03 / C3-A: The Conservation–Criticality–Positivity Trilemma and Divergence of Bihelical Pair Production

Having shown in C2 that a scalar, additive energy budget alone cannot rule out a critical geometric cascade, this round formally brings in finer structure from the true N–S nonlinearity. Starting from three natural quadratic quantities — kinetic energy $\|u\|_2^2$ (positive, subcritical, nonlinearly conserved), helicity $H$ (scale-critical, nonlinearly conserved, but of indefinite sign), and the critical quantity $\|u\|_{\dot H^{1/2}}^2$ (positive, scale-critical, but with no exact nonlinear conservation law) — it tabulates these and finds that the three properties (positive / critical / nonlinearly conserved) cannot all be obtained simultaneously from any one of these three most natural quantities, constituting the Conservation–Criticality–Positivity Trilemma. Using the helical projection $u=u^++u^-$, it defines the critical size of each sign sector $H_\pm=\|D^{1/2}u^\pm\|_2^2$, and derives from nonlinear helicity conservation that the two sectors' production rates must be equal, $\mathcal R_+=\mathcal R_-$; the common value $\mathcal R$ is called the critical helical pair-production rate. Main theorem (12.1): if $T_\ast$ is a finite blow-up time, then $\int_0^{T_\ast}[\mathcal R]_+\,dt=\infty$ — the proof needs only $\dot H^{1/2}\hookrightarrow L^3$ to convert C1's $L^3$ escape into divergence of a critical quadratic quantity, then uses the pair-production identity to rule out a finite budget. The document stresses that this is the first genuinely unavoidable condition to use the full structure of $B(u,u)$ rather than the energy identity alone — finer than C2's critical toll, since $\mathcal R$ is not an energy flux but a specific projection in helical coordinates. It cites Biferale–Titi's global-regularity result for sign-definite, helically decimated N–S as an external point of comparison: once the opposite-chirality degrees of freedom are removed, helicity becomes sign-definite and equal to the critical size, the trilemma is genuinely lifted, and global regularity can be proved — showing that the mixed-helicity degrees of freedom are not a decoration that can be dropped at will, but the genuine source of the difficulty. The document also proposes a candidate minority-factor estimate (bounding $\mathcal R$ by the smaller of the two sectors), but explicitly flags it as a CANDIDATE LEMMA, not to be used as a theorem.