{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:ZX3BMMFY2FZACH47LEBYYEOJQO","short_pith_number":"pith:ZX3BMMFY","canonical_record":{"source":{"id":"1803.06239","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-16T14:02:42Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"7b56e0f153b8cf6a0e8ad69624a7472587d6c7267b9653de025a1d987566486e","abstract_canon_sha256":"fb8a851ae8033153393aae8879b3d5c914de7e1949264df6a8324a0d20e382b0"},"schema_version":"1.0"},"canonical_sha256":"cdf61630b8d172011f9f59038c11c983b9b5166281a3795272fad0866c115716","source":{"kind":"arxiv","id":"1803.06239","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.06239","created_at":"2026-05-18T00:20:50Z"},{"alias_kind":"arxiv_version","alias_value":"1803.06239v1","created_at":"2026-05-18T00:20:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.06239","created_at":"2026-05-18T00:20:50Z"},{"alias_kind":"pith_short_12","alias_value":"ZX3BMMFY2FZA","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZX3BMMFY2FZACH47","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZX3BMMFY","created_at":"2026-05-18T12:33:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:ZX3BMMFY2FZACH47LEBYYEOJQO","target":"record","payload":{"canonical_record":{"source":{"id":"1803.06239","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-16T14:02:42Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"7b56e0f153b8cf6a0e8ad69624a7472587d6c7267b9653de025a1d987566486e","abstract_canon_sha256":"fb8a851ae8033153393aae8879b3d5c914de7e1949264df6a8324a0d20e382b0"},"schema_version":"1.0"},"canonical_sha256":"cdf61630b8d172011f9f59038c11c983b9b5166281a3795272fad0866c115716","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:50.177299Z","signature_b64":"YnDBbOOSV3yhzuvxgj2cSOGYYD56EXmCtnWPAUYkVj9Qij99BILSPuvPTC79Yy/73WnXk9aHY3M9tITHHQSLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cdf61630b8d172011f9f59038c11c983b9b5166281a3795272fad0866c115716","last_reissued_at":"2026-05-18T00:20:50.176544Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:50.176544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.06239","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":"2026-05-18T00:20:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IwJ0mkyc1e0zVxcOI8X5CV5VFSkUHQZrrTpJrGy+Rm8+GMGswrQtxbXg6I8HYpI0FUG4fkuuOukaS7P+W/iCCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:23:40.852173Z"},"content_sha256":"69137dcf38615fa1cab0cd46ccd62fe0ca52fe3c183000931743d3cb5d8cc53b","schema_version":"1.0","event_id":"sha256:69137dcf38615fa1cab0cd46ccd62fe0ca52fe3c183000931743d3cb5d8cc53b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:ZX3BMMFY2FZACH47LEBYYEOJQO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Trianguloids and Triangulations of Root Polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Alexander Postnikov, Gleb Nenashev, Pavel Galashin","submitted_at":"2018-03-16T14:02:42Z","abstract_excerpt":"Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.06239","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":"2026-05-18T00:20:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JBsq0mXJA5vZyk5B70X+8bdLfVc3KHHyro8UfjQXkfNykpfOECopiH90lIjdnY39DshwKNTcciyCPauhjXr7Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:23:40.852832Z"},"content_sha256":"5e61b946670096129ce9ad961662f17785af9f2468f0bb0cee0e302ce9ae0e1b","schema_version":"1.0","event_id":"sha256:5e61b946670096129ce9ad961662f17785af9f2468f0bb0cee0e302ce9ae0e1b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZX3BMMFY2FZACH47LEBYYEOJQO/bundle.json","state_url":"https://pith.science/pith/ZX3BMMFY2FZACH47LEBYYEOJQO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZX3BMMFY2FZACH47LEBYYEOJQO/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":"2026-05-27T09:23:40Z","links":{"resolver":"https://pith.science/pith/ZX3BMMFY2FZACH47LEBYYEOJQO","bundle":"https://pith.science/pith/ZX3BMMFY2FZACH47LEBYYEOJQO/bundle.json","state":"https://pith.science/pith/ZX3BMMFY2FZACH47LEBYYEOJQO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZX3BMMFY2FZACH47LEBYYEOJQO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:ZX3BMMFY2FZACH47LEBYYEOJQO","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":"fb8a851ae8033153393aae8879b3d5c914de7e1949264df6a8324a0d20e382b0","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-16T14:02:42Z","title_canon_sha256":"7b56e0f153b8cf6a0e8ad69624a7472587d6c7267b9653de025a1d987566486e"},"schema_version":"1.0","source":{"id":"1803.06239","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.06239","created_at":"2026-05-18T00:20:50Z"},{"alias_kind":"arxiv_version","alias_value":"1803.06239v1","created_at":"2026-05-18T00:20:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.06239","created_at":"2026-05-18T00:20:50Z"},{"alias_kind":"pith_short_12","alias_value":"ZX3BMMFY2FZA","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZX3BMMFY2FZACH47","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZX3BMMFY","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:5e61b946670096129ce9ad961662f17785af9f2468f0bb0cee0e302ce9ae0e1b","target":"graph","created_at":"2026-05-18T00:20:50Z","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":"Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of r","authors_text":"Alexander Postnikov, Gleb Nenashev, Pavel Galashin","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-16T14:02:42Z","title":"Trianguloids and Triangulations of Root Polytopes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.06239","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":"sha256:69137dcf38615fa1cab0cd46ccd62fe0ca52fe3c183000931743d3cb5d8cc53b","target":"record","created_at":"2026-05-18T00:20:50Z","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":"fb8a851ae8033153393aae8879b3d5c914de7e1949264df6a8324a0d20e382b0","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-16T14:02:42Z","title_canon_sha256":"7b56e0f153b8cf6a0e8ad69624a7472587d6c7267b9653de025a1d987566486e"},"schema_version":"1.0","source":{"id":"1803.06239","kind":"arxiv","version":1}},"canonical_sha256":"cdf61630b8d172011f9f59038c11c983b9b5166281a3795272fad0866c115716","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cdf61630b8d172011f9f59038c11c983b9b5166281a3795272fad0866c115716","first_computed_at":"2026-05-18T00:20:50.176544Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:50.176544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YnDBbOOSV3yhzuvxgj2cSOGYYD56EXmCtnWPAUYkVj9Qij99BILSPuvPTC79Yy/73WnXk9aHY3M9tITHHQSLAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:50.177299Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.06239","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69137dcf38615fa1cab0cd46ccd62fe0ca52fe3c183000931743d3cb5d8cc53b","sha256:5e61b946670096129ce9ad961662f17785af9f2468f0bb0cee0e302ce9ae0e1b"],"state_sha256":"bc9b84ac240ad494aa72e5b5065fad562b6f0f9a952f06dda9f096a617caa835"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SFXRQvLuRm7nFUxiMOLZTo5e46FmKvcxBl7h+Pxxs3+SQ+7xYLGGjHBHulaXMbhKECq8RPVmReRN4h+X3liJCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T09:23:40.856268Z","bundle_sha256":"56cb7a3ae9b1f82afbc12d86763634f12eaab9d84ece7468407c893da9b88cf1"}}