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We show that a uniformly random degree-$n$ cover $X_n$ of $X$ has no eigenvalues below $\\lambda_0(\\widetilde{X})-\\varepsilon$ other than those of $X$ and with the same multiplicity, with probability tending to $1$ as $n\\to \\infty$. 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Let $\\lambda_0(\\widetilde{X})$ denote the bottom of the $L^2-$spectrum of the Laplacian on the universal cover $\\widetilde{X}$. We show that a uniformly random degree-$n$ cover $X_n$ of $X$ has no eigenvalues below $\\lambda_0(\\widetilde{X})-\\varepsilon$ other than those of $X$ and with the same multiplicity, with probability tending to $1$ as $n\\to \\infty$. This extends a result of Hide--Magee to metrics of pinched negative curvature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.07056","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/2505.07056/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":"2505.07056","created_at":"2026-07-05T11:01:41.257047+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.07056v1","created_at":"2026-07-05T11:01:41.257047+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.07056","created_at":"2026-07-05T11:01:41.257047+00:00"},{"alias_kind":"pith_short_12","alias_value":"V36JWMHVRQSM","created_at":"2026-07-05T11:01:41.257047+00:00"},{"alias_kind":"pith_short_16","alias_value":"V36JWMHVRQSMSK5P","created_at":"2026-07-05T11:01:41.257047+00:00"},{"alias_kind":"pith_short_8","alias_value":"V36JWMHV","created_at":"2026-07-05T11:01:41.257047+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.17466","citing_title":"On the spectral stability of finite coverings","ref_index":23,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY","json":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY.json","graph_json":"https://pith.science/api/pith-number/V36JWMHVRQSMSK5PDFTTC26DRY/graph.json","events_json":"https://pith.science/api/pith-number/V36JWMHVRQSMSK5PDFTTC26DRY/events.json","paper":"https://pith.science/paper/V36JWMHV"},"agent_actions":{"view_html":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY","download_json":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY.json","view_paper":"https://pith.science/paper/V36JWMHV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.07056&json=true","fetch_graph":"https://pith.science/api/pith-number/V36JWMHVRQSMSK5PDFTTC26DRY/graph.json","fetch_events":"https://pith.science/api/pith-number/V36JWMHVRQSMSK5PDFTTC26DRY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY/action/storage_attestation","attest_author":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY/action/author_attestation","sign_citation":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY/action/citation_signature","submit_replication":"https://pith.science/pith/V36JWMHVRQSMSK5PDFTTC26DRY/action/replication_record"}},"created_at":"2026-07-05T11:01:41.257047+00:00","updated_at":"2026-07-05T11:01:41.257047+00:00"}