinear operations), then we get a global classical solution.” That is not a resolution of the Clay problem unless you either build or justify those extra properties.