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While Stark's conjecture establishes theoretical connections between special units and class group structures, explicit constructions have remained elusive, and precise quantum complexity bounds for class group computations are lacking. Here we establish an integrated framework defining Stark-Coleman invariants $\\kappa_p(K) = \\log_p \\left( \\frac{\\varepsilon_{\\mathrm{St},p}}{\\sigma(\\varepsilon_{\\mathrm{St},p})} \\right) \\mod p^{\\mathrm{ord}_p(\\Delta_K)}$ "},"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":"2506.07640","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2025-06-09T11:06:17Z","cross_cats_sorted":["cs.CR","math.GR","quant-ph"],"title_canon_sha256":"b8a293f4dcd52e5bc80622fae5c35476c948466512f85bc65e8ff761af677957","abstract_canon_sha256":"33d21e132f880d6ce62a3e53c2b6d3d510cb06ed68dd68a5d59c0f3fe1bcf8c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:27:30.674470Z","signature_b64":"gogaUYm6XZC54tboVYGFOYB3n4PJxxI34XYjr2/aXk0a94v16dIX26WHJwb80UNzrXjLE+UkyA0q7pFoBJepCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80bca749ca3570fd7bdcdc62c3804584987262dfd260849742a0b720de042b4e","last_reissued_at":"2026-07-05T11:27:30.673916Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:27:30.673916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stark-Coleman Invariants and Quantum Lower Bounds: An Integrated Framework for Real Quadratic Fields","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.CR","math.GR","quant-ph"],"primary_cat":"math.NT","authors_text":"Chenglian Liu, Ruopengyu Xu","submitted_at":"2025-06-09T11:06:17Z","abstract_excerpt":"Class groups of real quadratic fields represent fundamental structures in algebraic number theory with significant computational implications. 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Here we establish an integrated framework defining Stark-Coleman invariants $\\kappa_p(K) = \\log_p \\left( \\frac{\\varepsilon_{\\mathrm{St},p}}{\\sigma(\\varepsilon_{\\mathrm{St},p})} \\right) \\mod p^{\\mathrm{ord}_p(\\Delta_K)}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07640","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/2506.07640/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":"2506.07640","created_at":"2026-07-05T11:27:30.673982+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.07640v1","created_at":"2026-07-05T11:27:30.673982+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.07640","created_at":"2026-07-05T11:27:30.673982+00:00"},{"alias_kind":"pith_short_12","alias_value":"QC6KOSOKGVYP","created_at":"2026-07-05T11:27:30.673982+00:00"},{"alias_kind":"pith_short_16","alias_value":"QC6KOSOKGVYP2664","created_at":"2026-07-05T11:27:30.673982+00:00"},{"alias_kind":"pith_short_8","alias_value":"QC6KOSOK","created_at":"2026-07-05T11:27:30.673982+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/QC6KOSOKGVYP26643RRMHACFQS","json":"https://pith.science/pith/QC6KOSOKGVYP26643RRMHACFQS.json","graph_json":"https://pith.science/api/pith-number/QC6KOSOKGVYP26643RRMHACFQS/graph.json","events_json":"https://pith.science/api/pith-number/QC6KOSOKGVYP26643RRMHACFQS/events.json","paper":"https://pith.science/paper/QC6KOSOK"},"agent_actions":{"view_html":"https://pith.science/pith/QC6KOSOKGVYP26643RRMHACFQS","download_json":"https://pith.science/pith/QC6KOSOKGVYP26643RRMHACFQS.json","view_paper":"https://pith.science/paper/QC6KOSOK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.07640&json=true","fetch_graph":"https://pith.science/api/pith-number/QC6KOSOKGVYP26643RRMHACFQS/graph.json","fetch_events":"https://pith.science/api/pith-number/QC6KOSOKGVYP26643RRMHACFQS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QC6KOSOKGVYP26643RRMHACFQS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QC6KOSOKGVYP26643RRMHACFQS/action/storage_attestation","attest_author":"https://pith.science/pith/QC6KOSOKGVYP26643RRMHACFQS/action/author_attestation","sign_citation":"https://pith.science/pith/QC6KOSOKGVYP26643RRMHACFQS/action/citation_signature","submit_replication":"https://pith.science/pith/QC6KOSOKGVYP26643RRMHACFQS/action/replication_record"}},"created_at":"2026-07-05T11:27:30.673982+00:00","updated_at":"2026-07-05T11:27:30.673982+00:00"}