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We also obtain exact estimations on the norm of the transfer matrices and the norm of the formal solutions for these energies. Especially, for $E\\in \\Sigma_{II}\\cup\\Sigma_{III}, $ the norms of the transfer matrices behave like $$ e^{c_1\\gamma\\sqrt{n}}\\le\\|T_{n}(E)\\|\\le e^{c_2\\gamma\\sqrt{n}}. $$ The local dimensions of "},"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":"1512.08011","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-12-25T15:40:47Z","cross_cats_sorted":["math.DS","math.MP","math.SP"],"title_canon_sha256":"45375677dd12a0b6975d12498d79d8d9c52287305d33a3afe79329d51d88e300","abstract_canon_sha256":"42fc56ede4239a105eb8976326d2e615a3e981daa04b9c70332376be2e05a6fc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:09.990778Z","signature_b64":"zYJoJSvCqKcQL1GbHeIkYy2wBO+vWpXCZ2HYz7wEDjJNPet7poP87unDq5cwfE7B0Wx61bONxqAERdoIYOWcDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c1b4bcd38b9302bf60e2d53bcad342f9de387eb72f3dc1016801ed187e6f722b","last_reissued_at":"2026-05-18T01:22:09.990175Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:09.990175Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"\"Mixed spectral nature\" of Thue--Morse Hamiltonian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Qinghui Liu, Xiao Yao, Yanhui Qu","submitted_at":"2015-12-25T15:40:47Z","abstract_excerpt":"We find three dense subsets $\\Sigma_I,\\Sigma_{II}$ and $\\Sigma_{III}$ of the spectrum of the Thue-Morse Hamiltonian, such that each energy in $\\Sigma_I$ is extended, each energy in $\\Sigma_{II}$ is pseudo-localized and each energy in $\\Sigma_{III}$ is one-sided pseudo-localized. We also obtain exact estimations on the norm of the transfer matrices and the norm of the formal solutions for these energies. Especially, for $E\\in \\Sigma_{II}\\cup\\Sigma_{III}, $ the norms of the transfer matrices behave like $$ e^{c_1\\gamma\\sqrt{n}}\\le\\|T_{n}(E)\\|\\le e^{c_2\\gamma\\sqrt{n}}. $$ The local dimensions of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.08011","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1512.08011","created_at":"2026-05-18T01:22:09.990252+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.08011v2","created_at":"2026-05-18T01:22:09.990252+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.08011","created_at":"2026-05-18T01:22:09.990252+00:00"},{"alias_kind":"pith_short_12","alias_value":"YG2LZU4LSMBL","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_16","alias_value":"YG2LZU4LSMBL6YHC","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_8","alias_value":"YG2LZU4L","created_at":"2026-05-18T12:29:50.041715+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/YG2LZU4LSMBL6YHC2U54VU2C7H","json":"https://pith.science/pith/YG2LZU4LSMBL6YHC2U54VU2C7H.json","graph_json":"https://pith.science/api/pith-number/YG2LZU4LSMBL6YHC2U54VU2C7H/graph.json","events_json":"https://pith.science/api/pith-number/YG2LZU4LSMBL6YHC2U54VU2C7H/events.json","paper":"https://pith.science/paper/YG2LZU4L"},"agent_actions":{"view_html":"https://pith.science/pith/YG2LZU4LSMBL6YHC2U54VU2C7H","download_json":"https://pith.science/pith/YG2LZU4LSMBL6YHC2U54VU2C7H.json","view_paper":"https://pith.science/paper/YG2LZU4L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.08011&json=true","fetch_graph":"https://pith.science/api/pith-number/YG2LZU4LSMBL6YHC2U54VU2C7H/graph.json","fetch_events":"https://pith.science/api/pith-number/YG2LZU4LSMBL6YHC2U54VU2C7H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YG2LZU4LSMBL6YHC2U54VU2C7H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YG2LZU4LSMBL6YHC2U54VU2C7H/action/storage_attestation","attest_author":"https://pith.science/pith/YG2LZU4LSMBL6YHC2U54VU2C7H/action/author_attestation","sign_citation":"https://pith.science/pith/YG2LZU4LSMBL6YHC2U54VU2C7H/action/citation_signature","submit_replication":"https://pith.science/pith/YG2LZU4LSMBL6YHC2U54VU2C7H/action/replication_record"}},"created_at":"2026-05-18T01:22:09.990252+00:00","updated_at":"2026-05-18T01:22:09.990252+00:00"}