{"as_of":"2026-08-12T05:56:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:86aaed8934757fc9a6194f700ad00031b8a5b0106dfa258e212c369f25919f9a","coverage":[{"denominator":15,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":15,"source":"paper_references, paper_reference_links","source_observed_at":"2026-07-14T17:21:13.397738Z","state":"measured"},{"denominator":15,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":15,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-11T06:34:44.6726+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2606.25715/citation-record","integrity":"/paper/2606.25715/integrity","json":"/paper/2606.25715/citation-record.json","paper":"/paper/2606.25715"},"outbound":[{"citation":{"cited_paper":{"arxiv_id":"2606.22451","last_updated":"2026-07-21T15:33:14Z","snapshot_observed_at":"2026-08-02T10:41:41.270921Z","submitted_at":"2026-06-21T11:51:35Z","title":"Exact Nonnegative Matrix Factorization via Cone-Ray Witnesses: Certificates, a One-Sided Solver, and a Findability Phase Transition","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2606.22451","snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Ramteke,Exact nonnegative matrix factorization via cone-ray witnesses: obtuseness ranking, saturation curves, and an augmented alt-LP breakthrough, Preprint, arXiv:2606.22451, 2026","venue":null,"work_id":null,"year":2026},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"cited_paper":"/paper/2606.22451","citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:a2a83ab76d6933f6e935025613ae04d2bac38273aa2f2e067c60f197a66f7dd7","observation_id":"7d0b1375-6b9b-4c76-93d0-cf08869a52cf","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":null,"venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:a1e27a27de7a161f8726cb46059551811baf637bd3a7e4b2aec1fe8f53ce6409","observation_id":"a490e998-cbdb-4efa-9a2c-52f48a22f3b0","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":null,"venue":null,"work_id":null,"year":1983},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:8871a0f19940b8caad1bcc682c1dceedbe9797656d48bce8885be7e79c69a74f","observation_id":"34a46ea7-f0f0-41ee-8733-55517688bfd0","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Hrubˇ eˇ s,On the nonnegative rank of distance matrices, Inform","venue":null,"work_id":null,"year":2012},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:5ad4743fd757258e8a7e0bdfa029201d625dc6e781d304e876cf468628980206","observation_id":"5caaf87c-d335-411c-9232-041787088185","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":null,"venue":null,"work_id":null,"year":2009},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:4e04b66793af26af9edcfb87226c8211d0716d186ac57d0ff15895ccb01544bd","observation_id":"56b745c0-7802-4fb4-a2c8-1804ab483611","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Yannakakis,Expressing combinatorial optimization problems by linear programs, J","venue":null,"work_id":null,"year":1991},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:e487a07bc65969645dfd067005446bc9e2d67b3ba70fdd4778cccb1ca469e3e4","observation_id":"8ea55144-7b38-4fdc-a53a-f5f96c6612ab","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Fiorini, S","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:73d22f1be1a68b6b1858edef5c061a9854ae801f950ee8f92e767dbcca497509","observation_id":"537fe266-d5eb-46c6-8453-3c7426407625","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Fiorini, T","venue":null,"work_id":null,"year":2012},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:277c5f501864f3c570f27adfa42703a8863ab382cb39732ce3cc525de875e13f","observation_id":"4f981821-6059-4fa7-9d26-c57a068431d4","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":null,"venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:2571eefc859ffde0d166ef4508a72705de7a01da7516fe6d507a9ef40b51ae52","observation_id":"6d8e5b7f-b9ac-4636-bd34-57cae8801075","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Arora, R","venue":null,"work_id":null,"year":2012},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:202cfecc740f5586ee89524a5986dd05a2ee9d535ec67a41264ea1ac549959aa","observation_id":"65200757-253c-4f63-8fba-d495050d089d","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Absil, R","venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:190a530ce14c20a057aa96dd9872d0483dc5fb3a14dbf19ea9ff5ffbd5213d65","observation_id":"29562303-36ba-4d09-9adc-61f439bc4f3c","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Edelman, T","venue":null,"work_id":null,"year":1998},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:11f236498090c8db43f2fddc3ef66ed412bb2a9e5c4f1a2efbc89c94c7d0faae","observation_id":"66ae8abb-95d5-4683-94e8-e0b5172bace7","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Diaconis,Patterns in eigenvalues, Bull","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:f4b0bab37ff84d85b01824b7747557d20b6cbcc64a74507d519533f885f5bb96","observation_id":"0aa40725-b362-4ecb-87f6-056e27226918","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":null,"venue":null,"work_id":null,"year":1953},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:216fd65ab29b8f5c9e60b9be62275ac33839cbfffcec99f40ead9ae2798f829a","observation_id":"a8795989-8260-4125-8b6e-37784d61e423","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-07-14T17:21:13.397738Z","title":"Fukuda,cddlib reference manual, available at https://people.inf.ethz.ch/fukudak/cdd home/","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-07-14T17:21:13.397738Z"},"links":{"citing_paper":"/paper/2606.25715"},"observation_digest":"sha256:ec5b8db732603ca7706f025e308b6d7c44e929414215796eacaaa4581b40023b","observation_id":"9d051bbb-a887-436e-81c8-adee990ef06e","resolution":{"observed_at":"2026-07-14T17:21:13.397738Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2606.25715","last_updated":"2026-07-12T09:10:41Z","latest_version":2,"primary_category":"math.NA","snapshot_observed_at":"2026-07-16T23:18:45.947797Z","submitted_at":"2026-06-24T11:31:05Z","title":"Gap-Aware Exact Nonnegative Matrix Factorization: A Two-Sided SVD Gauge and a Three-Regime W-Rank Taxonomy"},"reference_resolution":{"displayed":15,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":15,"verified_exact":0,"verified_fuzzy":0},"total_outbound_references":15},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-11T06:34:44.6726+00:00","source":"crossref"},{"observed_at":"2026-08-11T06:34:36.301508+00:00","source":"retraction_watch"}],"thesis":"As of 12 August 2026, this Paper Citation Record lists 15 of 15 outbound references and 0 inbound Pith citation observations for arXiv:2606.25715."}