{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:2EDXS5UMW3XA4VWW6SAGF3VO4N","short_pith_number":"pith:2EDXS5UM","canonical_record":{"source":{"id":"2605.13662","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-13T15:19:19Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b7575570dd8e3cad34953ad59beb6f5d23cd17671fbe915a7520510b2c3640b8","abstract_canon_sha256":"d6f5cb6463c67c6488c1129caac6637315610715bbdd53ab7d03db1ac524619d"},"schema_version":"1.0"},"canonical_sha256":"d10779768cb6ee0e56d6f48062eeaee374f972246788f6f0aa72be64e0f26a42","source":{"kind":"arxiv","id":"2605.13662","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13662","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13662v1","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13662","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"pith_short_12","alias_value":"2EDXS5UMW3XA","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"2EDXS5UMW3XA4VWW","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"2EDXS5UM","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:2EDXS5UMW3XA4VWW6SAGF3VO4N","target":"record","payload":{"canonical_record":{"source":{"id":"2605.13662","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-13T15:19:19Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b7575570dd8e3cad34953ad59beb6f5d23cd17671fbe915a7520510b2c3640b8","abstract_canon_sha256":"d6f5cb6463c67c6488c1129caac6637315610715bbdd53ab7d03db1ac524619d"},"schema_version":"1.0"},"canonical_sha256":"d10779768cb6ee0e56d6f48062eeaee374f972246788f6f0aa72be64e0f26a42","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:17.301665Z","signature_b64":"bGLhBBAmc30llidwNzLDP7eig3gD+I83x+SnaP8W3n7wvrKBWfMZbNi0TLiIRPVA0Ittgsp8EVtPcTmxKcdtDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d10779768cb6ee0e56d6f48062eeaee374f972246788f6f0aa72be64e0f26a42","last_reissued_at":"2026-05-18T02:44:17.301153Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:17.301153Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.13662","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":"2026-05-18T02:44:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wb0pPN2TmcnbPpqULU+LxoDuWI+eidQQObKu3XFdnXd94Um6O8NNpCVZ+za/BQ+43VfcVm0yxrB2yavavlDRBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T11:42:35.660612Z"},"content_sha256":"885faa0bcc9d55ac40668b47575f9a0aee081af2a5796bbeac773b435fb55be9","schema_version":"1.0","event_id":"sha256:885faa0bcc9d55ac40668b47575f9a0aee081af2a5796bbeac773b435fb55be9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:2EDXS5UMW3XA4VWW6SAGF3VO4N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distance Reduction in Bouquet Decompositions and Toric Ideals of Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For complete intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they reduce distance on the circuits.","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Alexander Milner, Dimitra Kosta, Oliver Clarke","submitted_at":"2026-05-13T15:19:19Z","abstract_excerpt":"The distance-reduction property for a generating set, i.e., a Markov basis, of a toric ideal is a condition that ensures tight connectivity of its fibres. In this paper, we study the distance-reduction property for toric ideals of graphs and move on to explore the relationship between the distance-reduction property and the bouquet structure of homogeneous toric ideals, which includes the class of toric ideals of graphs. For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. Under the condition of homogeneity, we show that, for toric ideals with the same bouquet structure and signature, the distance-reduction properties are preserved.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The toric ideals are homogeneous and the bouquet matrix is a monomial curve in A^3; without homogeneity the preservation across bouquets may fail.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For complete-intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they distance-reduce the circuits, and this property is preserved across homogeneous ideals sharing the same bouquet structure and signature.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For complete intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they reduce distance on the circuits.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0a907f4b524add87b4380f7faf560e5f9147d9e44db09828ad6cd7b8e8d2d61c"},"source":{"id":"2605.13662","kind":"arxiv","version":1},"verdict":{"id":"e9e43d8f-bd54-42ac-ab90-996f1807380c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:40:32.680397Z","strongest_claim":"For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. Under the condition of homogeneity, we show that, for toric ideals with the same bouquet structure and signature, the distance-reduction properties are preserved.","one_line_summary":"For complete-intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they distance-reduce the circuits, and this property is preserved across homogeneous ideals sharing the same bouquet structure and signature.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The toric ideals are homogeneous and the bouquet matrix is a monomial curve in A^3; without homogeneity the preservation across bouquets may fail.","pith_extraction_headline":"For complete intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they reduce distance on the circuits."},"references":{"count":22,"sample":[{"doi":"","year":2012,"title":"Springer Science & Business Media, 2012","work_id":"9e37f06b-d251-4895-a6de-63c445c0e80a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"Distance-reducing markov bases for sampling from a discrete sample space.Bernoulli, 11(5):793–813, 2005","work_id":"ca610a3e-54b9-4404-a9b8-31fe8534b19e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"Minimal systems of binomial generators and the indispensable complex of a toric ideal.Proceedings of the American Mathematical Society, pages 3443–3451, 2007","work_id":"f802f767-aca2-446a-9d33-0f1ac0e890a4","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"Markov complexity of monomial curves.Journal of Algebra, 417:391–411, 2014","work_id":"30679970-c014-49d3-8f14-981805f05492","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Distance reducing markov bases.Journal of Pure and Applied Algebra, 229(10):108057, 2025","work_id":"dee0d8e3-1de7-48c5-a814-ab54b6cca271","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":22,"snapshot_sha256":"033de0f4838eacd20af6c4f088a0b6953ac32dec09e5ff2906703c38c59f818b","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"274230eb35d6107ffdfd57ff801771fac456a756484866d23675c4c14412e9cc"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"e9e43d8f-bd54-42ac-ab90-996f1807380c"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U8/5cvIspc2yK/SrKF5nNxoVR8L9m3o17/PUSa+sM6YuwBLqDeEpehpNONwo4BJjC4xpwzbv+XxhhcXznsOMCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T11:42:35.661268Z"},"content_sha256":"bf29089170369af7d152a45b8e144cd8c5fd43ffbece94b701759669b575a6f3","schema_version":"1.0","event_id":"sha256:bf29089170369af7d152a45b8e144cd8c5fd43ffbece94b701759669b575a6f3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2EDXS5UMW3XA4VWW6SAGF3VO4N/bundle.json","state_url":"https://pith.science/pith/2EDXS5UMW3XA4VWW6SAGF3VO4N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2EDXS5UMW3XA4VWW6SAGF3VO4N/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":"2026-06-01T11:42:35Z","links":{"resolver":"https://pith.science/pith/2EDXS5UMW3XA4VWW6SAGF3VO4N","bundle":"https://pith.science/pith/2EDXS5UMW3XA4VWW6SAGF3VO4N/bundle.json","state":"https://pith.science/pith/2EDXS5UMW3XA4VWW6SAGF3VO4N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2EDXS5UMW3XA4VWW6SAGF3VO4N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2EDXS5UMW3XA4VWW6SAGF3VO4N","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":"d6f5cb6463c67c6488c1129caac6637315610715bbdd53ab7d03db1ac524619d","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-13T15:19:19Z","title_canon_sha256":"b7575570dd8e3cad34953ad59beb6f5d23cd17671fbe915a7520510b2c3640b8"},"schema_version":"1.0","source":{"id":"2605.13662","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13662","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13662v1","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13662","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"pith_short_12","alias_value":"2EDXS5UMW3XA","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"2EDXS5UMW3XA4VWW","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"2EDXS5UM","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:bf29089170369af7d152a45b8e144cd8c5fd43ffbece94b701759669b575a6f3","target":"graph","created_at":"2026-05-18T02:44:17Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. Under the condition of homogeneity, we show that, for toric ideals with the same bouquet structure and signature, the distance-reduction properties are preserved."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The toric ideals are homogeneous and the bouquet matrix is a monomial curve in A^3; without homogeneity the preservation across bouquets may fail."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"For complete-intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they distance-reduce the circuits, and this property is preserved across homogeneous ideals sharing the same bouquet structure and signature."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"For complete intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they reduce distance on the circuits."}],"snapshot_sha256":"0a907f4b524add87b4380f7faf560e5f9147d9e44db09828ad6cd7b8e8d2d61c"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"274230eb35d6107ffdfd57ff801771fac456a756484866d23675c4c14412e9cc"},"paper":{"abstract_excerpt":"The distance-reduction property for a generating set, i.e., a Markov basis, of a toric ideal is a condition that ensures tight connectivity of its fibres. In this paper, we study the distance-reduction property for toric ideals of graphs and move on to explore the relationship between the distance-reduction property and the bouquet structure of homogeneous toric ideals, which includes the class of toric ideals of graphs. For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the","authors_text":"Alexander Milner, Dimitra Kosta, Oliver Clarke","cross_cats":["math.CO"],"headline":"For complete intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they reduce distance on the circuits.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-13T15:19:19Z","title":"Distance Reduction in Bouquet Decompositions and Toric Ideals of Graphs"},"references":{"count":22,"internal_anchors":0,"resolved_work":22,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Springer Science & Business Media, 2012","work_id":"9e37f06b-d251-4895-a6de-63c445c0e80a","year":2012},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Distance-reducing markov bases for sampling from a discrete sample space.Bernoulli, 11(5):793–813, 2005","work_id":"ca610a3e-54b9-4404-a9b8-31fe8534b19e","year":2005},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Minimal systems of binomial generators and the indispensable complex of a toric ideal.Proceedings of the American Mathematical Society, pages 3443–3451, 2007","work_id":"f802f767-aca2-446a-9d33-0f1ac0e890a4","year":2007},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Markov complexity of monomial curves.Journal of Algebra, 417:391–411, 2014","work_id":"30679970-c014-49d3-8f14-981805f05492","year":2014},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Distance reducing markov bases.Journal of Pure and Applied Algebra, 229(10):108057, 2025","work_id":"dee0d8e3-1de7-48c5-a814-ab54b6cca271","year":2025}],"snapshot_sha256":"033de0f4838eacd20af6c4f088a0b6953ac32dec09e5ff2906703c38c59f818b"},"source":{"id":"2605.13662","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T17:40:32.680397Z","id":"e9e43d8f-bd54-42ac-ab90-996f1807380c","model_set":{"reader":"grok-4.3"},"one_line_summary":"For complete-intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they distance-reduce the circuits, and this property is preserved across homogeneous ideals sharing the same bouquet structure and signature.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"For complete intersection toric ideals of graphs, minimal Markov bases are distance-reducing exactly when they reduce distance on the circuits.","strongest_claim":"For toric ideals of graphs which are complete intersection, we show that the minimal Markov bases are distance-reducing if and only if they distance-reduce the circuits of the ideal. Under the condition of homogeneity, we show that, for toric ideals with the same bouquet structure and signature, the distance-reduction properties are preserved.","weakest_assumption":"The toric ideals are homogeneous and the bouquet matrix is a monomial curve in A^3; without homogeneity the preservation across bouquets may fail."}},"verdict_id":"e9e43d8f-bd54-42ac-ab90-996f1807380c"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:885faa0bcc9d55ac40668b47575f9a0aee081af2a5796bbeac773b435fb55be9","target":"record","created_at":"2026-05-18T02:44:17Z","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":"d6f5cb6463c67c6488c1129caac6637315610715bbdd53ab7d03db1ac524619d","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-05-13T15:19:19Z","title_canon_sha256":"b7575570dd8e3cad34953ad59beb6f5d23cd17671fbe915a7520510b2c3640b8"},"schema_version":"1.0","source":{"id":"2605.13662","kind":"arxiv","version":1}},"canonical_sha256":"d10779768cb6ee0e56d6f48062eeaee374f972246788f6f0aa72be64e0f26a42","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d10779768cb6ee0e56d6f48062eeaee374f972246788f6f0aa72be64e0f26a42","first_computed_at":"2026-05-18T02:44:17.301153Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:17.301153Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bGLhBBAmc30llidwNzLDP7eig3gD+I83x+SnaP8W3n7wvrKBWfMZbNi0TLiIRPVA0Ittgsp8EVtPcTmxKcdtDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:17.301665Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13662","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:885faa0bcc9d55ac40668b47575f9a0aee081af2a5796bbeac773b435fb55be9","sha256:bf29089170369af7d152a45b8e144cd8c5fd43ffbece94b701759669b575a6f3"],"state_sha256":"56df862ddca845e1e9df76b9a32629bd52cd5c2b209a6f623b2292a27b7113b6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hp255AHHLKJHVycJwJa6idSr5OiQ/sgrrNgYqXEfx3nxsYUHBfGUsQEb0o5W+sOzN6PN1Y5QELR02s54Qh3QCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T11:42:35.664003Z","bundle_sha256":"e7e78043e4085494fe5b1ed768432a612c177cab0feb6be5f295fa778f579c6b"}}