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We develop a new Gordon-type method that exploits approximate repetitions and palindromic symmetries simultaneously. Denote by $L(E)$ the Lyapunov exponent and let\n  \\[\\beta(\\alpha) =\n\\limsup_{|k|\\to\\infty} -\\frac{\\log\\|k\\alpha\\|_{\\mathbb R/\\mathbb Z}}{|k|}. \\] We prove that, for every completely resonant phase $2\\theta\\in\\alpha\\mathbb Z+\\mathb"},"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":"2607.24188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-27T09:07:52Z","cross_cats_sorted":["math.MP","math.SP"],"title_canon_sha256":"621f4f43d8d1ec785266d4269490ccaed81a2d0554c39488afe44e091cbe79dc","abstract_canon_sha256":"301e0e5aade227805a1176a8ef3ccf4f60577d36c3a10c3e380b3d72bc7cb4dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T02:23:49.928230Z","signature_b64":"gPZ0VnFVlRPws4QW8MvSKm0qFbPN9GLnd6UOG6yAz7OTGRGoylemTIg/hta8JT4KgpjW2JqX9j/+CE6xngjCCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d1d6b9548f0bde417eb1e03ef748df85c5c3f2800f9f2786f21127a6ac7536d3","last_reissued_at":"2026-07-28T02:23:49.927289Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T02:23:49.927289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A doubled Gordon threshold for palindromic quasiperiodic Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Wencai Liu","submitted_at":"2026-07-27T09:07:52Z","abstract_excerpt":"We consider one-frequency quasiperiodic Schr\\\"odinger operators \\[ (H_{v,\\alpha,\\theta}u)(n) =\nu(n+1)+u(n-1) +\nv(\\theta+n\\alpha)u(n) \\] acting on $\\ell^2(\\mathbb Z)$, where $\\alpha\\notin\\mathbb Q$ and $v\\in C^2(\\mathbb T,\\mathbb R)$ is an even function. We develop a new Gordon-type method that exploits approximate repetitions and palindromic symmetries simultaneously. Denote by $L(E)$ the Lyapunov exponent and let\n  \\[\\beta(\\alpha) =\n\\limsup_{|k|\\to\\infty} -\\frac{\\log\\|k\\alpha\\|_{\\mathbb R/\\mathbb Z}}{|k|}. \\] We prove that, for every completely resonant phase $2\\theta\\in\\alpha\\mathbb Z+\\mathb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.24188","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/2607.24188/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":"2607.24188","created_at":"2026-07-28T02:23:49.927743+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.24188v1","created_at":"2026-07-28T02:23:49.927743+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.24188","created_at":"2026-07-28T02:23:49.927743+00:00"},{"alias_kind":"pith_short_12","alias_value":"2HLLSVEPBPPE","created_at":"2026-07-28T02:23:49.927743+00:00"},{"alias_kind":"pith_short_16","alias_value":"2HLLSVEPBPPEC7VR","created_at":"2026-07-28T02:23:49.927743+00:00"},{"alias_kind":"pith_short_8","alias_value":"2HLLSVEP","created_at":"2026-07-28T02:23:49.927743+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/2HLLSVEPBPPEC7VR4A7POSG7QX","json":"https://pith.science/pith/2HLLSVEPBPPEC7VR4A7POSG7QX.json","graph_json":"https://pith.science/api/pith-number/2HLLSVEPBPPEC7VR4A7POSG7QX/graph.json","events_json":"https://pith.science/api/pith-number/2HLLSVEPBPPEC7VR4A7POSG7QX/events.json","paper":"https://pith.science/paper/2HLLSVEP"},"agent_actions":{"view_html":"https://pith.science/pith/2HLLSVEPBPPEC7VR4A7POSG7QX","download_json":"https://pith.science/pith/2HLLSVEPBPPEC7VR4A7POSG7QX.json","view_paper":"https://pith.science/paper/2HLLSVEP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.24188&json=true","fetch_graph":"https://pith.science/api/pith-number/2HLLSVEPBPPEC7VR4A7POSG7QX/graph.json","fetch_events":"https://pith.science/api/pith-number/2HLLSVEPBPPEC7VR4A7POSG7QX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2HLLSVEPBPPEC7VR4A7POSG7QX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2HLLSVEPBPPEC7VR4A7POSG7QX/action/storage_attestation","attest_author":"https://pith.science/pith/2HLLSVEPBPPEC7VR4A7POSG7QX/action/author_attestation","sign_citation":"https://pith.science/pith/2HLLSVEPBPPEC7VR4A7POSG7QX/action/citation_signature","submit_replication":"https://pith.science/pith/2HLLSVEPBPPEC7VR4A7POSG7QX/action/replication_record"}},"created_at":"2026-07-28T02:23:49.927743+00:00","updated_at":"2026-07-28T02:23:49.927743+00:00"}