15 / C3-M: Vorticity–Strain Coupling, Betchov Global Collapse, and the Directional-Geometry Debt

C3-L left two parallel channels that must diverge: spectral-moment escape and middle-strain escape. This round asks: can the two be forced to couple through genuine vortex-stretching geometry? The answer is not the simple claim that "vorticity must align with some strain eigenvector." The core is the Exact Stretching-Orientation Identity (Theorem 4.1): the instantaneous stretching rate along the vorticity direction $\alpha=\lambda_2+(\lambda_3-\lambda_2)c_3-(\lambda_2-\lambda_1)c_1$ — split exactly into three typed components: a middle-eigenvalue baseline, a principal-stretching-direction surplus, and a compressive-direction depletion. From this comes the Excess-Stretching Alignment Debt (Theorem 7.1): if the middle strain value alone cannot account for the stretching, vorticity must pay a debt in alignment with the most-stretching eigenvector, giving the Middle-Strain / Principal-Alignment Carrier Dichotomy (Theorem 10.1) — but the document explicitly warns that this is not the same as "blow-up forces vorticity to align with the most-stretching direction," since the first branch (borne by the middle strain) is already sufficient on its own. The paper's sharpest correction sits in §12-14: invoking the Betchov identity $\int\omega\cdot S\omega=-4\int\det S$, it proves that the global integral erases all directional information (Global Orientation Collapse) — local orientation collapses, through $\int dx$, into the global strain determinant, retaining no trace of $\xi$ whatsoever. Consequently "large global enstrophy growth $\Rightarrow$ vorticity aligns with the most-stretching eigenvector" is an illegitimate inference, formally refuting an intuition common in the numerical literature (NG-M1). The true global carrier is instead shown to be two-positive-eigenvalue strain geometry ($\lambda_2>0$, theorems/propositions §15-17). The document also carefully distinguishes "local eigenframe alignment" from "spatial coherence of the vorticity direction" as two different geometries — only the latter is the actual target of Constantin–Fefferman-type geometric-depletion theorems — and cites the latest 2026 Grujić log-BMO directional-control result as a conditional interface. Finally, it invokes Miller's 2024/2026 inverse-coupling orthogonality $\langle-\Delta S,\omega\otimes\omega\rangle=0$, proving that "vorticity acting on strain" by itself is not sufficient to account for blow-up. The research frontier formally turns to localized Betchov compensation, rather than searching for yet another global scalar identity.