Small Brownian Loops Hitting SLE${}_2$: An Exact Natural-Content Limit

Let $D$ be a bounded analytic Jordan domain and let $\gamma$ be chordal ${\mathrm{S}\mathrm{L}\mathrm{E}}_2$ in $D$, equipped with its $5\mathrm{/}4$-dimensional natural-content measure ${\mu }_{\gamma }$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let ${\mathcal{M}}_{\epsilon }^{\gamma }$ be the root-intensity measure of loops with duration in $[{\epsilon }^2,t_0]$ whose traces hit $\gamma$. For every $f\in C_c(D)$, we prove that ${\epsilon }^{5\mathrm{/}4}{\mathcal{M}}_{\epsilon }^{\gamma }(f)$ converges in $L^1$ to $\frac{4}{5\pi }{\bar{v}}_{\mathrm{B}\mathrm{B}}{\mu }_{\gamma }(f)$, where ${\bar{v}}_{\mathrm{B}\mathrm{B}}\in (0,\mathrm{\infty })$ is the mean specific area swept out by the Brownian-bridge offset of a natural-time two-sided whole-plane ${\mathrm{S}\mathrm{L}\mathrm{E}}_2$. In particular, the positive random measures converge vaguely in probability. The proof uses a duration-octave identity, a deterministic finite-$R$ reference coefficient obtained from a stopped two-arm Markov skeleton, a marked physical Palm tangent, an annular remote-return estimate, and a legal mesoscopic diagonal. As an application, the uniformly time-marked roots of an independent Brownian loop soup satisfy the corresponding vague law of large numbers. The analytic-boundary hypothesis enters only through a global finite-domain uniform-integrability estimate; its analogue for an arbitrary bounded Jordan domain remains open.