{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:GKY5PYDZIDUM3U3VPKW64U6KCN","short_pith_number":"pith:GKY5PYDZ","canonical_record":{"source":{"id":"0807.4159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2008-07-25T18:11:53Z","cross_cats_sorted":["math.AT","math.CO"],"title_canon_sha256":"31af79d103ac3b7473269087b62e58d307a58084243255ae26e91dbb55894e45","abstract_canon_sha256":"f1987fc407c371d9c50645047101b1c1268133768e73bac071bc09b326729847"},"schema_version":"1.0"},"canonical_sha256":"32b1d7e07940e8cdd3757aadee53ca1340fc3b2ca6869beb23f5d211a396a752","source":{"kind":"arxiv","id":"0807.4159","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.4159","created_at":"2026-05-18T01:49:54Z"},{"alias_kind":"arxiv_version","alias_value":"0807.4159v1","created_at":"2026-05-18T01:49:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.4159","created_at":"2026-05-18T01:49:54Z"},{"alias_kind":"pith_short_12","alias_value":"GKY5PYDZIDUM","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"GKY5PYDZIDUM3U3V","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"GKY5PYDZ","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:GKY5PYDZIDUM3U3VPKW64U6KCN","target":"record","payload":{"canonical_record":{"source":{"id":"0807.4159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2008-07-25T18:11:53Z","cross_cats_sorted":["math.AT","math.CO"],"title_canon_sha256":"31af79d103ac3b7473269087b62e58d307a58084243255ae26e91dbb55894e45","abstract_canon_sha256":"f1987fc407c371d9c50645047101b1c1268133768e73bac071bc09b326729847"},"schema_version":"1.0"},"canonical_sha256":"32b1d7e07940e8cdd3757aadee53ca1340fc3b2ca6869beb23f5d211a396a752","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:49:54.406795Z","signature_b64":"3T0PIgeNOs02L7lVM1CnCaA98DF7gdro23bu1zoBvTeWY+yRrYp2Xt2omOFCbWn0E1K42JgalMElXZ8O/IDkCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32b1d7e07940e8cdd3757aadee53ca1340fc3b2ca6869beb23f5d211a396a752","last_reissued_at":"2026-05-18T01:49:54.406294Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:49:54.406294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0807.4159","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":"2026-05-18T01:49:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vk8ltrgTV/p2M/KqyGAlRxcbuVFXfL5+Y2no/tpZc6ZtZW7AOCziHClhPBSVHrmiQ5fhaB6yL6XkEwf+Zv70Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T19:56:47.127847Z"},"content_sha256":"84d1bd6d14a70a74ba6765ee74c087c79911aad0f8c6275df0ec09efc471fdbd","schema_version":"1.0","event_id":"sha256:84d1bd6d14a70a74ba6765ee74c087c79911aad0f8c6275df0ec09efc471fdbd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:GKY5PYDZIDUM3U3VPKW64U6KCN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Marked tubes and the graph multiplihedron","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO"],"primary_cat":"math.QA","authors_text":"Satyan L. Devadoss, Stefan Forcey","submitted_at":"2008-07-25T18:11:53Z","abstract_excerpt":"Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces, and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.4159","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":"2026-05-18T01:49:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Lo9eT2PoWKG4krK0rRAAzPuwWqbm6R0lKJGUjzgSAeu0q//seYwovbmLtlYrdCIORXema1zFHCuq9bUK395VCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T19:56:47.128515Z"},"content_sha256":"eb88e0c14ed4cca942c727100fd50347d2f4f1b8cb7a7f3ee3e14d0fb934b6bd","schema_version":"1.0","event_id":"sha256:eb88e0c14ed4cca942c727100fd50347d2f4f1b8cb7a7f3ee3e14d0fb934b6bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GKY5PYDZIDUM3U3VPKW64U6KCN/bundle.json","state_url":"https://pith.science/pith/GKY5PYDZIDUM3U3VPKW64U6KCN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GKY5PYDZIDUM3U3VPKW64U6KCN/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":"2026-05-28T19:56:47Z","links":{"resolver":"https://pith.science/pith/GKY5PYDZIDUM3U3VPKW64U6KCN","bundle":"https://pith.science/pith/GKY5PYDZIDUM3U3VPKW64U6KCN/bundle.json","state":"https://pith.science/pith/GKY5PYDZIDUM3U3VPKW64U6KCN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GKY5PYDZIDUM3U3VPKW64U6KCN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:GKY5PYDZIDUM3U3VPKW64U6KCN","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":"f1987fc407c371d9c50645047101b1c1268133768e73bac071bc09b326729847","cross_cats_sorted":["math.AT","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2008-07-25T18:11:53Z","title_canon_sha256":"31af79d103ac3b7473269087b62e58d307a58084243255ae26e91dbb55894e45"},"schema_version":"1.0","source":{"id":"0807.4159","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0807.4159","created_at":"2026-05-18T01:49:54Z"},{"alias_kind":"arxiv_version","alias_value":"0807.4159v1","created_at":"2026-05-18T01:49:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0807.4159","created_at":"2026-05-18T01:49:54Z"},{"alias_kind":"pith_short_12","alias_value":"GKY5PYDZIDUM","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"GKY5PYDZIDUM3U3V","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"GKY5PYDZ","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:eb88e0c14ed4cca942c727100fd50347d2f4f1b8cb7a7f3ee3e14d0fb934b6bd","target":"graph","created_at":"2026-05-18T01:49:54Z","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":"Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces, and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron.","authors_text":"Satyan L. Devadoss, Stefan Forcey","cross_cats":["math.AT","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2008-07-25T18:11:53Z","title":"Marked tubes and the graph multiplihedron"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0807.4159","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":"sha256:84d1bd6d14a70a74ba6765ee74c087c79911aad0f8c6275df0ec09efc471fdbd","target":"record","created_at":"2026-05-18T01:49:54Z","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":"f1987fc407c371d9c50645047101b1c1268133768e73bac071bc09b326729847","cross_cats_sorted":["math.AT","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2008-07-25T18:11:53Z","title_canon_sha256":"31af79d103ac3b7473269087b62e58d307a58084243255ae26e91dbb55894e45"},"schema_version":"1.0","source":{"id":"0807.4159","kind":"arxiv","version":1}},"canonical_sha256":"32b1d7e07940e8cdd3757aadee53ca1340fc3b2ca6869beb23f5d211a396a752","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32b1d7e07940e8cdd3757aadee53ca1340fc3b2ca6869beb23f5d211a396a752","first_computed_at":"2026-05-18T01:49:54.406294Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:49:54.406294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3T0PIgeNOs02L7lVM1CnCaA98DF7gdro23bu1zoBvTeWY+yRrYp2Xt2omOFCbWn0E1K42JgalMElXZ8O/IDkCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:49:54.406795Z","signed_message":"canonical_sha256_bytes"},"source_id":"0807.4159","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84d1bd6d14a70a74ba6765ee74c087c79911aad0f8c6275df0ec09efc471fdbd","sha256:eb88e0c14ed4cca942c727100fd50347d2f4f1b8cb7a7f3ee3e14d0fb934b6bd"],"state_sha256":"b5578751efc8e884c9080ecc71a8452ac077588f5b94d0a679dff74088c75721"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pUGMIhgjku7HO/dBJ9bs15ZU39SWNec53dwj54Lh/I9zyyTUo9qWrvtQAK0jvfx32K5AvL4zcpoaStQCQu65Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T19:56:47.132217Z","bundle_sha256":"8e934ce0a86561c11d8781acbf584c8c94004189ffaed37c7bc31a86cdbd8ac4"}}