16 / C3-N: Localized Betchov Boundary Current and the Local Balance of Strain Self-Amplification

C3-M established only that the local Betchov surplus must be compensated outside the core. This round formally closes that gap: it first uses the purely kinematic identity $\operatorname{tr}(A^3)=3\det S+\frac34\omega\cdot S\omega$ to define $b_B=\omega\cdot S\omega+4\det S=\frac43\operatorname{tr}(A^3)$, then invokes the Carbone–Wilczek result $\operatorname{tr}(A^3)=\nabla\cdot F_B$ (where $F_B=(A^2-\frac12\operatorname{tr}(A^2)I)u$) — a purely kinematic identity with no time evolution involved. The Localized Betchov Boundary Theorem (Theorem 4.1): $\int\chi b_B=-\frac43\int\nabla\chi\cdot F_B$, which for a ball can be written as a boundary surface integral — the local Betchov mismatch is exactly a precise spatial divergence flux, which must cross the localization boundary rather than being an arbitrary far-field bookkeeping entry. The document also supplies a second, companion identity (Theorem 10.1, corresponding to $|S|^2-\frac12|\omega|^2=\nabla\cdot(Au)$). The core result is the Exact Local Strain Self-Amplification Balance (Theorem 17.1): taking the inner product of the strain equation with $\chi S$ gives, exactly, $\frac d{dt}E_S^\chi+\nu\int\chi|\nabla S|^2=-2\int\chi\det S+\mathcal C_\chi$ — the only cubic generation term retained inside the bulk is $-2\int\chi\det S$ (bulk self-amplification); vorticity mismatch, advection, the pressure Hessian, the viscous localization correction, and the moving-core gauge term all fall into the boundary/gauge correction package $\mathcal C_\chi$. The paper's most important NO-GO appears in the scaling audit (§25-28): it proves that $\int\chi_{R_\lambda}b_{B,\lambda}\,dx=\lambda^3\int\chi_Rb_B\,dx$ carries exactly the same instantaneous scaling $\lambda^3$ as the bulk self-amplification term — so $R\to0$ by itself does not make the boundary contribution negligible. §28 is stronger still: global kinetic energy controls only $\nu\int\|\nabla u\|_2^2dt$, with no uniform control on the boundary term's integral over the shrinking radius, so an exact boundary representation does not amount to a finite boundary budget — one of this round's most important no-gos. The document also carefully notes that $F_B$ is the kinematic spatial flux that makes the identity hold; it is not an energy flux, not a signed conserved quantity, and not an irreversible expenditure, so it cannot be treated directly as "a finite cost paid once per generation." The research frontier turns to C3-O: whether the asymptotic bulk-vs-boundary ratio can be forced into some impossible state.