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We prove that the plunge count $\\Lambda_\\varepsilon=\\#\\{n:\\varepsilon<\\lambda_n(S_{cA_0,B_0})<1-\\varepsilon\\}$ satisfies $\\Lambda_\\varepsilon \\le C(A_0,B_0)\\,\\widetilde{L}\\,(1+\\ln_+(ca/\\widetilde{L}))$, with $\\widetilde{L}=\\ln(1/(\\varepsilon(1-\\varepsilon)))$, for all $c>0$ and $0<\\varepsilon"},"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":"2607.23016","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-07-25T03:17:05Z","cross_cats_sorted":["math.CA","math.SP"],"title_canon_sha256":"cbffde0b2f983eb80a8393ce1498e2679aeac7dc28515c3f2a82ec1239dad377","abstract_canon_sha256":"b742182e8c41c363357224e2706a583a801fa0c2daaf5a43981097297d5dbabf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T00:22:07.701639Z","signature_b64":"qkoMEc5fwSt2H7J3BlBLNs2hDaWtcEEji+qufc5l1m4Xiq1JVGPWYRria5o2G4u4pGdBoJFrZrAAg6kkBNUnAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2de6641ed852589c8569d4aab80bf852a0210df67c9097bdd1bce3d96bd7f2fa","last_reissued_at":"2026-07-28T00:22:07.700798Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T00:22:07.700798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An independent proof of the plunge-region conjecture for time-frequency localization operators in dimension one","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.SP"],"primary_cat":"math.FA","authors_text":"Ahmadreza Azimifard","submitted_at":"2026-07-25T03:17:05Z","abstract_excerpt":"Let $A_0,B_0\\subset\\mathbb{R}$ be bounded measurable sets of positive measure with finite topological boundaries, and let $S_{cA_0,B_0}=P_{cA_0}Q_{B_0}P_{cA_0}$ be the associated time-frequency localization operator, where $P_E$ is multiplication by $\\mathbf{1}_E$ and $Q_E=\\mathcal{F}^{-1}P_E\\mathcal{F}$. We prove that the plunge count $\\Lambda_\\varepsilon=\\#\\{n:\\varepsilon<\\lambda_n(S_{cA_0,B_0})<1-\\varepsilon\\}$ satisfies $\\Lambda_\\varepsilon \\le C(A_0,B_0)\\,\\widetilde{L}\\,(1+\\ln_+(ca/\\widetilde{L}))$, with $\\widetilde{L}=\\ln(1/(\\varepsilon(1-\\varepsilon)))$, for all $c>0$ and $0<\\varepsilon"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23016","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/2607.23016/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":"2607.23016","created_at":"2026-07-28T00:22:07.701225+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.23016v1","created_at":"2026-07-28T00:22:07.701225+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.23016","created_at":"2026-07-28T00:22:07.701225+00:00"},{"alias_kind":"pith_short_12","alias_value":"FXTGIHWYKJMJ","created_at":"2026-07-28T00:22:07.701225+00:00"},{"alias_kind":"pith_short_16","alias_value":"FXTGIHWYKJMJZBLJ","created_at":"2026-07-28T00:22:07.701225+00:00"},{"alias_kind":"pith_short_8","alias_value":"FXTGIHWY","created_at":"2026-07-28T00:22:07.701225+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/FXTGIHWYKJMJZBLJ2SVLQC7YKK","json":"https://pith.science/pith/FXTGIHWYKJMJZBLJ2SVLQC7YKK.json","graph_json":"https://pith.science/api/pith-number/FXTGIHWYKJMJZBLJ2SVLQC7YKK/graph.json","events_json":"https://pith.science/api/pith-number/FXTGIHWYKJMJZBLJ2SVLQC7YKK/events.json","paper":"https://pith.science/paper/FXTGIHWY"},"agent_actions":{"view_html":"https://pith.science/pith/FXTGIHWYKJMJZBLJ2SVLQC7YKK","download_json":"https://pith.science/pith/FXTGIHWYKJMJZBLJ2SVLQC7YKK.json","view_paper":"https://pith.science/paper/FXTGIHWY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.23016&json=true","fetch_graph":"https://pith.science/api/pith-number/FXTGIHWYKJMJZBLJ2SVLQC7YKK/graph.json","fetch_events":"https://pith.science/api/pith-number/FXTGIHWYKJMJZBLJ2SVLQC7YKK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FXTGIHWYKJMJZBLJ2SVLQC7YKK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FXTGIHWYKJMJZBLJ2SVLQC7YKK/action/storage_attestation","attest_author":"https://pith.science/pith/FXTGIHWYKJMJZBLJ2SVLQC7YKK/action/author_attestation","sign_citation":"https://pith.science/pith/FXTGIHWYKJMJZBLJ2SVLQC7YKK/action/citation_signature","submit_replication":"https://pith.science/pith/FXTGIHWYKJMJZBLJ2SVLQC7YKK/action/replication_record"}},"created_at":"2026-07-28T00:22:07.701225+00:00","updated_at":"2026-07-28T00:22:07.701225+00:00"}