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We first show that the ordinary scalar theory is insensitive to the doubled origin: every continuous map from \\(\\Ltwo\\) to a Hausdorff space factors through the quotient \\(q:\\Ltwo\\to\\R\\), while, for the natural measure, \\[\n  C^\\infty(\\Ltwo)\\cong C^\\infty(\\R),\n  \\qquad\n  L^2(\\Ltwo)\\cong L^2(\\R). \\] Accordingly, the natural scalar free Hamiltonian is unitaril"},"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":"2607.26080","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"quant-ph","submitted_at":"2026-07-24T16:53:14Z","cross_cats_sorted":["math-ph","math.GN","math.MP","math.OA"],"title_canon_sha256":"f34560bd6ee69fb3c44b6f02c8de48f0785a9204fd94951091ec8b4243b32898","abstract_canon_sha256":"5fe639867b53648c700eef3c8f4945f1735c5c3c7614d15260c09bb02fd72df1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5acacd31bac99df828fef004f8b5ae1e9c0cd1b96cf5225102cf7ceea280baf7","last_reissued_at":"2026-07-30T00:08:18.757284Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T00:08:18.757284Z"},"graph_snapshot":{"paper":{"title":"Quantum mechanics on the line with two origins","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.GN","math.MP","math.OA"],"primary_cat":"quant-ph","authors_text":"Abhiram Sripat","submitted_at":"2026-07-24T16:53:14Z","abstract_excerpt":"We study scalar and spinorial quantum dynamics on the standard line with two origins, \\[\n  \\Ltwo=(\\R_1\\sqcup\\R_2)/\\!\\sim,\n  \\qquad\n  (x,1)\\sim(x,2)\\quad\\text{for }x\\neq0, \\] equipped with its identity-glued smooth structure and flat metric. We first show that the ordinary scalar theory is insensitive to the doubled origin: every continuous map from \\(\\Ltwo\\) to a Hausdorff space factors through the quotient \\(q:\\Ltwo\\to\\R\\), while, for the natural measure, \\[\n  C^\\infty(\\Ltwo)\\cong C^\\infty(\\R),\n  \\qquad\n  L^2(\\Ltwo)\\cong L^2(\\R). \\] Accordingly, the natural scalar free Hamiltonian is unitaril"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26080","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/2607.26080/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":"2607.26080","created_at":"2026-07-30T00:08:18.760915+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.26080v1","created_at":"2026-07-30T00:08:18.760915+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26080","created_at":"2026-07-30T00:08:18.760915+00:00"},{"alias_kind":"pith_short_12","alias_value":"LLFM2MN2ZGO7","created_at":"2026-07-30T00:08:18.760915+00:00"},{"alias_kind":"pith_short_16","alias_value":"LLFM2MN2ZGO7QKH6","created_at":"2026-07-30T00:08:18.760915+00:00"},{"alias_kind":"pith_short_8","alias_value":"LLFM2MN2","created_at":"2026-07-30T00:08:18.760915+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/LLFM2MN2ZGO7QKH66ACPRNNOD2","json":"https://pith.science/pith/LLFM2MN2ZGO7QKH66ACPRNNOD2.json","graph_json":"https://pith.science/api/pith-number/LLFM2MN2ZGO7QKH66ACPRNNOD2/graph.json","events_json":"https://pith.science/api/pith-number/LLFM2MN2ZGO7QKH66ACPRNNOD2/events.json","paper":"https://pith.science/paper/LLFM2MN2"},"agent_actions":{"view_html":"https://pith.science/pith/LLFM2MN2ZGO7QKH66ACPRNNOD2","download_json":"https://pith.science/pith/LLFM2MN2ZGO7QKH66ACPRNNOD2.json","view_paper":"https://pith.science/paper/LLFM2MN2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.26080&json=true","fetch_graph":"https://pith.science/api/pith-number/LLFM2MN2ZGO7QKH66ACPRNNOD2/graph.json","fetch_events":"https://pith.science/api/pith-number/LLFM2MN2ZGO7QKH66ACPRNNOD2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LLFM2MN2ZGO7QKH66ACPRNNOD2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LLFM2MN2ZGO7QKH66ACPRNNOD2/action/storage_attestation","attest_author":"https://pith.science/pith/LLFM2MN2ZGO7QKH66ACPRNNOD2/action/author_attestation","sign_citation":"https://pith.science/pith/LLFM2MN2ZGO7QKH66ACPRNNOD2/action/citation_signature","submit_replication":"https://pith.science/pith/LLFM2MN2ZGO7QKH66ACPRNNOD2/action/replication_record"}},"created_at":"2026-07-30T00:08:18.760915+00:00","updated_at":"2026-07-30T00:08:18.760915+00:00"}