{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:TCAE2NLYC2NYIGR3HTAQ474ZT5","short_pith_number":"pith:TCAE2NLY","canonical_record":{"source":{"id":"2510.21001","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AG","submitted_at":"2025-10-23T21:21:03Z","cross_cats_sorted":[],"title_canon_sha256":"8de8e8dc7b83cd7bf8ad09875069b938929bc2e196dc95c8bc70eb5e28946e1d","abstract_canon_sha256":"66710366c4f96cdb5408fc6076046d25f9be819fad347af0d0402a9417c41337"},"schema_version":"1.0"},"canonical_sha256":"98804d3578169b841a3b3cc10e7f999f6394ff2fec36c5e4f3b14e74a71f866f","source":{"kind":"arxiv","id":"2510.21001","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2510.21001","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"arxiv_version","alias_value":"2510.21001v3","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.21001","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"pith_short_12","alias_value":"TCAE2NLYC2NY","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"pith_short_16","alias_value":"TCAE2NLYC2NYIGR3","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"pith_short_8","alias_value":"TCAE2NLY","created_at":"2026-05-20T00:04:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:TCAE2NLYC2NYIGR3HTAQ474ZT5","target":"record","payload":{"canonical_record":{"source":{"id":"2510.21001","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AG","submitted_at":"2025-10-23T21:21:03Z","cross_cats_sorted":[],"title_canon_sha256":"8de8e8dc7b83cd7bf8ad09875069b938929bc2e196dc95c8bc70eb5e28946e1d","abstract_canon_sha256":"66710366c4f96cdb5408fc6076046d25f9be819fad347af0d0402a9417c41337"},"schema_version":"1.0"},"canonical_sha256":"98804d3578169b841a3b3cc10e7f999f6394ff2fec36c5e4f3b14e74a71f866f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:04:15.590471Z","signature_b64":"k6DwGO5tJzHyXkdtkwajjkNdbCS3adgHqSf8APZNlIAjU3r3gPtsfBvY8XEFVjbc72GU1IQek5rqB7G16ZS9Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98804d3578169b841a3b3cc10e7f999f6394ff2fec36c5e4f3b14e74a71f866f","last_reissued_at":"2026-05-20T00:04:15.589728Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:04:15.589728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2510.21001","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":"2026-05-20T00:04:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ud6HsCqDhTWQSzKERkFUV2tIM4ZAuTm+vh6I06az+XvmfuSUTNuUBCL6/v4k7RtiBJOsCMJmZnsIO+ADnHGdCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T19:41:52.735715Z"},"content_sha256":"b6b8efe0ae34b98437b1294112ffb0352eb1fdb2bdfdb776b5cfb50f484bc062","schema_version":"1.0","event_id":"sha256:b6b8efe0ae34b98437b1294112ffb0352eb1fdb2bdfdb776b5cfb50f484bc062"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:TCAE2NLYC2NYIGR3HTAQ474ZT5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Grauert's Approximation Theorem in any Characteristic and Applications","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Gerhard Pfister, Gert-Martin Greuel","submitted_at":"2025-10-23T21:21:03Z","abstract_excerpt":"In his seminal Inventiones paper from 1972 Grauert proved the existence of a semiuniversal deformation of an arbitrary complex analytic isolated singularity. For the proof he invented an approximation theorem for solving a system of \"nested\" analytic equations, which is now called Grauert's approximation theorem. To prove this, Grauert introduced standard bases for ideals in power series rings and proved a generalized Weiertrass division theorem. All this was done for convergent power series over the complex numbers.\n  The purpose of this article is to extend Grauert's division and approximati"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.21001","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/2510.21001/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":"2026-05-20T00:04:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZZy8+iVvvvdD8yb3oaUdIc1T61nwGmPvsrV2CE2cBbYNL6DK8tGW43mMBFLRuw/4osWeZQ+E/86J8yRi90EVBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T19:41:52.736456Z"},"content_sha256":"cf33870e26f37fd9e7ffe63ba101c6dc0359983713b906f6b110529f893ce473","schema_version":"1.0","event_id":"sha256:cf33870e26f37fd9e7ffe63ba101c6dc0359983713b906f6b110529f893ce473"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TCAE2NLYC2NYIGR3HTAQ474ZT5/bundle.json","state_url":"https://pith.science/pith/TCAE2NLYC2NYIGR3HTAQ474ZT5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TCAE2NLYC2NYIGR3HTAQ474ZT5/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":"2026-06-08T19:41:52Z","links":{"resolver":"https://pith.science/pith/TCAE2NLYC2NYIGR3HTAQ474ZT5","bundle":"https://pith.science/pith/TCAE2NLYC2NYIGR3HTAQ474ZT5/bundle.json","state":"https://pith.science/pith/TCAE2NLYC2NYIGR3HTAQ474ZT5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TCAE2NLYC2NYIGR3HTAQ474ZT5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:TCAE2NLYC2NYIGR3HTAQ474ZT5","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":"66710366c4f96cdb5408fc6076046d25f9be819fad347af0d0402a9417c41337","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AG","submitted_at":"2025-10-23T21:21:03Z","title_canon_sha256":"8de8e8dc7b83cd7bf8ad09875069b938929bc2e196dc95c8bc70eb5e28946e1d"},"schema_version":"1.0","source":{"id":"2510.21001","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2510.21001","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"arxiv_version","alias_value":"2510.21001v3","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.21001","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"pith_short_12","alias_value":"TCAE2NLYC2NY","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"pith_short_16","alias_value":"TCAE2NLYC2NYIGR3","created_at":"2026-05-20T00:04:15Z"},{"alias_kind":"pith_short_8","alias_value":"TCAE2NLY","created_at":"2026-05-20T00:04:15Z"}],"graph_snapshots":[{"event_id":"sha256:cf33870e26f37fd9e7ffe63ba101c6dc0359983713b906f6b110529f893ce473","target":"graph","created_at":"2026-05-20T00:04:15Z","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/2510.21001/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In his seminal Inventiones paper from 1972 Grauert proved the existence of a semiuniversal deformation of an arbitrary complex analytic isolated singularity. For the proof he invented an approximation theorem for solving a system of \"nested\" analytic equations, which is now called Grauert's approximation theorem. To prove this, Grauert introduced standard bases for ideals in power series rings and proved a generalized Weiertrass division theorem. All this was done for convergent power series over the complex numbers.\n  The purpose of this article is to extend Grauert's division and approximati","authors_text":"Gerhard Pfister, Gert-Martin Greuel","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AG","submitted_at":"2025-10-23T21:21:03Z","title":"Grauert's Approximation Theorem in any Characteristic and Applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.21001","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:b6b8efe0ae34b98437b1294112ffb0352eb1fdb2bdfdb776b5cfb50f484bc062","target":"record","created_at":"2026-05-20T00:04:15Z","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":"66710366c4f96cdb5408fc6076046d25f9be819fad347af0d0402a9417c41337","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AG","submitted_at":"2025-10-23T21:21:03Z","title_canon_sha256":"8de8e8dc7b83cd7bf8ad09875069b938929bc2e196dc95c8bc70eb5e28946e1d"},"schema_version":"1.0","source":{"id":"2510.21001","kind":"arxiv","version":3}},"canonical_sha256":"98804d3578169b841a3b3cc10e7f999f6394ff2fec36c5e4f3b14e74a71f866f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98804d3578169b841a3b3cc10e7f999f6394ff2fec36c5e4f3b14e74a71f866f","first_computed_at":"2026-05-20T00:04:15.589728Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:15.589728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k6DwGO5tJzHyXkdtkwajjkNdbCS3adgHqSf8APZNlIAjU3r3gPtsfBvY8XEFVjbc72GU1IQek5rqB7G16ZS9Bw==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:15.590471Z","signed_message":"canonical_sha256_bytes"},"source_id":"2510.21001","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b6b8efe0ae34b98437b1294112ffb0352eb1fdb2bdfdb776b5cfb50f484bc062","sha256:cf33870e26f37fd9e7ffe63ba101c6dc0359983713b906f6b110529f893ce473"],"state_sha256":"5638849040021c5a3de098c192340132d417987968ab8de13063edccf3cc3968"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iRtmv6MEVXPfg2VzLzVX+vkWf9RAMFwoOHeEuuAUHCa0dD2dnf6h96mqU398D2uZCmpBoYi2R/VLLW7t8TfFDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T19:41:52.740592Z","bundle_sha256":"f8a39e46b25652629b004b2067c8c35b8ba47032a43a9d1bb74cf84169c6b4ca"}}