{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:FJAWDIPA6M6EKC3Q6UD5MULDBC","short_pith_number":"pith:FJAWDIPA","canonical_record":{"source":{"id":"1205.6256","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-05-29T04:11:09Z","cross_cats_sorted":[],"title_canon_sha256":"047ab7a82a71b5630d0dc6a505e3de47d52fbbb737c4bfb0803f68d14c6d7825","abstract_canon_sha256":"dffed89e134d013b217bf2ee8f4a969b237cb5c1628f1008838320cfae1c8120"},"schema_version":"1.0"},"canonical_sha256":"2a4161a1e0f33c450b70f507d6516308a9d7ed360e9443d0da193ad9ade21086","source":{"kind":"arxiv","id":"1205.6256","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.6256","created_at":"2026-05-18T03:10:58Z"},{"alias_kind":"arxiv_version","alias_value":"1205.6256v3","created_at":"2026-05-18T03:10:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.6256","created_at":"2026-05-18T03:10:58Z"},{"alias_kind":"pith_short_12","alias_value":"FJAWDIPA6M6E","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FJAWDIPA6M6EKC3Q","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FJAWDIPA","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:FJAWDIPA6M6EKC3Q6UD5MULDBC","target":"record","payload":{"canonical_record":{"source":{"id":"1205.6256","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-05-29T04:11:09Z","cross_cats_sorted":[],"title_canon_sha256":"047ab7a82a71b5630d0dc6a505e3de47d52fbbb737c4bfb0803f68d14c6d7825","abstract_canon_sha256":"dffed89e134d013b217bf2ee8f4a969b237cb5c1628f1008838320cfae1c8120"},"schema_version":"1.0"},"canonical_sha256":"2a4161a1e0f33c450b70f507d6516308a9d7ed360e9443d0da193ad9ade21086","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:58.424128Z","signature_b64":"NXzVDHxBMi+ExTm0ELmg0xBrLQEz/gqWOC5fpUCLFIeMkfOmuY4hGXMLBAge+VD/6qRoQzRYayVw9QPAthlECw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a4161a1e0f33c450b70f507d6516308a9d7ed360e9443d0da193ad9ade21086","last_reissued_at":"2026-05-18T03:10:58.423387Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:58.423387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1205.6256","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:10:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tiZudjttCSdh5t5Q1DYDiRc/A4O41AgmQZqCfzboHzX3+3L7gQoDvAveT263MWQckH4gH4eZC3IueAA0DuKUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T21:00:34.263398Z"},"content_sha256":"215d5a609094283561e39bda066318ae1fb877bb6d6ea537269e21fe8a4e50a4","schema_version":"1.0","event_id":"sha256:215d5a609094283561e39bda066318ae1fb877bb6d6ea537269e21fe8a4e50a4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:FJAWDIPA6M6EKC3Q6UD5MULDBC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lattices generated by Chip Firing Game models: criteria and recognition algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Thi Ha Duong Phan, Trung Van Pham","submitted_at":"2012-05-29T04:11:09Z","abstract_excerpt":"It is well-known that the class of lattices generated by Chip Firing games (CFGs) is strictly included in the class of upper locally distributive lattices (ULD). However a necessary and sufficient criterion for this class is still an open question. In this paper we settle this problem by giving such a criterion. This criterion provides a polynomial-time algorithm for constructing a CFG which generates a given lattice if such a CFG exists. Going further we solve the same problem on two other classes of lattices which are generated by CFGs on the classes of undirected graphs and directed acyclic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.6256","kind":"arxiv","version":3},"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":"2026-05-18T03:10:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"08kOu8RQDDdrx+XPChxvPSEZaHs6jp4BAZNe01bbNCtggJ+EFnVJAo3h+MQm0q0MfJas3ngu0yfA0Ksh/MYWBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T21:00:34.264044Z"},"content_sha256":"cdeb0d47f7f483ec934dfd74122884af586ebced7e3964b9d0153bbe49e39e52","schema_version":"1.0","event_id":"sha256:cdeb0d47f7f483ec934dfd74122884af586ebced7e3964b9d0153bbe49e39e52"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FJAWDIPA6M6EKC3Q6UD5MULDBC/bundle.json","state_url":"https://pith.science/pith/FJAWDIPA6M6EKC3Q6UD5MULDBC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FJAWDIPA6M6EKC3Q6UD5MULDBC/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":"2026-05-25T21:00:34Z","links":{"resolver":"https://pith.science/pith/FJAWDIPA6M6EKC3Q6UD5MULDBC","bundle":"https://pith.science/pith/FJAWDIPA6M6EKC3Q6UD5MULDBC/bundle.json","state":"https://pith.science/pith/FJAWDIPA6M6EKC3Q6UD5MULDBC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FJAWDIPA6M6EKC3Q6UD5MULDBC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FJAWDIPA6M6EKC3Q6UD5MULDBC","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":"dffed89e134d013b217bf2ee8f4a969b237cb5c1628f1008838320cfae1c8120","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-05-29T04:11:09Z","title_canon_sha256":"047ab7a82a71b5630d0dc6a505e3de47d52fbbb737c4bfb0803f68d14c6d7825"},"schema_version":"1.0","source":{"id":"1205.6256","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.6256","created_at":"2026-05-18T03:10:58Z"},{"alias_kind":"arxiv_version","alias_value":"1205.6256v3","created_at":"2026-05-18T03:10:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.6256","created_at":"2026-05-18T03:10:58Z"},{"alias_kind":"pith_short_12","alias_value":"FJAWDIPA6M6E","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FJAWDIPA6M6EKC3Q","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FJAWDIPA","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:cdeb0d47f7f483ec934dfd74122884af586ebced7e3964b9d0153bbe49e39e52","target":"graph","created_at":"2026-05-18T03:10:58Z","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":"It is well-known that the class of lattices generated by Chip Firing games (CFGs) is strictly included in the class of upper locally distributive lattices (ULD). However a necessary and sufficient criterion for this class is still an open question. In this paper we settle this problem by giving such a criterion. This criterion provides a polynomial-time algorithm for constructing a CFG which generates a given lattice if such a CFG exists. Going further we solve the same problem on two other classes of lattices which are generated by CFGs on the classes of undirected graphs and directed acyclic","authors_text":"Thi Ha Duong Phan, Trung Van Pham","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-05-29T04:11:09Z","title":"Lattices generated by Chip Firing Game models: criteria and recognition algorithm"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.6256","kind":"arxiv","version":3},"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":"sha256:215d5a609094283561e39bda066318ae1fb877bb6d6ea537269e21fe8a4e50a4","target":"record","created_at":"2026-05-18T03:10:58Z","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":"dffed89e134d013b217bf2ee8f4a969b237cb5c1628f1008838320cfae1c8120","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-05-29T04:11:09Z","title_canon_sha256":"047ab7a82a71b5630d0dc6a505e3de47d52fbbb737c4bfb0803f68d14c6d7825"},"schema_version":"1.0","source":{"id":"1205.6256","kind":"arxiv","version":3}},"canonical_sha256":"2a4161a1e0f33c450b70f507d6516308a9d7ed360e9443d0da193ad9ade21086","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2a4161a1e0f33c450b70f507d6516308a9d7ed360e9443d0da193ad9ade21086","first_computed_at":"2026-05-18T03:10:58.423387Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:10:58.423387Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NXzVDHxBMi+ExTm0ELmg0xBrLQEz/gqWOC5fpUCLFIeMkfOmuY4hGXMLBAge+VD/6qRoQzRYayVw9QPAthlECw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:10:58.424128Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.6256","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:215d5a609094283561e39bda066318ae1fb877bb6d6ea537269e21fe8a4e50a4","sha256:cdeb0d47f7f483ec934dfd74122884af586ebced7e3964b9d0153bbe49e39e52"],"state_sha256":"ca8e016f031907095eec6553493f7b48a9615a6e10bed407fb1a10420fb98e73"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PsLF9DKrR/R5qcQsiz4UNQCuUCvw3TkIQ2VkHNizQoOGW/KWJaVpzio6F2r9Ipr1EGJ3tfLpYVzsGBQ+BG2lCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T21:00:34.267747Z","bundle_sha256":"2ad36a7a2f0a5a6f02e1373d4293b27d8393bff73f375f96b6aae3945e2b48c9"}}