{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DGC75JGEGFXSCRSPT3VSNBSRYD","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":"c9c1f9bad8225df440a9b23bbd318c14a0bbcd540977d18b9b659c887f266348","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-06-19T04:38:29Z","title_canon_sha256":"13df3e07388c42c4e8b6ed27d3ced6e87b0f1a341407d95bba7313fa624a764b"},"schema_version":"1.0","source":{"id":"2606.21091","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.21091","created_at":"2026-06-23T01:12:29Z"},{"alias_kind":"arxiv_version","alias_value":"2606.21091v1","created_at":"2026-06-23T01:12:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.21091","created_at":"2026-06-23T01:12:29Z"},{"alias_kind":"pith_short_12","alias_value":"DGC75JGEGFXS","created_at":"2026-06-23T01:12:29Z"},{"alias_kind":"pith_short_16","alias_value":"DGC75JGEGFXSCRSP","created_at":"2026-06-23T01:12:29Z"},{"alias_kind":"pith_short_8","alias_value":"DGC75JGE","created_at":"2026-06-23T01:12:29Z"}],"graph_snapshots":[{"event_id":"sha256:966bcf2a1af84fe66f21c592ef6566be395345b3d5e5423342ca2e96d97163b4","target":"graph","created_at":"2026-06-23T01:12:29Z","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/2606.21091/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"It is well known that there is a bijective correspondence between the set of positive integer solutions to the Markov equation and the set of Fano triangles mutation equivalent to the Fano triangle of $\\mathbb{P}^2$. In this paper, we establish a higher dimensional generalization of this correspondence for arbitrary Fano simplices of any dimension. On the polyhedral side, we introduce a distinguished class of facets, called admissible facets, and show that their number is preserved under facet mutation. As a consequence, facet mutation classes of Fano simplices carry natural exchange graph str","authors_text":"Yunhyung Cho","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-06-19T04:38:29Z","title":"Mutation of Fano Simplices and Markov type equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21091","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:f13092dbbcaf5be035eb3486de02e297f64a0eb5d64436c7114095797feeff00","target":"record","created_at":"2026-06-23T01:12:29Z","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":"c9c1f9bad8225df440a9b23bbd318c14a0bbcd540977d18b9b659c887f266348","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-06-19T04:38:29Z","title_canon_sha256":"13df3e07388c42c4e8b6ed27d3ced6e87b0f1a341407d95bba7313fa624a764b"},"schema_version":"1.0","source":{"id":"2606.21091","kind":"arxiv","version":1}},"canonical_sha256":"1985fea4c4316f21464f9eeb268651c0e0493519fdf4659ed60f28119ec33f4d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1985fea4c4316f21464f9eeb268651c0e0493519fdf4659ed60f28119ec33f4d","first_computed_at":"2026-06-23T01:12:29.563609Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T01:12:29.563609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T+kKd/0Zqwdha8E7GXYaGvZHrX2O+jD66Ypj1shf3whyScP3FGxxyTEZCHZpLqLKD1sRiE/BlaPXRRd8pc5MAg==","signature_status":"signed_v1","signed_at":"2026-06-23T01:12:29.564063Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.21091","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f13092dbbcaf5be035eb3486de02e297f64a0eb5d64436c7114095797feeff00","sha256:966bcf2a1af84fe66f21c592ef6566be395345b3d5e5423342ca2e96d97163b4"],"state_sha256":"514e8faff375e97ab2c04222831e23893548cd9a198d9fa84c973cf4f9adcc1d"}