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We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let $\\mathcal M_\\varepsilon^\\gamma$ be the root-intensity measure of loops with duration in $[\\varepsilon^2,t_0]$ whose traces hit $\\gamma$. For every $f\\in C_c(D)$, we prove that $\\varepsilon^{5/4}\\mathcal M_\\varepsilon^\\gamma(f)$ converges in $L^1$ to $\\frac{4}{5\\pi}\\bar v_{\\mathrm{BB}}\\mu_\\gamma(f)$, where $\\bar v_{\\ma"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.26439","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-29T03:37:21Z","cross_cats_sorted":[],"title_canon_sha256":"592a4418deaad554525137fc9fdac3e53413dece8f67471a4b41b23c74f9760c","abstract_canon_sha256":"5e562eac2357eb85096478978392f17297f74da41494d6dc665b66c5dca5e609"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"13394d73742e143cac9cf921aa58b1705622e6e630c26a8ea1e0e9b95cde8c97","last_reissued_at":"2026-07-30T01:18:25.840768Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:18:25.840768Z"},"graph_snapshot":{"paper":{"title":"Small Brownian Loops Hitting SLE$_2$: An Exact Natural-Content Limit","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Zhengwen Qiao","submitted_at":"2026-07-29T03:37:21Z","abstract_excerpt":"Let $D$ be a bounded analytic Jordan domain and let $\\gamma$ be chordal $\\mathrm{SLE}_2$ in $D$, equipped with its $5/4$-dimensional natural-content measure $\\mu_\\gamma$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let $\\mathcal M_\\varepsilon^\\gamma$ be the root-intensity measure of loops with duration in $[\\varepsilon^2,t_0]$ whose traces hit $\\gamma$. 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