{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:QVA7KKJVCXS2KZB5J7B5XHHOAO","short_pith_number":"pith:QVA7KKJV","schema_version":"1.0","canonical_sha256":"8541f5293515e5a5643d4fc3db9cee03b69e5ece2472d07068f2c2bacdc350f4","source":{"kind":"arxiv","id":"1602.04726","version":1},"attestation_state":"computed","paper":{"title":"The Rank Theorem and $L^2$-invariants in Free Entropy: Global Upper Bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Kenley Jung","submitted_at":"2016-02-15T16:24:33Z","abstract_excerpt":"Using an analogy with the rank theorem in differential geometry, it is shown that for a finite $n$-tuple $X$ in a tracial von Neumann algebra and any finite $m$-tuple $F$ of $*$-polynomials in $n$ noncommuting indeterminates, \\begin{eqnarray*} \\delta_0(X) & \\leq & \\text{Nullity}(D^sF(X)) + \\delta_0(F(X):X) \\end{eqnarray*} where $\\delta_0$ is the (modified) microstates free entropy dimension and $D^sF(X)$ is a kind of derivative of $F$ evaluated at $X$. When $F(X) =0$ and $|D^sF(X)|$ has nonzero Fuglede-Kadison-L\\\"uck determinant, then $X$ is $\\alpha$-bounded in the sense of \\cite{j3} where $\\a"},"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":"1602.04726","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-02-15T16:24:33Z","cross_cats_sorted":[],"title_canon_sha256":"c988ce26cd6b39fefdb031564763e1788750d517872cc5d0294c3093f7808100","abstract_canon_sha256":"f2bb052dc7edf9b743e144fd5b2ec6057f5d7fbdadcecfe97bab893307abea2b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:49.322056Z","signature_b64":"A/T6xr0XFkMRNGlPBZ91qwXWsZbI+evZwqGHipaW8CCRkDBMuBzzQ1fPlLpVe1ems57l4sezknQN7urBmQ3dDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8541f5293515e5a5643d4fc3db9cee03b69e5ece2472d07068f2c2bacdc350f4","last_reissued_at":"2026-05-18T01:20:49.321555Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:49.321555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Rank Theorem and $L^2$-invariants in Free Entropy: Global Upper Bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Kenley Jung","submitted_at":"2016-02-15T16:24:33Z","abstract_excerpt":"Using an analogy with the rank theorem in differential geometry, it is shown that for a finite $n$-tuple $X$ in a tracial von Neumann algebra and any finite $m$-tuple $F$ of $*$-polynomials in $n$ noncommuting indeterminates, \\begin{eqnarray*} \\delta_0(X) & \\leq & \\text{Nullity}(D^sF(X)) + \\delta_0(F(X):X) \\end{eqnarray*} where $\\delta_0$ is the (modified) microstates free entropy dimension and $D^sF(X)$ is a kind of derivative of $F$ evaluated at $X$. When $F(X) =0$ and $|D^sF(X)|$ has nonzero Fuglede-Kadison-L\\\"uck determinant, then $X$ is $\\alpha$-bounded in the sense of \\cite{j3} where $\\a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04726","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":""},"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":"1602.04726","created_at":"2026-05-18T01:20:49.321634+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.04726v1","created_at":"2026-05-18T01:20:49.321634+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04726","created_at":"2026-05-18T01:20:49.321634+00:00"},{"alias_kind":"pith_short_12","alias_value":"QVA7KKJVCXS2","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"QVA7KKJVCXS2KZB5","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"QVA7KKJV","created_at":"2026-05-18T12:30:41.710351+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.20397","citing_title":"An $\\ell^2$ Obstruction for Elementary Embeddings of Hyperbolic Groups","ref_index":10,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO","json":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO.json","graph_json":"https://pith.science/api/pith-number/QVA7KKJVCXS2KZB5J7B5XHHOAO/graph.json","events_json":"https://pith.science/api/pith-number/QVA7KKJVCXS2KZB5J7B5XHHOAO/events.json","paper":"https://pith.science/paper/QVA7KKJV"},"agent_actions":{"view_html":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO","download_json":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO.json","view_paper":"https://pith.science/paper/QVA7KKJV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.04726&json=true","fetch_graph":"https://pith.science/api/pith-number/QVA7KKJVCXS2KZB5J7B5XHHOAO/graph.json","fetch_events":"https://pith.science/api/pith-number/QVA7KKJVCXS2KZB5J7B5XHHOAO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO/action/storage_attestation","attest_author":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO/action/author_attestation","sign_citation":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO/action/citation_signature","submit_replication":"https://pith.science/pith/QVA7KKJVCXS2KZB5J7B5XHHOAO/action/replication_record"}},"created_at":"2026-05-18T01:20:49.321634+00:00","updated_at":"2026-05-18T01:20:49.321634+00:00"}