{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:WQRCKDWJGWJRARK5ZVIZIMY7M6","short_pith_number":"pith:WQRCKDWJ","canonical_record":{"source":{"id":"2102.08560","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-02-17T04:01:42Z","cross_cats_sorted":["cs.DM","cs.GT"],"title_canon_sha256":"cc184eaa79b814e86c9d3edf4ae951dccc5464ac1da2db309e2caad2dd2f533b","abstract_canon_sha256":"d1103c222c98b735ffed4ccfa1b6e0fc454f36822085b9ea41da034f9429da3d"},"schema_version":"1.0"},"canonical_sha256":"b422250ec9359310455dcd5194331f67a842968c3f856564f7131465eac988e0","source":{"kind":"arxiv","id":"2102.08560","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.08560","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"arxiv_version","alias_value":"2102.08560v1","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.08560","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"pith_short_12","alias_value":"WQRCKDWJGWJR","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"pith_short_16","alias_value":"WQRCKDWJGWJRARK5","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"pith_short_8","alias_value":"WQRCKDWJ","created_at":"2026-07-05T02:16:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:WQRCKDWJGWJRARK5ZVIZIMY7M6","target":"record","payload":{"canonical_record":{"source":{"id":"2102.08560","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-02-17T04:01:42Z","cross_cats_sorted":["cs.DM","cs.GT"],"title_canon_sha256":"cc184eaa79b814e86c9d3edf4ae951dccc5464ac1da2db309e2caad2dd2f533b","abstract_canon_sha256":"d1103c222c98b735ffed4ccfa1b6e0fc454f36822085b9ea41da034f9429da3d"},"schema_version":"1.0"},"canonical_sha256":"b422250ec9359310455dcd5194331f67a842968c3f856564f7131465eac988e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:16:04.601426Z","signature_b64":"xJ5R8WlHtRoN42df8kRGF2tj/4kKMwX8SFdxUn0lIKp6CizcLrRiRsAfbcPq2Zthx+JkWMwzLzfRUh2qKRSDCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b422250ec9359310455dcd5194331f67a842968c3f856564f7131465eac988e0","last_reissued_at":"2026-07-05T02:16:04.600601Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:16:04.600601Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2102.08560","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":"2026-07-05T02:16:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vQTkWiDKx0AiYFn9uns0vYuX19jfgeGE3V2jjcr3+/rbwrqyHb9u4a5becFN++KGDP72B9f3dmLeuNFDUm4YCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T03:52:03.672153Z"},"content_sha256":"9c79392b9bc9086a797616d1237b1dc3faf25a66c585bcdbbf7a4cb2b5876b83","schema_version":"1.0","event_id":"sha256:9c79392b9bc9086a797616d1237b1dc3faf25a66c585bcdbbf7a4cb2b5876b83"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:WQRCKDWJGWJRARK5ZVIZIMY7M6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fair division of graphs and of tangled cakes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.GT"],"primary_cat":"math.CO","authors_text":"Ayumi Igarashi, William S. Zwicker","submitted_at":"2021-02-17T04:01:42Z","abstract_excerpt":"A tangle is a connected topological space constructed by gluing several copies of the unit interval $[0, 1]$. We explore which tangles guarantee envy-free allocations of connected shares for n agents, meaning that such allocations exist no matter which monotonic and continuous functions represent agents' valuations. Each single tangle $\\mathcal{T}$ corresponds in a natural way to an infinite topological class $\\mathcal{G}(\\mathcal{T})$ of multigraphs, many of which are graphs. This correspondence links EF fair division of tangles to EFk$_{outer}$ fair division of graphs. We know from Bil\\`o et"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.08560","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2102.08560/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:16:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FhJXfw4X/yojW5I1oFZUZEnhDrMQEQ8QxJYEUpuw9XzB/L/L9mEF4JnfJFM6NmLdJ+GmQyt4GlD8LUcn4coKBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T03:52:03.673136Z"},"content_sha256":"7acbfe008de4d6e9c93d6473f794634ce1424c85b0319497eb1de2a723368520","schema_version":"1.0","event_id":"sha256:7acbfe008de4d6e9c93d6473f794634ce1424c85b0319497eb1de2a723368520"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WQRCKDWJGWJRARK5ZVIZIMY7M6/bundle.json","state_url":"https://pith.science/pith/WQRCKDWJGWJRARK5ZVIZIMY7M6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WQRCKDWJGWJRARK5ZVIZIMY7M6/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":"2026-08-18T03:52:03Z","links":{"resolver":"https://pith.science/pith/WQRCKDWJGWJRARK5ZVIZIMY7M6","bundle":"https://pith.science/pith/WQRCKDWJGWJRARK5ZVIZIMY7M6/bundle.json","state":"https://pith.science/pith/WQRCKDWJGWJRARK5ZVIZIMY7M6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WQRCKDWJGWJRARK5ZVIZIMY7M6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:WQRCKDWJGWJRARK5ZVIZIMY7M6","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":"d1103c222c98b735ffed4ccfa1b6e0fc454f36822085b9ea41da034f9429da3d","cross_cats_sorted":["cs.DM","cs.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-02-17T04:01:42Z","title_canon_sha256":"cc184eaa79b814e86c9d3edf4ae951dccc5464ac1da2db309e2caad2dd2f533b"},"schema_version":"1.0","source":{"id":"2102.08560","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.08560","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"arxiv_version","alias_value":"2102.08560v1","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.08560","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"pith_short_12","alias_value":"WQRCKDWJGWJR","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"pith_short_16","alias_value":"WQRCKDWJGWJRARK5","created_at":"2026-07-05T02:16:04Z"},{"alias_kind":"pith_short_8","alias_value":"WQRCKDWJ","created_at":"2026-07-05T02:16:04Z"}],"graph_snapshots":[{"event_id":"sha256:7acbfe008de4d6e9c93d6473f794634ce1424c85b0319497eb1de2a723368520","target":"graph","created_at":"2026-07-05T02:16:04Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2102.08560/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A tangle is a connected topological space constructed by gluing several copies of the unit interval $[0, 1]$. We explore which tangles guarantee envy-free allocations of connected shares for n agents, meaning that such allocations exist no matter which monotonic and continuous functions represent agents' valuations. Each single tangle $\\mathcal{T}$ corresponds in a natural way to an infinite topological class $\\mathcal{G}(\\mathcal{T})$ of multigraphs, many of which are graphs. This correspondence links EF fair division of tangles to EFk$_{outer}$ fair division of graphs. We know from Bil\\`o et","authors_text":"Ayumi Igarashi, William S. Zwicker","cross_cats":["cs.DM","cs.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-02-17T04:01:42Z","title":"Fair division of graphs and of tangled cakes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.08560","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":"sha256:9c79392b9bc9086a797616d1237b1dc3faf25a66c585bcdbbf7a4cb2b5876b83","target":"record","created_at":"2026-07-05T02:16:04Z","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":"d1103c222c98b735ffed4ccfa1b6e0fc454f36822085b9ea41da034f9429da3d","cross_cats_sorted":["cs.DM","cs.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-02-17T04:01:42Z","title_canon_sha256":"cc184eaa79b814e86c9d3edf4ae951dccc5464ac1da2db309e2caad2dd2f533b"},"schema_version":"1.0","source":{"id":"2102.08560","kind":"arxiv","version":1}},"canonical_sha256":"b422250ec9359310455dcd5194331f67a842968c3f856564f7131465eac988e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b422250ec9359310455dcd5194331f67a842968c3f856564f7131465eac988e0","first_computed_at":"2026-07-05T02:16:04.600601Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:16:04.600601Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xJ5R8WlHtRoN42df8kRGF2tj/4kKMwX8SFdxUn0lIKp6CizcLrRiRsAfbcPq2Zthx+JkWMwzLzfRUh2qKRSDCA==","signature_status":"signed_v1","signed_at":"2026-07-05T02:16:04.601426Z","signed_message":"canonical_sha256_bytes"},"source_id":"2102.08560","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9c79392b9bc9086a797616d1237b1dc3faf25a66c585bcdbbf7a4cb2b5876b83","sha256:7acbfe008de4d6e9c93d6473f794634ce1424c85b0319497eb1de2a723368520"],"state_sha256":"6285d1026bdbd4776f5df070d8800ea1aa939995ed669107185dd52ee2dd068d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A/kI0SSqDSDW050axdBZqa7ScjCLI+JDbD0dJa7KNOp5eswXdxfBnxrXpQ7BfdQSq6Wg/Y2+sDnSgyXN+y5pAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T03:52:03.681738Z","bundle_sha256":"bffe654fa4e47bd0e1ab2356e55c482f5f22f1711724c4bc4581d1faf849a500"}}