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For graph domains \\[r<R-V(\\theta)\\], the first band satisfies \\[ \\lambda_{j+1}=\\alpha^2+\\frac{\\pi^2}{R^2} +\\frac{2\\pi^2}{R^3}\\bigl(\\eta_j(T_n+V)-b_0\\bigr)+o(R^{-3}), \\qquad j=0,1, \\] where \\[T_n\\] is the nonlocal spherical operator with multiplier \\[\\psi(\\ell+\\alpha)-\\psi(\\alpha)\\]. Consequently the horoconvex fundamental gap has the sharp \\"},"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":"2606.10416","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2026-06-09T04:43:08Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"b43ad6ce56f56573b0a8d9b79904951537e9256ddae32dc1dabf3f5af7206b50","abstract_canon_sha256":"5599134d06b8152b9159cafa72a195b4a8198941f5e59449ae98a11c6ea83af0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:10:17.802162Z","signature_b64":"Ul0ENRUKULA/LaDaH8ppG8LVaRaAlvvVes+B1n/o37ZD8+l5N6Qa8iXRisFBQgi0KOQp/L+NpFte+CGSiHOWDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d2d9f360a20ca0c3d2875a7de7d68b1e6aa05d67171222290fc1674bf31af92","last_reissued_at":"2026-06-10T01:10:17.801425Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:10:17.801425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Effective Angular Asymptotics and the Sharp $D^{-3}$ Horoconvex Gap Scale","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.SP","authors_text":"Sanghyun Park","submitted_at":"2026-06-09T04:43:08Z","abstract_excerpt":"We prove first-band large-diameter asymptotics for the Dirichlet spectrum on horoconvex domains in real hyperbolic space. After Chebyshev centering, a divergent sequence compactifies to a horospherical support-envelope deficit \\[V\\] on \\[\\mathbb S^{n-1}\\]. For graph domains \\[r<R-V(\\theta)\\], the first band satisfies \\[ \\lambda_{j+1}=\\alpha^2+\\frac{\\pi^2}{R^2} +\\frac{2\\pi^2}{R^3}\\bigl(\\eta_j(T_n+V)-b_0\\bigr)+o(R^{-3}), \\qquad j=0,1, \\] where \\[T_n\\] is the nonlocal spherical operator with multiplier \\[\\psi(\\ell+\\alpha)-\\psi(\\alpha)\\]. Consequently the horoconvex fundamental gap has the sharp \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10416","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.10416/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.10416","created_at":"2026-06-10T01:10:17.801540+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.10416v1","created_at":"2026-06-10T01:10:17.801540+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.10416","created_at":"2026-06-10T01:10:17.801540+00:00"},{"alias_kind":"pith_short_12","alias_value":"BUWZ6NQKEDFA","created_at":"2026-06-10T01:10:17.801540+00:00"},{"alias_kind":"pith_short_16","alias_value":"BUWZ6NQKEDFAYPJI","created_at":"2026-06-10T01:10:17.801540+00:00"},{"alias_kind":"pith_short_8","alias_value":"BUWZ6NQK","created_at":"2026-06-10T01:10:17.801540+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH","json":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH.json","graph_json":"https://pith.science/api/pith-number/BUWZ6NQKEDFAYPJIOWT547LIWH/graph.json","events_json":"https://pith.science/api/pith-number/BUWZ6NQKEDFAYPJIOWT547LIWH/events.json","paper":"https://pith.science/paper/BUWZ6NQK"},"agent_actions":{"view_html":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH","download_json":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH.json","view_paper":"https://pith.science/paper/BUWZ6NQK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.10416&json=true","fetch_graph":"https://pith.science/api/pith-number/BUWZ6NQKEDFAYPJIOWT547LIWH/graph.json","fetch_events":"https://pith.science/api/pith-number/BUWZ6NQKEDFAYPJIOWT547LIWH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH/action/storage_attestation","attest_author":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH/action/author_attestation","sign_citation":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH/action/citation_signature","submit_replication":"https://pith.science/pith/BUWZ6NQKEDFAYPJIOWT547LIWH/action/replication_record"}},"created_at":"2026-06-10T01:10:17.801540+00:00","updated_at":"2026-06-10T01:10:17.801540+00:00"}