← NS-X72 / 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.
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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