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The motivic vanishing cycle $MF_{U,f}^\\phi$ is an element of the $\\hat\\mu$-equivariant motivic Grothendieck ring ${\\mathcal M}^{\\hat\\mu}_X$ defined by Denef and Loeser math.AG/0006050 and Looijenga math.AG/0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435.\n  We prove three main results:\n  (a) $MF_{U,f}^\\phi$ depends only on the third-order"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.6428","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-05-28T09:37:50Z","cross_cats_sorted":[],"title_canon_sha256":"24c560a2835e193df13dda2e34da5b59a5b70066f8def96a0cc7909668ebc0d3","abstract_canon_sha256":"529c56274a4a04c8cdecddb48ac051aa79ef08425a27b6431c5f3fa4d598f828"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:02.878455Z","signature_b64":"p/C4GlyboI9wtRaJGZimVn76q65s69gQ+Hq38ltrzW9yrmkAdpHlu6PwQjr2wDwUwzIm22iGFyXx+CVRt+PjBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0089ce0e7f877ae84f402115a814bdeeea6c0c2e297e5950cc729296612612ee","last_reissued_at":"2026-05-18T00:00:02.877946Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:02.877946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On motivic vanishing cycles of critical loci","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Dominic Joyce, Sven Meinhardt, Vittoria Bussi","submitted_at":"2013-05-28T09:37:50Z","abstract_excerpt":"Let $U$ be a smooth scheme over an algebraically closed field $\\mathbb K$ of characteristic zero and $f:U\\to{\\mathbb A}^1$ a regular function, and write $X=$Crit$(f)$, as a closed subscheme of $U$. The motivic vanishing cycle $MF_{U,f}^\\phi$ is an element of the $\\hat\\mu$-equivariant motivic Grothendieck ring ${\\mathcal M}^{\\hat\\mu}_X$ defined by Denef and Loeser math.AG/0006050 and Looijenga math.AG/0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435.\n  We prove three main results:\n  (a) $MF_{U,f}^\\phi$ depends only on the third-order"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6428","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1305.6428","created_at":"2026-05-18T00:00:02.878031+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.6428v3","created_at":"2026-05-18T00:00:02.878031+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6428","created_at":"2026-05-18T00:00:02.878031+00:00"},{"alias_kind":"pith_short_12","alias_value":"ACE44DT7Q55O","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"ACE44DT7Q55OQT2A","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"ACE44DT7","created_at":"2026-05-18T12:27:38.830355+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553","json":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553.json","graph_json":"https://pith.science/api/pith-number/ACE44DT7Q55OQT2AEEK2QFF553/graph.json","events_json":"https://pith.science/api/pith-number/ACE44DT7Q55OQT2AEEK2QFF553/events.json","paper":"https://pith.science/paper/ACE44DT7"},"agent_actions":{"view_html":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553","download_json":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553.json","view_paper":"https://pith.science/paper/ACE44DT7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.6428&json=true","fetch_graph":"https://pith.science/api/pith-number/ACE44DT7Q55OQT2AEEK2QFF553/graph.json","fetch_events":"https://pith.science/api/pith-number/ACE44DT7Q55OQT2AEEK2QFF553/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553/action/storage_attestation","attest_author":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553/action/author_attestation","sign_citation":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553/action/citation_signature","submit_replication":"https://pith.science/pith/ACE44DT7Q55OQT2AEEK2QFF553/action/replication_record"}},"created_at":"2026-05-18T00:00:02.878031+00:00","updated_at":"2026-05-18T00:00:02.878031+00:00"}