{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:CGVKWQ4XMV7COJ74VLHTZA6JYV","short_pith_number":"pith:CGVKWQ4X","canonical_record":{"source":{"id":"1309.0660","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-03T12:53:39Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"a7c37334a0913164bb6dbb8a6f9e53af32184dc57c67d24d6a7741b8a3b205b8","abstract_canon_sha256":"c2ef222a678cd31bb70db2566c2d313b8448cd3e2ff6c1984eba44f0a78ee19b"},"schema_version":"1.0"},"canonical_sha256":"11aaab4397657e2727fcaacf3c83c9c56afbe4ddc4606d79aed888f0fb17d45c","source":{"kind":"arxiv","id":"1309.0660","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0660","created_at":"2026-05-18T01:25:47Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0660v1","created_at":"2026-05-18T01:25:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0660","created_at":"2026-05-18T01:25:47Z"},{"alias_kind":"pith_short_12","alias_value":"CGVKWQ4XMV7C","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CGVKWQ4XMV7COJ74","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CGVKWQ4X","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:CGVKWQ4XMV7COJ74VLHTZA6JYV","target":"record","payload":{"canonical_record":{"source":{"id":"1309.0660","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-03T12:53:39Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"a7c37334a0913164bb6dbb8a6f9e53af32184dc57c67d24d6a7741b8a3b205b8","abstract_canon_sha256":"c2ef222a678cd31bb70db2566c2d313b8448cd3e2ff6c1984eba44f0a78ee19b"},"schema_version":"1.0"},"canonical_sha256":"11aaab4397657e2727fcaacf3c83c9c56afbe4ddc4606d79aed888f0fb17d45c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:47.822508Z","signature_b64":"fUfTTQEWJTakv/iuhSc4oNJdyAVegXSddM/QeFphDTI+b83hN2E5dS5TV9G73S6wkyHeoIrVAt7asRbJ4uc0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11aaab4397657e2727fcaacf3c83c9c56afbe4ddc4606d79aed888f0fb17d45c","last_reissued_at":"2026-05-18T01:25:47.821846Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:47.821846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.0660","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:25:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wUPwzX3NyA+OJ7aaqwOTfRwshWRARMjMA13T6BIruKHDu7BF9GpSpdxRMDJSu1EHIalLnxhideocpxV7N9d5AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:58:12.695402Z"},"content_sha256":"148e8448e153ce3aafe0bd71e7c4735b6e20e49b5b29ad9d7d5c7f4e656200bb","schema_version":"1.0","event_id":"sha256:148e8448e153ce3aafe0bd71e7c4735b6e20e49b5b29ad9d7d5c7f4e656200bb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:CGVKWQ4XMV7COJ74VLHTZA6JYV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two-dimensional Einstein numbers and associativity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"J\\'an Halu\\v{s}ka, Tom\\'a\\v{s} Gregor","submitted_at":"2013-09-03T12:53:39Z","abstract_excerpt":"In this paper, we deal with generalizations of real Einstein numbers to various spaces and dimensions. We search operations and their properties in generalized settings. Especially, we are interested in the generalized operation of hyperbolic addition to more-dimensional spaces, which is associative and commutative. We extend the theory to some abstract spaces, especially to Hilbert-like ones. Further, we bring two different two-dimensional generalizations of Einstein numbers and study properties of new-defined operations -- mainly associativity, commutativity, and distributive laws."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0660","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:25:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"17p9gDMN7A1tfcHGfH12OmDRb5reTb+SeNJomy3ub8ZbCDQswGwYXhhMeLHxg1OR527eqZIOOi05T9tasjkvCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:58:12.695768Z"},"content_sha256":"de1203ecc844a1654be1ddbdb1975a693fe5c5ff09e7474ee47d076068652c44","schema_version":"1.0","event_id":"sha256:de1203ecc844a1654be1ddbdb1975a693fe5c5ff09e7474ee47d076068652c44"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CGVKWQ4XMV7COJ74VLHTZA6JYV/bundle.json","state_url":"https://pith.science/pith/CGVKWQ4XMV7COJ74VLHTZA6JYV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CGVKWQ4XMV7COJ74VLHTZA6JYV/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T23:58:12Z","links":{"resolver":"https://pith.science/pith/CGVKWQ4XMV7COJ74VLHTZA6JYV","bundle":"https://pith.science/pith/CGVKWQ4XMV7COJ74VLHTZA6JYV/bundle.json","state":"https://pith.science/pith/CGVKWQ4XMV7COJ74VLHTZA6JYV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CGVKWQ4XMV7COJ74VLHTZA6JYV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:CGVKWQ4XMV7COJ74VLHTZA6JYV","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":"c2ef222a678cd31bb70db2566c2d313b8448cd3e2ff6c1984eba44f0a78ee19b","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-03T12:53:39Z","title_canon_sha256":"a7c37334a0913164bb6dbb8a6f9e53af32184dc57c67d24d6a7741b8a3b205b8"},"schema_version":"1.0","source":{"id":"1309.0660","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0660","created_at":"2026-05-18T01:25:47Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0660v1","created_at":"2026-05-18T01:25:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0660","created_at":"2026-05-18T01:25:47Z"},{"alias_kind":"pith_short_12","alias_value":"CGVKWQ4XMV7C","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CGVKWQ4XMV7COJ74","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CGVKWQ4X","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:de1203ecc844a1654be1ddbdb1975a693fe5c5ff09e7474ee47d076068652c44","target":"graph","created_at":"2026-05-18T01:25:47Z","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":"In this paper, we deal with generalizations of real Einstein numbers to various spaces and dimensions. We search operations and their properties in generalized settings. Especially, we are interested in the generalized operation of hyperbolic addition to more-dimensional spaces, which is associative and commutative. We extend the theory to some abstract spaces, especially to Hilbert-like ones. Further, we bring two different two-dimensional generalizations of Einstein numbers and study properties of new-defined operations -- mainly associativity, commutativity, and distributive laws.","authors_text":"J\\'an Halu\\v{s}ka, Tom\\'a\\v{s} Gregor","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-03T12:53:39Z","title":"Two-dimensional Einstein numbers and associativity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0660","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":"sha256:148e8448e153ce3aafe0bd71e7c4735b6e20e49b5b29ad9d7d5c7f4e656200bb","target":"record","created_at":"2026-05-18T01:25:47Z","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":"c2ef222a678cd31bb70db2566c2d313b8448cd3e2ff6c1984eba44f0a78ee19b","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-03T12:53:39Z","title_canon_sha256":"a7c37334a0913164bb6dbb8a6f9e53af32184dc57c67d24d6a7741b8a3b205b8"},"schema_version":"1.0","source":{"id":"1309.0660","kind":"arxiv","version":1}},"canonical_sha256":"11aaab4397657e2727fcaacf3c83c9c56afbe4ddc4606d79aed888f0fb17d45c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"11aaab4397657e2727fcaacf3c83c9c56afbe4ddc4606d79aed888f0fb17d45c","first_computed_at":"2026-05-18T01:25:47.821846Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:47.821846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fUfTTQEWJTakv/iuhSc4oNJdyAVegXSddM/QeFphDTI+b83hN2E5dS5TV9G73S6wkyHeoIrVAt7asRbJ4uc0BA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:47.822508Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.0660","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:148e8448e153ce3aafe0bd71e7c4735b6e20e49b5b29ad9d7d5c7f4e656200bb","sha256:de1203ecc844a1654be1ddbdb1975a693fe5c5ff09e7474ee47d076068652c44"],"state_sha256":"4c3c5edc3fbd2379c1c5eeaf5c2c056419c59a4bcbe90f64fcdf7791b505a1ae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"72LJBnGNhcwIfrdulfOCHQL9Pj9IP8DQ2nrRCsN5f4Yv54hjmVOOi53weLxu0CSneFswkyy6YJuJMPzKq3BuAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T23:58:12.697789Z","bundle_sha256":"fc03e4ddafafc8b7a8c066f9ce3671694f726e3fb212c91ebbae0cb50a0afeee"}}