{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ZG7UQSO7C224HWZ2ZVDLLYPS6H","short_pith_number":"pith:ZG7UQSO7","canonical_record":{"source":{"id":"1307.6491","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-24T17:01:19Z","cross_cats_sorted":[],"title_canon_sha256":"555bab69e2a20d8e9a08c6a60ac422a5611ec098e7f4c494303245045cb0decd","abstract_canon_sha256":"940b3922ceafd32de2232c386f73d21ca8aa86c67605cac913b700e8dea5311b"},"schema_version":"1.0"},"canonical_sha256":"c9bf4849df16b5c3db3acd46b5e1f2f1f896ded8eb7c0d63cd12774efb1076c4","source":{"kind":"arxiv","id":"1307.6491","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.6491","created_at":"2026-05-18T01:18:18Z"},{"alias_kind":"arxiv_version","alias_value":"1307.6491v2","created_at":"2026-05-18T01:18:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.6491","created_at":"2026-05-18T01:18:18Z"},{"alias_kind":"pith_short_12","alias_value":"ZG7UQSO7C224","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZG7UQSO7C224HWZ2","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZG7UQSO7","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ZG7UQSO7C224HWZ2ZVDLLYPS6H","target":"record","payload":{"canonical_record":{"source":{"id":"1307.6491","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-24T17:01:19Z","cross_cats_sorted":[],"title_canon_sha256":"555bab69e2a20d8e9a08c6a60ac422a5611ec098e7f4c494303245045cb0decd","abstract_canon_sha256":"940b3922ceafd32de2232c386f73d21ca8aa86c67605cac913b700e8dea5311b"},"schema_version":"1.0"},"canonical_sha256":"c9bf4849df16b5c3db3acd46b5e1f2f1f896ded8eb7c0d63cd12774efb1076c4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:18.625285Z","signature_b64":"9earu3sfcmi0m+t7PZ/MmVv0j2P2XYzwUiw96oq5Qm2epmtvenL7a6ilt8RHqiUt92oQSfwKzN7m/1Nv2y/2CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9bf4849df16b5c3db3acd46b5e1f2f1f896ded8eb7c0d63cd12774efb1076c4","last_reissued_at":"2026-05-18T01:18:18.624536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:18.624536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.6491","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:18:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vQqY+QysHi947o9h/vwelaeCCz9LHlFILcV7C8Y0v3XWu1D25MVQawfKE31hibYWFYeyui1HiDMMwUs3fGt1Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T20:27:41.851353Z"},"content_sha256":"30e28d009b3a17e049aaa00512e17c163adc80709fd6851a058b4a6bd410f30c","schema_version":"1.0","event_id":"sha256:30e28d009b3a17e049aaa00512e17c163adc80709fd6851a058b4a6bd410f30c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ZG7UQSO7C224HWZ2ZVDLLYPS6H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Milnor and Tjurina numbers for smoothings of surface singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jonathan Wahl","submitted_at":"2013-07-24T17:01:19Z","abstract_excerpt":"For an isolated hypersurface singularity $f=0$, the Milnor number $\\mu$ is greater than or equal to the Tjurina number $\\tau$ (the dimension of the base of the semi-universal deformation), with equality if $f$ is quasi-homogeneous. K. Saito proved the converse. The same result is true for complete intersections, but is much harder. For a Gorenstein surface singularity $(V,0)$, the difference $\\mu - \\tau$ can be defined whether or not $V$ is smoothable; it was proved in [23] that it is non-negative, and equal to 0 iff $(V,0)$ is quasi-homogeneous. We conjecture a similar result for non-Gorenste"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6491","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T01:18:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AGGDQGZz7K2bOObNqKuFkYX01WtYP7u1iZPP/X/XUMviOhnNmdwP7ikbRG6zbM6rgq8WzcauIqL8GiaPEFQjCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T20:27:41.852058Z"},"content_sha256":"bacecf38cc6a1c39f95f3145051e6793db6b2efd663d748b8f27b5669d7786f7","schema_version":"1.0","event_id":"sha256:bacecf38cc6a1c39f95f3145051e6793db6b2efd663d748b8f27b5669d7786f7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZG7UQSO7C224HWZ2ZVDLLYPS6H/bundle.json","state_url":"https://pith.science/pith/ZG7UQSO7C224HWZ2ZVDLLYPS6H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZG7UQSO7C224HWZ2ZVDLLYPS6H/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":"2026-06-08T20:27:41Z","links":{"resolver":"https://pith.science/pith/ZG7UQSO7C224HWZ2ZVDLLYPS6H","bundle":"https://pith.science/pith/ZG7UQSO7C224HWZ2ZVDLLYPS6H/bundle.json","state":"https://pith.science/pith/ZG7UQSO7C224HWZ2ZVDLLYPS6H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZG7UQSO7C224HWZ2ZVDLLYPS6H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ZG7UQSO7C224HWZ2ZVDLLYPS6H","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":"940b3922ceafd32de2232c386f73d21ca8aa86c67605cac913b700e8dea5311b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-24T17:01:19Z","title_canon_sha256":"555bab69e2a20d8e9a08c6a60ac422a5611ec098e7f4c494303245045cb0decd"},"schema_version":"1.0","source":{"id":"1307.6491","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.6491","created_at":"2026-05-18T01:18:18Z"},{"alias_kind":"arxiv_version","alias_value":"1307.6491v2","created_at":"2026-05-18T01:18:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.6491","created_at":"2026-05-18T01:18:18Z"},{"alias_kind":"pith_short_12","alias_value":"ZG7UQSO7C224","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZG7UQSO7C224HWZ2","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZG7UQSO7","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:bacecf38cc6a1c39f95f3145051e6793db6b2efd663d748b8f27b5669d7786f7","target":"graph","created_at":"2026-05-18T01:18:18Z","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"},"paper":{"abstract_excerpt":"For an isolated hypersurface singularity $f=0$, the Milnor number $\\mu$ is greater than or equal to the Tjurina number $\\tau$ (the dimension of the base of the semi-universal deformation), with equality if $f$ is quasi-homogeneous. K. Saito proved the converse. The same result is true for complete intersections, but is much harder. For a Gorenstein surface singularity $(V,0)$, the difference $\\mu - \\tau$ can be defined whether or not $V$ is smoothable; it was proved in [23] that it is non-negative, and equal to 0 iff $(V,0)$ is quasi-homogeneous. We conjecture a similar result for non-Gorenste","authors_text":"Jonathan Wahl","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-24T17:01:19Z","title":"Milnor and Tjurina numbers for smoothings of surface singularities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6491","kind":"arxiv","version":2},"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":"sha256:30e28d009b3a17e049aaa00512e17c163adc80709fd6851a058b4a6bd410f30c","target":"record","created_at":"2026-05-18T01:18:18Z","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":"940b3922ceafd32de2232c386f73d21ca8aa86c67605cac913b700e8dea5311b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-07-24T17:01:19Z","title_canon_sha256":"555bab69e2a20d8e9a08c6a60ac422a5611ec098e7f4c494303245045cb0decd"},"schema_version":"1.0","source":{"id":"1307.6491","kind":"arxiv","version":2}},"canonical_sha256":"c9bf4849df16b5c3db3acd46b5e1f2f1f896ded8eb7c0d63cd12774efb1076c4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9bf4849df16b5c3db3acd46b5e1f2f1f896ded8eb7c0d63cd12774efb1076c4","first_computed_at":"2026-05-18T01:18:18.624536Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:18.624536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9earu3sfcmi0m+t7PZ/MmVv0j2P2XYzwUiw96oq5Qm2epmtvenL7a6ilt8RHqiUt92oQSfwKzN7m/1Nv2y/2CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:18.625285Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.6491","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:30e28d009b3a17e049aaa00512e17c163adc80709fd6851a058b4a6bd410f30c","sha256:bacecf38cc6a1c39f95f3145051e6793db6b2efd663d748b8f27b5669d7786f7"],"state_sha256":"e682884fab162f027d37340ebda4e08484d0f28eac3f045af253a38a5d084c14"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kTINtXAmLyAWNPIEYhPruCxFh3MuTo/lVpcRQB+BJXQkb51KeSZPpUsNO/m3zweQJJ2d19GExSTo27blBe1kCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T20:27:41.855916Z","bundle_sha256":"879739651adf3554df5e0717908d5fdab696be950d50c3ef2e8473a10f01079c"}}