{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:D3NCJ4L5I5CSF4Y3KUEZ4VQKHL","short_pith_number":"pith:D3NCJ4L5","canonical_record":{"source":{"id":"1207.3487","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-07-15T09:48:31Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"e3b74c1132a0b62987c82c32df566cbd02f00b8326d0d225b27cafb9eb30c134","abstract_canon_sha256":"26e9c1469418d4197cd5ddbb6eeceb2b9005434ffe458a4c9b2f9b8b6fa7ab9b"},"schema_version":"1.0"},"canonical_sha256":"1eda24f17d474522f31b55099e560a3ac240b4b0dff1894dda4a86ecf96fd2c7","source":{"kind":"arxiv","id":"1207.3487","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3487","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3487v1","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3487","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"pith_short_12","alias_value":"D3NCJ4L5I5CS","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"D3NCJ4L5I5CSF4Y3","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"D3NCJ4L5","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:D3NCJ4L5I5CSF4Y3KUEZ4VQKHL","target":"record","payload":{"canonical_record":{"source":{"id":"1207.3487","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-07-15T09:48:31Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"e3b74c1132a0b62987c82c32df566cbd02f00b8326d0d225b27cafb9eb30c134","abstract_canon_sha256":"26e9c1469418d4197cd5ddbb6eeceb2b9005434ffe458a4c9b2f9b8b6fa7ab9b"},"schema_version":"1.0"},"canonical_sha256":"1eda24f17d474522f31b55099e560a3ac240b4b0dff1894dda4a86ecf96fd2c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:00.963704Z","signature_b64":"m+rQOyrHwOIw353yjpJsVOPVL3jiC80R2SptN7WLHCS6FG9llgSFQoPmD3LQNTdSBMj+zgJtODaAatDYaB4fBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1eda24f17d474522f31b55099e560a3ac240b4b0dff1894dda4a86ecf96fd2c7","last_reissued_at":"2026-05-18T03:51:00.962970Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:00.962970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.3487","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":"2026-05-18T03:51:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vt2Lwylripc4QWOpKEfjVXNYpskRuiMizxWRpVjV4ttU7QtDvwNP81CIKIhMWbnjqdhlmpTO+jP340DhWayaDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T15:03:24.971864Z"},"content_sha256":"375e2af0ab2999e855a2d6f5e1016f7a055628eddf01bf5ea6a2fd008ad40fba","schema_version":"1.0","event_id":"sha256:375e2af0ab2999e855a2d6f5e1016f7a055628eddf01bf5ea6a2fd008ad40fba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:D3NCJ4L5I5CSF4Y3KUEZ4VQKHL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Categorical notions of layered tropical algebra and geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RA","authors_text":"Louis Rowen, Manfred Knebusch, Zur Izhakian","submitted_at":"2012-07-15T09:48:31Z","abstract_excerpt":"This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories, together with a \"tropicalization functor\" which permits us to pass from usual algebraic geometry to the tropical world. We consider tropical varieties from this categorical viewpoint, with emphasis on polynomial functions and their roots."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3487","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":"2026-05-18T03:51:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IlP9x4DLNjco369Y2W/FucHC1s+iPXh73/Yfwu/PTc6pUwvh9v4hzL+d+aFNXHyOXebQcRgfzxXIM0Z2efn0BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T15:03:24.972642Z"},"content_sha256":"2f21382d05529685c488bd6ff37f75e23c73d867f12fec2d30c87704e9d6dce2","schema_version":"1.0","event_id":"sha256:2f21382d05529685c488bd6ff37f75e23c73d867f12fec2d30c87704e9d6dce2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/D3NCJ4L5I5CSF4Y3KUEZ4VQKHL/bundle.json","state_url":"https://pith.science/pith/D3NCJ4L5I5CSF4Y3KUEZ4VQKHL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/D3NCJ4L5I5CSF4Y3KUEZ4VQKHL/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":"2026-06-11T15:03:24Z","links":{"resolver":"https://pith.science/pith/D3NCJ4L5I5CSF4Y3KUEZ4VQKHL","bundle":"https://pith.science/pith/D3NCJ4L5I5CSF4Y3KUEZ4VQKHL/bundle.json","state":"https://pith.science/pith/D3NCJ4L5I5CSF4Y3KUEZ4VQKHL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/D3NCJ4L5I5CSF4Y3KUEZ4VQKHL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:D3NCJ4L5I5CSF4Y3KUEZ4VQKHL","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":"26e9c1469418d4197cd5ddbb6eeceb2b9005434ffe458a4c9b2f9b8b6fa7ab9b","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-07-15T09:48:31Z","title_canon_sha256":"e3b74c1132a0b62987c82c32df566cbd02f00b8326d0d225b27cafb9eb30c134"},"schema_version":"1.0","source":{"id":"1207.3487","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3487","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3487v1","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3487","created_at":"2026-05-18T03:51:00Z"},{"alias_kind":"pith_short_12","alias_value":"D3NCJ4L5I5CS","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"D3NCJ4L5I5CSF4Y3","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"D3NCJ4L5","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:2f21382d05529685c488bd6ff37f75e23c73d867f12fec2d30c87704e9d6dce2","target":"graph","created_at":"2026-05-18T03:51:00Z","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":"This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories, together with a \"tropicalization functor\" which permits us to pass from usual algebraic geometry to the tropical world. We consider tropical varieties from this categorical viewpoint, with emphasis on polynomial functions and their roots.","authors_text":"Louis Rowen, Manfred Knebusch, Zur Izhakian","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-07-15T09:48:31Z","title":"Categorical notions of layered tropical algebra and geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3487","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":"sha256:375e2af0ab2999e855a2d6f5e1016f7a055628eddf01bf5ea6a2fd008ad40fba","target":"record","created_at":"2026-05-18T03:51:00Z","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":"26e9c1469418d4197cd5ddbb6eeceb2b9005434ffe458a4c9b2f9b8b6fa7ab9b","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-07-15T09:48:31Z","title_canon_sha256":"e3b74c1132a0b62987c82c32df566cbd02f00b8326d0d225b27cafb9eb30c134"},"schema_version":"1.0","source":{"id":"1207.3487","kind":"arxiv","version":1}},"canonical_sha256":"1eda24f17d474522f31b55099e560a3ac240b4b0dff1894dda4a86ecf96fd2c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1eda24f17d474522f31b55099e560a3ac240b4b0dff1894dda4a86ecf96fd2c7","first_computed_at":"2026-05-18T03:51:00.962970Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:51:00.962970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m+rQOyrHwOIw353yjpJsVOPVL3jiC80R2SptN7WLHCS6FG9llgSFQoPmD3LQNTdSBMj+zgJtODaAatDYaB4fBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:51:00.963704Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.3487","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:375e2af0ab2999e855a2d6f5e1016f7a055628eddf01bf5ea6a2fd008ad40fba","sha256:2f21382d05529685c488bd6ff37f75e23c73d867f12fec2d30c87704e9d6dce2"],"state_sha256":"0cfeb523ab1629627f16a96d61acd5f87e7c9341fac1f484b9251406752e7f02"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QiZWN38JfCVR+m0xgGQtjw5fRxHP1M9qPJ39mpeF6dQHJaKKpEyKSuRJknIhwu/+NHNISoTkvuZKaG//3740DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T15:03:24.977058Z","bundle_sha256":"b668bafaa07228d25a605acbf10bdc1651d5d9be8b87edc87b6703a00e1384a3"}}