{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:HL7T3XGR4IHG324XL5MV2U4U3B","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4c26a4f2432f476e2333737655eccaee4046fafbdf8a810af517786201c9c249","cross_cats_sorted":["math.MP","physics.atom-ph","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-07T15:15:39Z","title_canon_sha256":"819301a3b92e41461b96aa8ad5d9b246f5bbb399d32d87f085d80cb98b1b4778"},"schema_version":"1.0","source":{"id":"1004.1109","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.1109","created_at":"2026-05-18T02:08:08Z"},{"alias_kind":"arxiv_version","alias_value":"1004.1109v1","created_at":"2026-05-18T02:08:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.1109","created_at":"2026-05-18T02:08:08Z"},{"alias_kind":"pith_short_12","alias_value":"HL7T3XGR4IHG","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"HL7T3XGR4IHG324X","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"HL7T3XGR","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:eb1b7c386ff44285caa82dfade8c0ccccaa3dea0d20e4a64119104cf2a4e589f","target":"graph","created_at":"2026-05-18T02:08:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Comparison theorems are established for the Dirac and Klein--Gordon equations. We suppose that V^{(1)}(r) and V^{(2)}(r) are two real attractive central potentials in  d  dimensions that support discrete Dirac eigenvalues E^{(1)}_{k_d\\nu} and E^{(2)}_{k_d\\nu}. We prove that if V^{(1)}(r) \\leq V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered  E^{(1)}_{k_d\\nu} \\leq E^{(2)}_{k_d\\nu}. This result generalizes an earlier more restrictive theorem that required the wave functions to be node free. For the  the Klein--Gordon equation, similar reasoning also leads to a comp","authors_text":"Richard L. Hall","cross_cats":["math.MP","physics.atom-ph","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-07T15:15:39Z","title":"Relativistic Comparison Theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.1109","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f71dfecb8d7489d4d4630268de5ccc86c49b8d7832166d65cc8019eb603392e4","target":"record","created_at":"2026-05-18T02:08:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4c26a4f2432f476e2333737655eccaee4046fafbdf8a810af517786201c9c249","cross_cats_sorted":["math.MP","physics.atom-ph","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-07T15:15:39Z","title_canon_sha256":"819301a3b92e41461b96aa8ad5d9b246f5bbb399d32d87f085d80cb98b1b4778"},"schema_version":"1.0","source":{"id":"1004.1109","kind":"arxiv","version":1}},"canonical_sha256":"3aff3ddcd1e20e6deb975f595d5394d87ae4b32c77e64c4933ef847151e6f72c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3aff3ddcd1e20e6deb975f595d5394d87ae4b32c77e64c4933ef847151e6f72c","first_computed_at":"2026-05-18T02:08:08.427303Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:08:08.427303Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i+BcC4YfOzLVZMe6QgRCqgx5U0f5iifryA4JSYHIaflse2p3cb258CyaikETK4S0VVMGhugCxdH8Si5rfs2xCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:08:08.427920Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.1109","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f71dfecb8d7489d4d4630268de5ccc86c49b8d7832166d65cc8019eb603392e4","sha256:eb1b7c386ff44285caa82dfade8c0ccccaa3dea0d20e4a64119104cf2a4e589f"],"state_sha256":"f52e59c473364dd5aaf354b080ea2e4e132fb4c1ce8ae31ccc9623ed3a72eb38"}