{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:2UMRC3RNQIX7M2UOKQ5D4FZZ2Q","short_pith_number":"pith:2UMRC3RN","canonical_record":{"source":{"id":"2401.06025","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-01-11T16:33:45Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"ea733571314c925002715a064591de9deef34988999abbda522449026682644f","abstract_canon_sha256":"668540acfc1508663a046d6ebe24f06e62acd837551822d7a6e86d2f78e939e3"},"schema_version":"1.0"},"canonical_sha256":"d519116e2d822ff66a8e543a3e1739d416ad267f8f74b5991871f4e82911c06b","source":{"kind":"arxiv","id":"2401.06025","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.06025","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"arxiv_version","alias_value":"2401.06025v3","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.06025","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"pith_short_12","alias_value":"2UMRC3RNQIX7","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"pith_short_16","alias_value":"2UMRC3RNQIX7M2UO","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"pith_short_8","alias_value":"2UMRC3RN","created_at":"2026-07-05T10:19:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:2UMRC3RNQIX7M2UOKQ5D4FZZ2Q","target":"record","payload":{"canonical_record":{"source":{"id":"2401.06025","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-01-11T16:33:45Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"ea733571314c925002715a064591de9deef34988999abbda522449026682644f","abstract_canon_sha256":"668540acfc1508663a046d6ebe24f06e62acd837551822d7a6e86d2f78e939e3"},"schema_version":"1.0"},"canonical_sha256":"d519116e2d822ff66a8e543a3e1739d416ad267f8f74b5991871f4e82911c06b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:19:21.831058Z","signature_b64":"lh0l+ChowxvxZk1rXe2Z5fqtgQ+onHe8gTqNmBhoB+oE8sUxn6Zd4rKgfaf9Q+yD5vTbPpa+yfYWD1WrmEaTCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d519116e2d822ff66a8e543a3e1739d416ad267f8f74b5991871f4e82911c06b","last_reissued_at":"2026-07-05T10:19:21.830550Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:19:21.830550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2401.06025","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:19:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kq2NkekIgPFE9iMXLwIQ+1I1C6wA/LdZptTLz9IKIVPrGEuwHiTIDlVYc9aFEssCAKXAvV9Du4MRUuAjpPfAAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-22T18:03:26.154856Z"},"content_sha256":"0050b5344f0055cc23ec2f111cf55c046650b3fdfdcbad1b010ad39c6e1fb557","schema_version":"1.0","event_id":"sha256:0050b5344f0055cc23ec2f111cf55c046650b3fdfdcbad1b010ad39c6e1fb557"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:2UMRC3RNQIX7M2UOKQ5D4FZZ2Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Christopher O'Neill, Cole Brower, Joseph McDonough","submitted_at":"2024-01-11T16:33:45Z","abstract_excerpt":"A numerical semigroup is a cofinite subset of $\\mathbb Z_{\\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\\mathcal C_m \\subseteq \\mathbb R^{m-1}$, and the face of $\\mathcal C_m$ containing that integer point determines certain algebraic properties of $S$. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of $\\mathcal C_m$. We characterize several aspects of the Kunz fan in terms of the combinatorics of Ku"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.06025","kind":"arxiv","version":3},"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/2401.06025/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":"2026-07-05T10:19:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZYJm5AA/eLYSoqUGmtyw0um3f2Y2CZ/FEQPk+bRfAl/iuUmpXzEdCTGcFCIW6kCdTYrJ67nA82hjd/EYYjEOAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-22T18:03:26.155218Z"},"content_sha256":"adce617576b21bdc782d47774c11d434d8895b8165661792736aa7a9d68a046f","schema_version":"1.0","event_id":"sha256:adce617576b21bdc782d47774c11d434d8895b8165661792736aa7a9d68a046f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2UMRC3RNQIX7M2UOKQ5D4FZZ2Q/bundle.json","state_url":"https://pith.science/pith/2UMRC3RNQIX7M2UOKQ5D4FZZ2Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2UMRC3RNQIX7M2UOKQ5D4FZZ2Q/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":"2026-07-22T18:03:26Z","links":{"resolver":"https://pith.science/pith/2UMRC3RNQIX7M2UOKQ5D4FZZ2Q","bundle":"https://pith.science/pith/2UMRC3RNQIX7M2UOKQ5D4FZZ2Q/bundle.json","state":"https://pith.science/pith/2UMRC3RNQIX7M2UOKQ5D4FZZ2Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2UMRC3RNQIX7M2UOKQ5D4FZZ2Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:2UMRC3RNQIX7M2UOKQ5D4FZZ2Q","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":"668540acfc1508663a046d6ebe24f06e62acd837551822d7a6e86d2f78e939e3","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-01-11T16:33:45Z","title_canon_sha256":"ea733571314c925002715a064591de9deef34988999abbda522449026682644f"},"schema_version":"1.0","source":{"id":"2401.06025","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.06025","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"arxiv_version","alias_value":"2401.06025v3","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.06025","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"pith_short_12","alias_value":"2UMRC3RNQIX7","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"pith_short_16","alias_value":"2UMRC3RNQIX7M2UO","created_at":"2026-07-05T10:19:21Z"},{"alias_kind":"pith_short_8","alias_value":"2UMRC3RN","created_at":"2026-07-05T10:19:21Z"}],"graph_snapshots":[{"event_id":"sha256:adce617576b21bdc782d47774c11d434d8895b8165661792736aa7a9d68a046f","target":"graph","created_at":"2026-07-05T10:19:21Z","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/2401.06025/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A numerical semigroup is a cofinite subset of $\\mathbb Z_{\\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\\mathcal C_m \\subseteq \\mathbb R^{m-1}$, and the face of $\\mathcal C_m$ containing that integer point determines certain algebraic properties of $S$. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of $\\mathcal C_m$. We characterize several aspects of the Kunz fan in terms of the combinatorics of Ku","authors_text":"Christopher O'Neill, Cole Brower, Joseph McDonough","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-01-11T16:33:45Z","title":"Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.06025","kind":"arxiv","version":3},"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":"sha256:0050b5344f0055cc23ec2f111cf55c046650b3fdfdcbad1b010ad39c6e1fb557","target":"record","created_at":"2026-07-05T10:19:21Z","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":"668540acfc1508663a046d6ebe24f06e62acd837551822d7a6e86d2f78e939e3","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-01-11T16:33:45Z","title_canon_sha256":"ea733571314c925002715a064591de9deef34988999abbda522449026682644f"},"schema_version":"1.0","source":{"id":"2401.06025","kind":"arxiv","version":3}},"canonical_sha256":"d519116e2d822ff66a8e543a3e1739d416ad267f8f74b5991871f4e82911c06b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d519116e2d822ff66a8e543a3e1739d416ad267f8f74b5991871f4e82911c06b","first_computed_at":"2026-07-05T10:19:21.830550Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:19:21.830550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lh0l+ChowxvxZk1rXe2Z5fqtgQ+onHe8gTqNmBhoB+oE8sUxn6Zd4rKgfaf9Q+yD5vTbPpa+yfYWD1WrmEaTCA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:19:21.831058Z","signed_message":"canonical_sha256_bytes"},"source_id":"2401.06025","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0050b5344f0055cc23ec2f111cf55c046650b3fdfdcbad1b010ad39c6e1fb557","sha256:adce617576b21bdc782d47774c11d434d8895b8165661792736aa7a9d68a046f"],"state_sha256":"e74d0985c5619f0a065614e127f577c73c5cf4511bf8bbf1d06dd0924d935e01"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VhOvkMFS6AvQejCnwLMm3E7qfBTV+J8tUeGPGO7RAZGQS4rS/3UQXJz0mzOh7LUTYR150VYKiOLarX68V/pICQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-22T18:03:26.157285Z","bundle_sha256":"35bc3ed3a985a024dc9f6d43495c37d570e83884ff797d7a354fb19ad3af6e9c"}}