{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:KCXFQTBFQPMTM2ASEA2Y2MZ5WV","short_pith_number":"pith:KCXFQTBF","canonical_record":{"source":{"id":"2608.01199","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-08-02T12:30:33Z","cross_cats_sorted":[],"title_canon_sha256":"bce5118582a09d9a32b5a3d1c89d62a7b22b4c338ad0e5932d764fb315bb2df8","abstract_canon_sha256":"6e08d33c33e1f14f5fe84b4cf54e9011b811baf84f93ba896bde00c5f7ecfc7f"},"schema_version":"1.0"},"canonical_sha256":"50ae584c2583d936681220358d333db54733f792949a3bf5239fe9316c67072b","source":{"kind":"arxiv","id":"2608.01199","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.01199","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"arxiv_version","alias_value":"2608.01199v1","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01199","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"pith_short_12","alias_value":"KCXFQTBFQPMT","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"pith_short_16","alias_value":"KCXFQTBFQPMTM2AS","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"pith_short_8","alias_value":"KCXFQTBF","created_at":"2026-08-04T02:01:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:KCXFQTBFQPMTM2ASEA2Y2MZ5WV","target":"record","payload":{"canonical_record":{"source":{"id":"2608.01199","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-08-02T12:30:33Z","cross_cats_sorted":[],"title_canon_sha256":"bce5118582a09d9a32b5a3d1c89d62a7b22b4c338ad0e5932d764fb315bb2df8","abstract_canon_sha256":"6e08d33c33e1f14f5fe84b4cf54e9011b811baf84f93ba896bde00c5f7ecfc7f"},"schema_version":"1.0"},"canonical_sha256":"50ae584c2583d936681220358d333db54733f792949a3bf5239fe9316c67072b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:01:55.306750Z","signature_b64":"Wn5+3vjnSP6+w204r5Wf3hNVBheeCUS/m1rRAowmMvof5eSC5r76zhrNsedWWeD122XyUrv9OMgaYRi+8vt+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50ae584c2583d936681220358d333db54733f792949a3bf5239fe9316c67072b","last_reissued_at":"2026-08-04T02:01:55.304598Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:01:55.304598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2608.01199","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":"2026-08-04T02:01:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xd6sXzqE1ZSvUCRYBVk4WFKAeBOA4/pAH3U21F2gxt8L9na5cS9mz1pFIhNk9PH8s5iVoqhJUp3bBvozeu7aAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T02:40:18.137442Z"},"content_sha256":"59f1c1ba0118793ae78fac2129bee950e2551ee465991535ba549ad22129344d","schema_version":"1.0","event_id":"sha256:59f1c1ba0118793ae78fac2129bee950e2551ee465991535ba549ad22129344d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:KCXFQTBFQPMTM2ASEA2Y2MZ5WV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\mathrm{v}$-number of Lov\\'asz-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Emiliano Liwski, Manohar Kumar","submitted_at":"2026-08-02T12:30:33Z","abstract_excerpt":"In this paper, we introduce a new framework for computing certain localized $\\mathrm{v}$-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lov\\'asz-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph $G$, we derive an explicit formula for the localized $\\mathrm{v}$-number of the LSS ideal $L_G^{\\mathbb{K}}(d)$, denoted by $\\mathrm{v}_{\\mathfrak{p}_{\\emptyset}(G)}(L_G^{\\mathbb{K}}(d))$, for all $d \\geq 2$, where $\\mathbb{K}$ is an algebraically closed field. As a consequence, we prove that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01199","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/2608.01199/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":"2026-08-04T02:01:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mo/rhNgFUC9icHgDuE++6dl0sOkm+CWY0PkaCpNcJ99pZqf0c6op5Axjc8on6bKdTzQC/KMD1X115fG48G2DBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T02:40:18.138565Z"},"content_sha256":"aa6670b6785a5542576979041625c2ccccf2dedf6da409926e234bf63dbb4f3e","schema_version":"1.0","event_id":"sha256:aa6670b6785a5542576979041625c2ccccf2dedf6da409926e234bf63dbb4f3e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KCXFQTBFQPMTM2ASEA2Y2MZ5WV/bundle.json","state_url":"https://pith.science/pith/KCXFQTBFQPMTM2ASEA2Y2MZ5WV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KCXFQTBFQPMTM2ASEA2Y2MZ5WV/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":"2026-08-16T02:40:18Z","links":{"resolver":"https://pith.science/pith/KCXFQTBFQPMTM2ASEA2Y2MZ5WV","bundle":"https://pith.science/pith/KCXFQTBFQPMTM2ASEA2Y2MZ5WV/bundle.json","state":"https://pith.science/pith/KCXFQTBFQPMTM2ASEA2Y2MZ5WV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KCXFQTBFQPMTM2ASEA2Y2MZ5WV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KCXFQTBFQPMTM2ASEA2Y2MZ5WV","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":"6e08d33c33e1f14f5fe84b4cf54e9011b811baf84f93ba896bde00c5f7ecfc7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-08-02T12:30:33Z","title_canon_sha256":"bce5118582a09d9a32b5a3d1c89d62a7b22b4c338ad0e5932d764fb315bb2df8"},"schema_version":"1.0","source":{"id":"2608.01199","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.01199","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"arxiv_version","alias_value":"2608.01199v1","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01199","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"pith_short_12","alias_value":"KCXFQTBFQPMT","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"pith_short_16","alias_value":"KCXFQTBFQPMTM2AS","created_at":"2026-08-04T02:01:55Z"},{"alias_kind":"pith_short_8","alias_value":"KCXFQTBF","created_at":"2026-08-04T02:01:55Z"}],"graph_snapshots":[{"event_id":"sha256:aa6670b6785a5542576979041625c2ccccf2dedf6da409926e234bf63dbb4f3e","target":"graph","created_at":"2026-08-04T02:01:55Z","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/2608.01199/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we introduce a new framework for computing certain localized $\\mathrm{v}$-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lov\\'asz-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph $G$, we derive an explicit formula for the localized $\\mathrm{v}$-number of the LSS ideal $L_G^{\\mathbb{K}}(d)$, denoted by $\\mathrm{v}_{\\mathfrak{p}_{\\emptyset}(G)}(L_G^{\\mathbb{K}}(d))$, for all $d \\geq 2$, where $\\mathbb{K}$ is an algebraically closed field. As a consequence, we prove that","authors_text":"Emiliano Liwski, Manohar Kumar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-08-02T12:30:33Z","title":"$\\mathrm{v}$-number of Lov\\'asz-Saks-Schrijver Ideals and (parity) binomial edge ideals of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01199","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":"sha256:59f1c1ba0118793ae78fac2129bee950e2551ee465991535ba549ad22129344d","target":"record","created_at":"2026-08-04T02:01:55Z","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":"6e08d33c33e1f14f5fe84b4cf54e9011b811baf84f93ba896bde00c5f7ecfc7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-08-02T12:30:33Z","title_canon_sha256":"bce5118582a09d9a32b5a3d1c89d62a7b22b4c338ad0e5932d764fb315bb2df8"},"schema_version":"1.0","source":{"id":"2608.01199","kind":"arxiv","version":1}},"canonical_sha256":"50ae584c2583d936681220358d333db54733f792949a3bf5239fe9316c67072b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50ae584c2583d936681220358d333db54733f792949a3bf5239fe9316c67072b","first_computed_at":"2026-08-04T02:01:55.304598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-04T02:01:55.304598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Wn5+3vjnSP6+w204r5Wf3hNVBheeCUS/m1rRAowmMvof5eSC5r76zhrNsedWWeD122XyUrv9OMgaYRi+8vt+Aw==","signature_status":"signed_v1","signed_at":"2026-08-04T02:01:55.306750Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.01199","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:59f1c1ba0118793ae78fac2129bee950e2551ee465991535ba549ad22129344d","sha256:aa6670b6785a5542576979041625c2ccccf2dedf6da409926e234bf63dbb4f3e"],"state_sha256":"e16a35f4e62a9aa991e0ab8475797b97fad497d561c3d1c3555351aab7ab03bd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FCo1VRGQOKLY8TGei2A2VNChvmaqcqrBFnVh4m7dNYf0NOyHiETB41VAHHliuEw48466xeXM5YUDL2INtk9FCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T02:40:18.145208Z","bundle_sha256":"2ce6edd1ca8364ab147d1fb6d15177d9a33d083f7753b5db2f4402d60c401548"}}