21 / C3-S: Strain-Cone Margin, Cross-Scale Separator Compactness, and Merger Rigidity

C3-R answered only a binary question: whether a common far-field matrix exists at all. This round upgrades it to a quantitative one. The core definition is the margin $\gamma(V):=\operatorname{dist}(0,\operatorname{conv}V)$ — the distance from the origin to the convex hull of the local mean-strain set $V$; $\gamma(V)>0$ is exactly equivalent to C3-R's positive-support criterion holding, but now how far it holds by is also quantified. The core is the Uniform Margin Compactness Theorem (Theorem 6.1): if the margin has a uniform lower bound $\gamma(V_R)\ge\gamma_0>0$ across all scales, then, by compactness of the unit sphere in $\operatorname{Sym}_0(3)$, there exists a single fixed cross-scale separating matrix $K^\ast$ that simultaneously witnesses positive support for the core set at every scale — there is no need to hunt for a new separating matrix scale by scale; a uniform margin directly yields one universal witness. Merger Inheritance (Theorem 14.1) then proves: when two core clusters merge, as long as each retains its own lower bound $\gamma_0$, the margin of the merged set does not collapse arbitrarily — it shrinks only within an explicit factor controlled by the angular geometry — merging does not destroy uniformity, though it does exact a computable cost. The degenerate branch treats the case $\gamma\to0$: the document proves this corresponds exactly to C3-R's Carathéodory obstruction becoming tight — the Six-Core Near-Balance Witness (Proposition 21.1) shows that as the margin tends to zero, six-core configurations approach a critical arrangement in which the origin lands exactly on the boundary of the convex hull — precisely the configuration at which Carathéodory's bound saturates, no more and no less. This round's strongest quantitative result is a refined far-field pressure enstrophy debt (Theorem 27.1): $\mathfrak E_R\gtrsim\kappa^2\gamma^{-2/3}$ — the smaller the margin, the higher the rescaled-enstrophy cost required to sustain decoupling, further refining C3-P/C3-Q's $\kappa^2$ debt by the margin. The document explicitly flags NG-S1: this bound does not exclude $\gamma\to0$; it merely attaches a diverging price tag to it — if the rescaled enstrophy really can grow without bound, a vanishing margin remains algebraically permissible on its own, and a genuine obstruction must come from an independent upper bound on $\mathfrak E_R$ found elsewhere, which remains OPEN.