{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:DO7KQPAS55LLNMAYXDDTDK4APK","short_pith_number":"pith:DO7KQPAS","canonical_record":{"source":{"id":"2602.20765","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-24T10:45:18Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"c663e5c6b1c98cf715b79adde3ce2c55e0a77eef2ee7b454364424719d9ae741","abstract_canon_sha256":"58a14419416d29167a6edb062db2ec6d3c8de4357e91fb369cf04232825924b3"},"schema_version":"1.0"},"canonical_sha256":"1bbea83c12ef56b6b018b8c731ab807a8dad45ccaf4d3731017559fc41162464","source":{"kind":"arxiv","id":"2602.20765","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.20765","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"arxiv_version","alias_value":"2602.20765v2","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.20765","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"pith_short_12","alias_value":"DO7KQPAS55LL","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"pith_short_16","alias_value":"DO7KQPAS55LLNMAY","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"pith_short_8","alias_value":"DO7KQPAS","created_at":"2026-06-24T01:15:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:DO7KQPAS55LLNMAYXDDTDK4APK","target":"record","payload":{"canonical_record":{"source":{"id":"2602.20765","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-24T10:45:18Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"c663e5c6b1c98cf715b79adde3ce2c55e0a77eef2ee7b454364424719d9ae741","abstract_canon_sha256":"58a14419416d29167a6edb062db2ec6d3c8de4357e91fb369cf04232825924b3"},"schema_version":"1.0"},"canonical_sha256":"1bbea83c12ef56b6b018b8c731ab807a8dad45ccaf4d3731017559fc41162464","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:15:01.377634Z","signature_b64":"5O+sHhAWusHf8+9DwjltzrTMjxy5I+aYgvg5xC1RkchPGy61hIJqP4uaOLm3dVjE7hoZWPldBQP9RtYhWmWDCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bbea83c12ef56b6b018b8c731ab807a8dad45ccaf4d3731017559fc41162464","last_reissued_at":"2026-06-24T01:15:01.377218Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:15:01.377218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2602.20765","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":"2026-06-24T01:15:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"stDlIxEuDH79iO349iFNx8Yr7IYSZQCwm7P8UHRoGLny+tWZqUTZ8WLcKu9JMZxkcwqX9TSRMc79Wo4PqsXOCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T19:31:42.551787Z"},"content_sha256":"e4c3c3afae6db730e17377b94bf492c910f17bd6377205beb7934099705b5739","schema_version":"1.0","event_id":"sha256:e4c3c3afae6db730e17377b94bf492c910f17bd6377205beb7934099705b5739"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:DO7KQPAS55LLNMAYXDDTDK4APK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Preserving Hodge Vectors of Lattice Polytopes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Benjamin Nill, Vadym Kurylenko","submitted_at":"2026-02-24T10:45:18Z","abstract_excerpt":"Given lattice polytopes $P_1, \\ldots, P_k$ contained in a $k$-dimensional subspace $U \\subseteq \\mathbb{R}^d$ and a $d$-dimensional lattice polytope $Q \\subset \\mathbb{R}^d$, we compute the Hodge vector of the Cayley polytope $P_1 * \\cdots * P_k * Q$, and show that it equals the mixed volume of $P_1, \\ldots, P_k$ times the Hodge vector of the projection of $Q$ along $U$. Here, the Hodge vector of a lattice polytope is its local $h^*$-vector with leading and trailing zeroes removed. This result allows finding infinitely many high-dimensional lattice polytopes with the same Hodge vector that are"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.20765","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/2602.20765/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":"2026-06-24T01:15:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LMGwaPFYGGGSnirbfBLjR9cymbOyjgXlNXBh99OCn6nJ+X5TW9aNOfoK9tuUdlI54zVlhr2MdUVdNSuSkdxACA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T19:31:42.552395Z"},"content_sha256":"b61f916c6028755d918827888c1f08342049f5e56d6aa4349a8d230cf786e9e6","schema_version":"1.0","event_id":"sha256:b61f916c6028755d918827888c1f08342049f5e56d6aa4349a8d230cf786e9e6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DO7KQPAS55LLNMAYXDDTDK4APK/bundle.json","state_url":"https://pith.science/pith/DO7KQPAS55LLNMAYXDDTDK4APK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DO7KQPAS55LLNMAYXDDTDK4APK/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":"2026-08-13T19:31:42Z","links":{"resolver":"https://pith.science/pith/DO7KQPAS55LLNMAYXDDTDK4APK","bundle":"https://pith.science/pith/DO7KQPAS55LLNMAYXDDTDK4APK/bundle.json","state":"https://pith.science/pith/DO7KQPAS55LLNMAYXDDTDK4APK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DO7KQPAS55LLNMAYXDDTDK4APK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DO7KQPAS55LLNMAYXDDTDK4APK","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":"58a14419416d29167a6edb062db2ec6d3c8de4357e91fb369cf04232825924b3","cross_cats_sorted":["math.AG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-24T10:45:18Z","title_canon_sha256":"c663e5c6b1c98cf715b79adde3ce2c55e0a77eef2ee7b454364424719d9ae741"},"schema_version":"1.0","source":{"id":"2602.20765","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.20765","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"arxiv_version","alias_value":"2602.20765v2","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.20765","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"pith_short_12","alias_value":"DO7KQPAS55LL","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"pith_short_16","alias_value":"DO7KQPAS55LLNMAY","created_at":"2026-06-24T01:15:01Z"},{"alias_kind":"pith_short_8","alias_value":"DO7KQPAS","created_at":"2026-06-24T01:15:01Z"}],"graph_snapshots":[{"event_id":"sha256:b61f916c6028755d918827888c1f08342049f5e56d6aa4349a8d230cf786e9e6","target":"graph","created_at":"2026-06-24T01:15:01Z","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/2602.20765/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given lattice polytopes $P_1, \\ldots, P_k$ contained in a $k$-dimensional subspace $U \\subseteq \\mathbb{R}^d$ and a $d$-dimensional lattice polytope $Q \\subset \\mathbb{R}^d$, we compute the Hodge vector of the Cayley polytope $P_1 * \\cdots * P_k * Q$, and show that it equals the mixed volume of $P_1, \\ldots, P_k$ times the Hodge vector of the projection of $Q$ along $U$. Here, the Hodge vector of a lattice polytope is its local $h^*$-vector with leading and trailing zeroes removed. This result allows finding infinitely many high-dimensional lattice polytopes with the same Hodge vector that are","authors_text":"Benjamin Nill, Vadym Kurylenko","cross_cats":["math.AG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-24T10:45:18Z","title":"Preserving Hodge Vectors of Lattice Polytopes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.20765","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:e4c3c3afae6db730e17377b94bf492c910f17bd6377205beb7934099705b5739","target":"record","created_at":"2026-06-24T01:15:01Z","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":"58a14419416d29167a6edb062db2ec6d3c8de4357e91fb369cf04232825924b3","cross_cats_sorted":["math.AG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-02-24T10:45:18Z","title_canon_sha256":"c663e5c6b1c98cf715b79adde3ce2c55e0a77eef2ee7b454364424719d9ae741"},"schema_version":"1.0","source":{"id":"2602.20765","kind":"arxiv","version":2}},"canonical_sha256":"1bbea83c12ef56b6b018b8c731ab807a8dad45ccaf4d3731017559fc41162464","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1bbea83c12ef56b6b018b8c731ab807a8dad45ccaf4d3731017559fc41162464","first_computed_at":"2026-06-24T01:15:01.377218Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-24T01:15:01.377218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5O+sHhAWusHf8+9DwjltzrTMjxy5I+aYgvg5xC1RkchPGy61hIJqP4uaOLm3dVjE7hoZWPldBQP9RtYhWmWDCw==","signature_status":"signed_v1","signed_at":"2026-06-24T01:15:01.377634Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.20765","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e4c3c3afae6db730e17377b94bf492c910f17bd6377205beb7934099705b5739","sha256:b61f916c6028755d918827888c1f08342049f5e56d6aa4349a8d230cf786e9e6"],"state_sha256":"0bf7f2214346cf74453b1125acf77affc7c84fcfa680dd071b26a14ee805b111"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aWfRrHi7P5hSYSx51+PWh5pZAhJNER2nTBGug8j4TnJ+/eBxqHCZ2akFEXBR+iiJOqe+j/aqz8vL7Zf5mtTaCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T19:31:42.557337Z","bundle_sha256":"88f1dbebb9b99046b9d5024d426b25561a5c454a6dca4112d80dd9cf8b516b99"}}