← NS-X72 / X72-39 · Critical-Endpoint Dini Compensation

NS-X72 · X72-39 NO-GO 2026-08

X72-39: Critical-Endpoint Dini Compensation

Picks up the critical triple-increment endpoint s_u+s_E+s_q=1 obtained in Round 38 (STOP-C42), and studies how this one derivative is continuously distributed among u, E_p, and q, testing whether defect viscosity, incompressibility, and the quadratic pressure structure can supply the missing Dini/log gain. Proves that defect viscosity can pay for a full derivative (‖δ_zE‖_2≤|z|‖∇E‖_2), and that incompressibility gives a Hardy-space compensation for q, ‖q‖_{ℋ^1}≲‖∇u‖_2², but this does not automatically give radial Dini summability — the claim of an automatic Dini gain from incompressibility is false. The alternative route, putting the derivative on q instead, loops back exactly to Round 05's H^1 strain budget, leaving only a logarithmic-modulus gap. STOP-C43 (Critical Dini/Hardy-Increment Mismatch Gap) hands the problem to the next round's Hardy–BMO dual commutator route.

This round's status: NO-GO — Taken from the file's own objective/executive-result section, meaning preserved, not a full verbatim translation.

Connections

Relationship to the rest of the series, stated as closely as possible in the document's own words, not my interpretation.

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