{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:VRGOQ645XHSL25VB2AIOAMY5AF","short_pith_number":"pith:VRGOQ645","canonical_record":{"source":{"id":"2509.08144","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-09-09T20:53:28Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"52ee3bd0a3944d3f274bb8f4dc2555ea19dabe378c97ab93fe0ae05eab72d189","abstract_canon_sha256":"9d78a1ed9cc468167dbb387a0e4096a91b567e0fd62677f542fb1b7a204b894f"},"schema_version":"1.0"},"canonical_sha256":"ac4ce87b9db9e4bd76a1d010e0331d015de651d65093d0ef0898fddd00337c0d","source":{"kind":"arxiv","id":"2509.08144","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08144","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08144v1","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08144","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"pith_short_12","alias_value":"VRGOQ645XHSL","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"pith_short_16","alias_value":"VRGOQ645XHSL25VB","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"pith_short_8","alias_value":"VRGOQ645","created_at":"2026-07-05T12:08:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:VRGOQ645XHSL25VB2AIOAMY5AF","target":"record","payload":{"canonical_record":{"source":{"id":"2509.08144","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-09-09T20:53:28Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"52ee3bd0a3944d3f274bb8f4dc2555ea19dabe378c97ab93fe0ae05eab72d189","abstract_canon_sha256":"9d78a1ed9cc468167dbb387a0e4096a91b567e0fd62677f542fb1b7a204b894f"},"schema_version":"1.0"},"canonical_sha256":"ac4ce87b9db9e4bd76a1d010e0331d015de651d65093d0ef0898fddd00337c0d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:08:23.740171Z","signature_b64":"2iLICrGqTeU461yQsvPNMT9QpIZL75fahycPb3xEgZKP+FqBOqxfl35X8LYATsfg/sZSpvYwpZEMAU7HZ5zBDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac4ce87b9db9e4bd76a1d010e0331d015de651d65093d0ef0898fddd00337c0d","last_reissued_at":"2026-07-05T12:08:23.739621Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:08:23.739621Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.08144","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":"2026-07-05T12:08:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O24doleMRJ2ORqfYdKOB+p9bpJPgpvbxL43kscaG2G4aiuH2PZwi49REh3vGiuHr4w27vj5CoBgZaI7phP0VBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T19:51:18.644677Z"},"content_sha256":"7dd83cffceed4ff2150202b2ff693d14b63edcf27add0164b41c16d44ce772f8","schema_version":"1.0","event_id":"sha256:7dd83cffceed4ff2150202b2ff693d14b63edcf27add0164b41c16d44ce772f8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:VRGOQ645XHSL25VB2AIOAMY5AF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO"],"primary_cat":"math.CT","authors_text":"Alex Sistko, Cameron Wright, Jaiung Jun","submitted_at":"2025-09-09T20:53:28Z","abstract_excerpt":"We study the category $F$-$\\textbf{Mat}_\\bullet$ of matroids over an idyll $F$. We show that $F$-$\\textbf{Mat}_\\bullet$ is a proto-exact category, a non-additive generalization of an exact category by Dyckerhoff and Kapranov. We further show that $F$-$\\textbf{Mat}_\\bullet$ is proto-abelian in the sense of Andr\\'e. As an application, we establish that the category $\\textbf{TRS}_\\bullet^\\Sigma$ of tropical toric reflexive sheaves associated to a fan $\\Sigma$, introduced by Khan and Maclagan, is also proto-exact and proto-abelian. We then investigate the stability of modular tropical toric reflex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08144","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/2509.08144/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":"2026-07-05T12:08:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/G35WFP4jJ/0RuSSP1KciIHdEXmi1YBiDYgIzEr/HSjjfHjrk/l0UWwsJNYpYyGO9YZYnhcBMmpm2exwn3DGBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T19:51:18.645405Z"},"content_sha256":"250a258b4aa27ecb0bbf3a5b0a7687c99ffbbbfb8b5301ecc7d8727d9eef7f2d","schema_version":"1.0","event_id":"sha256:250a258b4aa27ecb0bbf3a5b0a7687c99ffbbbfb8b5301ecc7d8727d9eef7f2d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VRGOQ645XHSL25VB2AIOAMY5AF/bundle.json","state_url":"https://pith.science/pith/VRGOQ645XHSL25VB2AIOAMY5AF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VRGOQ645XHSL25VB2AIOAMY5AF/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":"2026-08-10T19:51:18Z","links":{"resolver":"https://pith.science/pith/VRGOQ645XHSL25VB2AIOAMY5AF","bundle":"https://pith.science/pith/VRGOQ645XHSL25VB2AIOAMY5AF/bundle.json","state":"https://pith.science/pith/VRGOQ645XHSL25VB2AIOAMY5AF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VRGOQ645XHSL25VB2AIOAMY5AF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:VRGOQ645XHSL25VB2AIOAMY5AF","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":"9d78a1ed9cc468167dbb387a0e4096a91b567e0fd62677f542fb1b7a204b894f","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-09-09T20:53:28Z","title_canon_sha256":"52ee3bd0a3944d3f274bb8f4dc2555ea19dabe378c97ab93fe0ae05eab72d189"},"schema_version":"1.0","source":{"id":"2509.08144","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08144","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08144v1","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08144","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"pith_short_12","alias_value":"VRGOQ645XHSL","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"pith_short_16","alias_value":"VRGOQ645XHSL25VB","created_at":"2026-07-05T12:08:23Z"},{"alias_kind":"pith_short_8","alias_value":"VRGOQ645","created_at":"2026-07-05T12:08:23Z"}],"graph_snapshots":[{"event_id":"sha256:250a258b4aa27ecb0bbf3a5b0a7687c99ffbbbfb8b5301ecc7d8727d9eef7f2d","target":"graph","created_at":"2026-07-05T12:08:23Z","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/2509.08144/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the category $F$-$\\textbf{Mat}_\\bullet$ of matroids over an idyll $F$. We show that $F$-$\\textbf{Mat}_\\bullet$ is a proto-exact category, a non-additive generalization of an exact category by Dyckerhoff and Kapranov. We further show that $F$-$\\textbf{Mat}_\\bullet$ is proto-abelian in the sense of Andr\\'e. As an application, we establish that the category $\\textbf{TRS}_\\bullet^\\Sigma$ of tropical toric reflexive sheaves associated to a fan $\\Sigma$, introduced by Khan and Maclagan, is also proto-exact and proto-abelian. We then investigate the stability of modular tropical toric reflex","authors_text":"Alex Sistko, Cameron Wright, Jaiung Jun","cross_cats":["math.AG","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-09-09T20:53:28Z","title":"Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08144","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":"sha256:7dd83cffceed4ff2150202b2ff693d14b63edcf27add0164b41c16d44ce772f8","target":"record","created_at":"2026-07-05T12:08:23Z","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":"9d78a1ed9cc468167dbb387a0e4096a91b567e0fd62677f542fb1b7a204b894f","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2025-09-09T20:53:28Z","title_canon_sha256":"52ee3bd0a3944d3f274bb8f4dc2555ea19dabe378c97ab93fe0ae05eab72d189"},"schema_version":"1.0","source":{"id":"2509.08144","kind":"arxiv","version":1}},"canonical_sha256":"ac4ce87b9db9e4bd76a1d010e0331d015de651d65093d0ef0898fddd00337c0d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac4ce87b9db9e4bd76a1d010e0331d015de651d65093d0ef0898fddd00337c0d","first_computed_at":"2026-07-05T12:08:23.739621Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:08:23.739621Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2iLICrGqTeU461yQsvPNMT9QpIZL75fahycPb3xEgZKP+FqBOqxfl35X8LYATsfg/sZSpvYwpZEMAU7HZ5zBDg==","signature_status":"signed_v1","signed_at":"2026-07-05T12:08:23.740171Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.08144","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7dd83cffceed4ff2150202b2ff693d14b63edcf27add0164b41c16d44ce772f8","sha256:250a258b4aa27ecb0bbf3a5b0a7687c99ffbbbfb8b5301ecc7d8727d9eef7f2d"],"state_sha256":"473a9d98ea5dae1344207233dc3f2573480175ad970af2d7850df93d73114dfa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EimUHFpxxO1zLtdAUXBpiFTdvYWqD5uKAIPKsP6CniGCg25G9WWwKssK0gV8zjImOQukRwEZG0BR88gFItvFCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T19:51:18.651790Z","bundle_sha256":"2f96635b0ba33cc17a9e71435927fd8b27c8ace89f30c1f9a7f23cfd89e330d9"}}