{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:XYBSELRPUI6QSFTY7GA6XEPF4Q","short_pith_number":"pith:XYBSELRP","schema_version":"1.0","canonical_sha256":"be03222e2fa23d091678f981eb91e5e4383de85fa32fd89cf29e0d930e3cea5f","source":{"kind":"arxiv","id":"2605.31456","version":1},"attestation_state":"computed","paper":{"title":"Geometric Analysis of the Damped Harmonic Oscillator via the Lambert W Function","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Arpan Sharma, Bhargava Jogi, Ken Roberts, Muralikrishna Molli, S.R. Valluri","submitted_at":"2026-05-29T15:50:13Z","abstract_excerpt":"The underdamped harmonic oscillator is analyzed through the complex mapping $\\zeta = e^{-i\\varphi}we^{-w}$ with $w = \\beta t + i\\Omega t$, which transforms the dynamics into a logarithmic spiral. Within this framework, the displacement extrema correspond to crossings of the imaginary axis by $\\zeta(t)$, yielding the explicit times $t_n = (\\theta - \\varphi - \\pi/2 + n\\pi)/\\Omega$, where $\\theta = \\arctan(\\Omega/\\beta)$. The Lambert $W$ function provides closed-form solutions $t = -\\beta^{-1}W_k(-\\beta A/\\omega_0)$ for the times at which the spiral radius attains a given threshold $A$, covering "},"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":"2605.31456","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math-ph","submitted_at":"2026-05-29T15:50:13Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"6e2a5b8d68e2784af5ca97c0c61c37d62f4a39fbc9b36a68f7e77edf86e4100c","abstract_canon_sha256":"4dc55e6598f177ca630ea220ea5879cbea1904526969c9b32e4858cbb1bd5238"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T02:04:06.814731Z","signature_b64":"D7svzHlLdV60ITAmL3NZY3dTybiuGrysfRi4uJuffcWl5CLkBIrQLdnWbbe+Vxff0XfKzSvS9D+Tbg2CNF3rCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be03222e2fa23d091678f981eb91e5e4383de85fa32fd89cf29e0d930e3cea5f","last_reissued_at":"2026-06-01T02:04:06.813922Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T02:04:06.813922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric Analysis of the Damped Harmonic Oscillator via the Lambert W Function","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Arpan Sharma, Bhargava Jogi, Ken Roberts, Muralikrishna Molli, S.R. Valluri","submitted_at":"2026-05-29T15:50:13Z","abstract_excerpt":"The underdamped harmonic oscillator is analyzed through the complex mapping $\\zeta = e^{-i\\varphi}we^{-w}$ with $w = \\beta t + i\\Omega t$, which transforms the dynamics into a logarithmic spiral. Within this framework, the displacement extrema correspond to crossings of the imaginary axis by $\\zeta(t)$, yielding the explicit times $t_n = (\\theta - \\varphi - \\pi/2 + n\\pi)/\\Omega$, where $\\theta = \\arctan(\\Omega/\\beta)$. The Lambert $W$ function provides closed-form solutions $t = -\\beta^{-1}W_k(-\\beta A/\\omega_0)$ for the times at which the spiral radius attains a given threshold $A$, covering "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.31456","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/2605.31456/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":"2605.31456","created_at":"2026-06-01T02:04:06.814064+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.31456v1","created_at":"2026-06-01T02:04:06.814064+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.31456","created_at":"2026-06-01T02:04:06.814064+00:00"},{"alias_kind":"pith_short_12","alias_value":"XYBSELRPUI6Q","created_at":"2026-06-01T02:04:06.814064+00:00"},{"alias_kind":"pith_short_16","alias_value":"XYBSELRPUI6QSFTY","created_at":"2026-06-01T02:04:06.814064+00:00"},{"alias_kind":"pith_short_8","alias_value":"XYBSELRP","created_at":"2026-06-01T02:04:06.814064+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/XYBSELRPUI6QSFTY7GA6XEPF4Q","json":"https://pith.science/pith/XYBSELRPUI6QSFTY7GA6XEPF4Q.json","graph_json":"https://pith.science/api/pith-number/XYBSELRPUI6QSFTY7GA6XEPF4Q/graph.json","events_json":"https://pith.science/api/pith-number/XYBSELRPUI6QSFTY7GA6XEPF4Q/events.json","paper":"https://pith.science/paper/XYBSELRP"},"agent_actions":{"view_html":"https://pith.science/pith/XYBSELRPUI6QSFTY7GA6XEPF4Q","download_json":"https://pith.science/pith/XYBSELRPUI6QSFTY7GA6XEPF4Q.json","view_paper":"https://pith.science/paper/XYBSELRP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.31456&json=true","fetch_graph":"https://pith.science/api/pith-number/XYBSELRPUI6QSFTY7GA6XEPF4Q/graph.json","fetch_events":"https://pith.science/api/pith-number/XYBSELRPUI6QSFTY7GA6XEPF4Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XYBSELRPUI6QSFTY7GA6XEPF4Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XYBSELRPUI6QSFTY7GA6XEPF4Q/action/storage_attestation","attest_author":"https://pith.science/pith/XYBSELRPUI6QSFTY7GA6XEPF4Q/action/author_attestation","sign_citation":"https://pith.science/pith/XYBSELRPUI6QSFTY7GA6XEPF4Q/action/citation_signature","submit_replication":"https://pith.science/pith/XYBSELRPUI6QSFTY7GA6XEPF4Q/action/replication_record"}},"created_at":"2026-06-01T02:04:06.814064+00:00","updated_at":"2026-06-01T02:04:06.814064+00:00"}