{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:F5WFM4THV7T3XSGT2HNWU3AP5D","short_pith_number":"pith:F5WFM4TH","canonical_record":{"source":{"id":"2502.06714","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2025-02-10T17:47:29Z","cross_cats_sorted":[],"title_canon_sha256":"c7a86bda551a31f7adbfc33c4ca7c78c028f9d5cc8116c5b031e64faaaab299d","abstract_canon_sha256":"102a427bbe03571136004e3033ad79f6d603e757b2e32f332b3bda5ceb3e5cab"},"schema_version":"1.0"},"canonical_sha256":"2f6c567267afe7bbc8d3d1db6a6c0fe8fc1e62886ce9ecfe6ff2a323db070603","source":{"kind":"arxiv","id":"2502.06714","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.06714","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"arxiv_version","alias_value":"2502.06714v2","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.06714","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"pith_short_12","alias_value":"F5WFM4THV7T3","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"pith_short_16","alias_value":"F5WFM4THV7T3XSGT","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"pith_short_8","alias_value":"F5WFM4TH","created_at":"2026-07-05T10:16:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:F5WFM4THV7T3XSGT2HNWU3AP5D","target":"record","payload":{"canonical_record":{"source":{"id":"2502.06714","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2025-02-10T17:47:29Z","cross_cats_sorted":[],"title_canon_sha256":"c7a86bda551a31f7adbfc33c4ca7c78c028f9d5cc8116c5b031e64faaaab299d","abstract_canon_sha256":"102a427bbe03571136004e3033ad79f6d603e757b2e32f332b3bda5ceb3e5cab"},"schema_version":"1.0"},"canonical_sha256":"2f6c567267afe7bbc8d3d1db6a6c0fe8fc1e62886ce9ecfe6ff2a323db070603","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:16:41.522769Z","signature_b64":"vgYtCIM6/eOSZV0pao7XVCU90aT1Cwv08B5iTKIUeNlHHWM/Ll2YY6Vq2ApgRaI00oVgdk8V8n+qR4kIjx/qCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f6c567267afe7bbc8d3d1db6a6c0fe8fc1e62886ce9ecfe6ff2a323db070603","last_reissued_at":"2026-07-05T10:16:41.522240Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:16:41.522240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2502.06714","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:16:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vNfXGhXGWGbpl3v/SpB+9XA4FF0juX2hLfmIfnzPS0n7a8xgAfF3rd9L/UGMKdFr63UxTmJVXDZpj7ClH2uTDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T09:06:37.143819Z"},"content_sha256":"bbd05dd341474cf931d405f4c470ab93ff76de13c52673f20292a1ae0e92b73f","schema_version":"1.0","event_id":"sha256:bbd05dd341474cf931d405f4c470ab93ff76de13c52673f20292a1ae0e92b73f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:F5WFM4THV7T3XSGT2HNWU3AP5D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tensor Product of Polymatroids and Common Information","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carles Padr\\'o","submitted_at":"2025-02-10T17:47:29Z","abstract_excerpt":"A new connection between two different necessary conditions for a polymatroid to be linearly representable is presented. Specifically, we prove that the existence of a tensor product with the uniform matroid of rank two on three elements implies the existence of a common information extension for every pair of subsets of the ground set."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.06714","kind":"arxiv","version":2},"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/2502.06714/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":"2026-07-05T10:16:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rl3d9HGTtq6TATO9qris8dzHmL4vQi3wOmikZVbFSXIY48PaJWiMiDMp/sg2uC1v56RFOzEfhTY3ESSJJCNpAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T09:06:37.144346Z"},"content_sha256":"16922863576763e463d6c1f323ce55e04bd1e987e33659704e2aca20c04161c1","schema_version":"1.0","event_id":"sha256:16922863576763e463d6c1f323ce55e04bd1e987e33659704e2aca20c04161c1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F5WFM4THV7T3XSGT2HNWU3AP5D/bundle.json","state_url":"https://pith.science/pith/F5WFM4THV7T3XSGT2HNWU3AP5D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F5WFM4THV7T3XSGT2HNWU3AP5D/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":"2026-08-13T09:06:37Z","links":{"resolver":"https://pith.science/pith/F5WFM4THV7T3XSGT2HNWU3AP5D","bundle":"https://pith.science/pith/F5WFM4THV7T3XSGT2HNWU3AP5D/bundle.json","state":"https://pith.science/pith/F5WFM4THV7T3XSGT2HNWU3AP5D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F5WFM4THV7T3XSGT2HNWU3AP5D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:F5WFM4THV7T3XSGT2HNWU3AP5D","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":"102a427bbe03571136004e3033ad79f6d603e757b2e32f332b3bda5ceb3e5cab","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2025-02-10T17:47:29Z","title_canon_sha256":"c7a86bda551a31f7adbfc33c4ca7c78c028f9d5cc8116c5b031e64faaaab299d"},"schema_version":"1.0","source":{"id":"2502.06714","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.06714","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"arxiv_version","alias_value":"2502.06714v2","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.06714","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"pith_short_12","alias_value":"F5WFM4THV7T3","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"pith_short_16","alias_value":"F5WFM4THV7T3XSGT","created_at":"2026-07-05T10:16:41Z"},{"alias_kind":"pith_short_8","alias_value":"F5WFM4TH","created_at":"2026-07-05T10:16:41Z"}],"graph_snapshots":[{"event_id":"sha256:16922863576763e463d6c1f323ce55e04bd1e987e33659704e2aca20c04161c1","target":"graph","created_at":"2026-07-05T10:16:41Z","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/2502.06714/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A new connection between two different necessary conditions for a polymatroid to be linearly representable is presented. Specifically, we prove that the existence of a tensor product with the uniform matroid of rank two on three elements implies the existence of a common information extension for every pair of subsets of the ground set.","authors_text":"Carles Padr\\'o","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2025-02-10T17:47:29Z","title":"Tensor Product of Polymatroids and Common Information"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.06714","kind":"arxiv","version":2},"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:bbd05dd341474cf931d405f4c470ab93ff76de13c52673f20292a1ae0e92b73f","target":"record","created_at":"2026-07-05T10:16:41Z","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":"102a427bbe03571136004e3033ad79f6d603e757b2e32f332b3bda5ceb3e5cab","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2025-02-10T17:47:29Z","title_canon_sha256":"c7a86bda551a31f7adbfc33c4ca7c78c028f9d5cc8116c5b031e64faaaab299d"},"schema_version":"1.0","source":{"id":"2502.06714","kind":"arxiv","version":2}},"canonical_sha256":"2f6c567267afe7bbc8d3d1db6a6c0fe8fc1e62886ce9ecfe6ff2a323db070603","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f6c567267afe7bbc8d3d1db6a6c0fe8fc1e62886ce9ecfe6ff2a323db070603","first_computed_at":"2026-07-05T10:16:41.522240Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:16:41.522240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vgYtCIM6/eOSZV0pao7XVCU90aT1Cwv08B5iTKIUeNlHHWM/Ll2YY6Vq2ApgRaI00oVgdk8V8n+qR4kIjx/qCg==","signature_status":"signed_v1","signed_at":"2026-07-05T10:16:41.522769Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.06714","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bbd05dd341474cf931d405f4c470ab93ff76de13c52673f20292a1ae0e92b73f","sha256:16922863576763e463d6c1f323ce55e04bd1e987e33659704e2aca20c04161c1"],"state_sha256":"d68f61a2ed3f93cb8d582c56530f9a0999aef42b33d025c2e8b6b3da61c183ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EFtrQ6OfFOyEZhIViVxuzp29Gh5mPWskyKj+iRFl8cWACJEo51+VEll2akeFnjIr4MrKD5gFh//L7PxlBA3DDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T09:06:37.149509Z","bundle_sha256":"61461d0027c10d87b27309c2ab24b813c1608f2aa456363813596b0cb6adc6ef"}}