{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NO3K3IAYTNCTZ2TJCJJCC6H7G2","short_pith_number":"pith:NO3K3IAY","canonical_record":{"source":{"id":"2606.29882","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-06-29T07:20:24Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"3b493ce8f54dcf464be08c544f1aa93bf7bdd70e7cc4962eda99aa3b49daf0c9","abstract_canon_sha256":"5fcd830d73587c277652ceba97cfb75537c962ef49391241207f5502339fef11"},"schema_version":"1.0"},"canonical_sha256":"6bb6ada0189b453cea6912522178ff36bf550fc556318d695c3d9729d9c1acb5","source":{"kind":"arxiv","id":"2606.29882","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.29882","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"arxiv_version","alias_value":"2606.29882v1","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29882","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"pith_short_12","alias_value":"NO3K3IAYTNCT","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"pith_short_16","alias_value":"NO3K3IAYTNCTZ2TJ","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"pith_short_8","alias_value":"NO3K3IAY","created_at":"2026-06-30T02:17:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NO3K3IAYTNCTZ2TJCJJCC6H7G2","target":"record","payload":{"canonical_record":{"source":{"id":"2606.29882","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-06-29T07:20:24Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"3b493ce8f54dcf464be08c544f1aa93bf7bdd70e7cc4962eda99aa3b49daf0c9","abstract_canon_sha256":"5fcd830d73587c277652ceba97cfb75537c962ef49391241207f5502339fef11"},"schema_version":"1.0"},"canonical_sha256":"6bb6ada0189b453cea6912522178ff36bf550fc556318d695c3d9729d9c1acb5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T02:17:39.456160Z","signature_b64":"n/o2w5fL36mdTHE3MBmC+M6DRmHGjhZc5YaDfqsg6n3EvqsMBoHV4dyYL8Nf3ZD+xFbRbrmaC5Xh2DFMwcnfBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6bb6ada0189b453cea6912522178ff36bf550fc556318d695c3d9729d9c1acb5","last_reissued_at":"2026-06-30T02:17:39.455440Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T02:17:39.455440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.29882","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":"2026-06-30T02:17:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A47G4IdG/6B6ckj+XNqYKpG7fA8q1CADWBXdZuoQLfhuD5k1PJPh5Eqrl7DMnFP9ZYjHVZ+KMBSLKEbyGD97Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:53:10.423114Z"},"content_sha256":"b9c1dc55cb79381ea4106799acc2a595190e4262166191ba646a591fa1db180a","schema_version":"1.0","event_id":"sha256:b9c1dc55cb79381ea4106799acc2a595190e4262166191ba646a591fa1db180a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NO3K3IAYTNCTZ2TJCJJCC6H7G2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Monopole triangle over integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Fan Ye, Haochen Qiu","submitted_at":"2026-06-29T07:20:24Z","abstract_excerpt":"We prove the surgery exact triangle for monopole (Seiberg--Witten) Floer homology over integer coefficients, extending the work of Kronheimer--Mrowka--Ozsv\\'{a}th--Szab\\'{o} over $\\mathbb{Z}/2$, Lin--Ruberman--Saveliev over $\\mathbb{Q}$, and Freeman over $\\mathbb{Z}[\\sqrt{-1}]$. Our proof is based on a modification of Kronheimer--Mrowka's local system on monopole Floer homology and an adaptation of Freeman's computation. As a standard application, following Bloom and Scaduto, we obtain a spectral sequence $\\widetilde{Kh}_{\\mathrm{odd}}(L)\\Rightarrow \\widetilde{HM}_\\bullet(-\\Sigma_2(L))$ over i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29882","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/2606.29882/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":"2026-06-30T02:17:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9l5cFf/VmE5jYuLiXkG1POt4xVtme4uU3pKXp08hUp+LWPpOYP1Wj5NmMDBZLGfEmau2RogN0s2xWu77FdCIAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:53:10.423621Z"},"content_sha256":"4d52f33f3901abc372cdf7f694228954e063bb6c4c73e1cdce2b9dd8cf9b216b","schema_version":"1.0","event_id":"sha256:4d52f33f3901abc372cdf7f694228954e063bb6c4c73e1cdce2b9dd8cf9b216b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NO3K3IAYTNCTZ2TJCJJCC6H7G2/bundle.json","state_url":"https://pith.science/pith/NO3K3IAYTNCTZ2TJCJJCC6H7G2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NO3K3IAYTNCTZ2TJCJJCC6H7G2/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":"2026-08-09T16:53:10Z","links":{"resolver":"https://pith.science/pith/NO3K3IAYTNCTZ2TJCJJCC6H7G2","bundle":"https://pith.science/pith/NO3K3IAYTNCTZ2TJCJJCC6H7G2/bundle.json","state":"https://pith.science/pith/NO3K3IAYTNCTZ2TJCJJCC6H7G2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NO3K3IAYTNCTZ2TJCJJCC6H7G2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NO3K3IAYTNCTZ2TJCJJCC6H7G2","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":"5fcd830d73587c277652ceba97cfb75537c962ef49391241207f5502339fef11","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-06-29T07:20:24Z","title_canon_sha256":"3b493ce8f54dcf464be08c544f1aa93bf7bdd70e7cc4962eda99aa3b49daf0c9"},"schema_version":"1.0","source":{"id":"2606.29882","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.29882","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"arxiv_version","alias_value":"2606.29882v1","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29882","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"pith_short_12","alias_value":"NO3K3IAYTNCT","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"pith_short_16","alias_value":"NO3K3IAYTNCTZ2TJ","created_at":"2026-06-30T02:17:39Z"},{"alias_kind":"pith_short_8","alias_value":"NO3K3IAY","created_at":"2026-06-30T02:17:39Z"}],"graph_snapshots":[{"event_id":"sha256:4d52f33f3901abc372cdf7f694228954e063bb6c4c73e1cdce2b9dd8cf9b216b","target":"graph","created_at":"2026-06-30T02:17:39Z","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/2606.29882/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove the surgery exact triangle for monopole (Seiberg--Witten) Floer homology over integer coefficients, extending the work of Kronheimer--Mrowka--Ozsv\\'{a}th--Szab\\'{o} over $\\mathbb{Z}/2$, Lin--Ruberman--Saveliev over $\\mathbb{Q}$, and Freeman over $\\mathbb{Z}[\\sqrt{-1}]$. Our proof is based on a modification of Kronheimer--Mrowka's local system on monopole Floer homology and an adaptation of Freeman's computation. As a standard application, following Bloom and Scaduto, we obtain a spectral sequence $\\widetilde{Kh}_{\\mathrm{odd}}(L)\\Rightarrow \\widetilde{HM}_\\bullet(-\\Sigma_2(L))$ over i","authors_text":"Fan Ye, Haochen Qiu","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-06-29T07:20:24Z","title":"Monopole triangle over integers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29882","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:b9c1dc55cb79381ea4106799acc2a595190e4262166191ba646a591fa1db180a","target":"record","created_at":"2026-06-30T02:17:39Z","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":"5fcd830d73587c277652ceba97cfb75537c962ef49391241207f5502339fef11","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2026-06-29T07:20:24Z","title_canon_sha256":"3b493ce8f54dcf464be08c544f1aa93bf7bdd70e7cc4962eda99aa3b49daf0c9"},"schema_version":"1.0","source":{"id":"2606.29882","kind":"arxiv","version":1}},"canonical_sha256":"6bb6ada0189b453cea6912522178ff36bf550fc556318d695c3d9729d9c1acb5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6bb6ada0189b453cea6912522178ff36bf550fc556318d695c3d9729d9c1acb5","first_computed_at":"2026-06-30T02:17:39.455440Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-30T02:17:39.455440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n/o2w5fL36mdTHE3MBmC+M6DRmHGjhZc5YaDfqsg6n3EvqsMBoHV4dyYL8Nf3ZD+xFbRbrmaC5Xh2DFMwcnfBg==","signature_status":"signed_v1","signed_at":"2026-06-30T02:17:39.456160Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.29882","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9c1dc55cb79381ea4106799acc2a595190e4262166191ba646a591fa1db180a","sha256:4d52f33f3901abc372cdf7f694228954e063bb6c4c73e1cdce2b9dd8cf9b216b"],"state_sha256":"855abddc9fb6f559e648fdc82a066b57e7d7409057bd12d2fa085d2712b17f8d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LDRm2Ous/CUJBnqfDz3ZFkxJ1rPIC5O5XK3Duw58mwhaTCfrpek99rxalEoyyu5l+d09wdXRk+meK8TJl9anCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T16:53:10.429080Z","bundle_sha256":"2befdefd6982ef69b2977085b23bc9676b64aaf2035ed6cf85b56b34aef9f9de"}}