{"as_of":"2026-08-15T20:22:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:5f3c847074a3687a5b7e0bf6ccec5432e170b822ed2bbc2099225dabe675c866","coverage":[{"denominator":21,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":21,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-07T10:20:48.559782Z","state":"measured"},{"denominator":25,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":25,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-15T06:32:42.880941+00:00","state":"measured"},{"denominator":4,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":4,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-15T19:09:25.689077Z","state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"pith","source_observed_at":"2026-08-07T00:51:34.572507Z","state":"measured"}],"external_citation_measurements":[],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.05915","snapshot_observed_at":"2026-08-07T05:40:43.310354Z","title":"Spencer-riemann-roch theory: Mirror symmetry of hodge decompositions and characteristic classes in constrained geometry","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2506.07410","last_updated":"2025-06-11T06:43:26Z","snapshot_observed_at":"2026-08-14T08:08:17.002033Z","submitted_at":"2025-06-09T04:13:57Z","title":"Spencer Differential Degeneration Theory and Its Applications in Algebraic Geometry","version":2},"reference_index":25,"source":"arxiv_source","source_observed_at":"2026-08-07T05:40:43.310354Z"},"links":{"cited_paper":"/paper/2506.05915","citing_paper":"/paper/2506.07410"},"observation_digest":"sha256:bea457c4455254094c5fce0eef25d63a14a90d9033008ce66b0d0ef391c89373","observation_id":"82408e61-bba7-46a1-9650-57e66aafd3bc","resolution":{"observed_at":"2026-08-07T05:40:43.310354Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.05915","snapshot_observed_at":"2026-08-07T04:12:57.886576Z","title":"Spencer-riemann-roch theory: Mirror symmetry of hodge decompositions and characteristic classes in constrained geometry","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2506.11593","last_updated":"2025-06-13T09:01:35Z","snapshot_observed_at":"2026-08-14T02:14:50.675249Z","submitted_at":"2025-06-13T09:01:35Z","title":"Extension Research of Principal Bundle Constraint System Theory on Ricci-flat K\\\"ahler Manifolds","version":1},"reference_index":28,"source":"arxiv_source","source_observed_at":"2026-08-07T04:12:57.886576Z"},"links":{"cited_paper":"/paper/2506.05915","citing_paper":"/paper/2506.11593"},"observation_digest":"sha256:8a5bf7684071772a730ed05ea6399e61a8dc38615b498ef12ac19eb9d7313cc9","observation_id":"85ff7c31-76ad-4e53-b005-14accdec020d","resolution":{"observed_at":"2026-08-07T04:12:57.886576Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"cited_work":{"arxiv_id":"2506.05915","doi":null,"metadata_source":"pith","pith_arxiv_id":"2506.05915","snapshot_observed_at":"2026-08-07T00:51:34.572507Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","venue":"math.GM","work_id":"6d8a6cd7-95a0-40dc-a1d1-3c86c10af080","year":2025},"citing_paper":{"arxiv_id":"2506.12720","last_updated":"2025-06-15T04:55:03Z","snapshot_observed_at":"2026-08-14T21:07:04.445735Z","submitted_at":"2025-06-15T04:55:03Z","title":"The Rigidity of Constraint: A Spencer-Hodge Theoretic Approach to the Hodge Conjecture","version":1},"reference_index":31,"source":"arxiv_source","source_observed_at":"2026-08-07T00:51:34.544181Z"},"links":{"cited_paper":"/paper/2506.05915","citing_paper":"/paper/2506.12720"},"observation_digest":"sha256:bd88332a0185d0728c7ca7d00f205e9a029964bae22b62c57f7424453ca5cb30","observation_id":"cfa724c0-3fc1-4dfa-878f-6a43292b2bf8","resolution":{"observed_at":"2026-08-07T00:51:34.577948Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.05915","snapshot_observed_at":"2026-08-15T19:09:25.689077Z","title":"Spencer-riemann-roch theory: Mirror symmetry of hodge decompositions and characteristic classes in constrained geometry","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2506.17754","last_updated":"2025-07-03T02:07:10Z","snapshot_observed_at":"2026-08-15T18:59:38.063516Z","submitted_at":"2025-06-21T16:18:48Z","title":"Exceptional Symmetry as a Source of Algebraic Cycles: Non-Constructive Methods for the Hodge Conjecture for a Special Class of Calabi-Yau 5-Folds","version":3},"reference_index":35,"source":"arxiv_source","source_observed_at":"2026-08-15T19:09:25.689077Z"},"links":{"cited_paper":"/paper/2506.05915","citing_paper":"/paper/2506.17754"},"observation_digest":"sha256:e9191b13f3c5258b265c64dfc5a33ee41fe504e1e36331592e4be8ec85898043","observation_id":"49a66b68-fd9b-4f12-b9ef-f0c19d06373e","resolution":{"observed_at":"2026-08-15T19:09:25.689077Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"links":{"evidence":"/evidence","html":"/paper/2506.05915/citation-record","integrity":"/paper/2506.05915/integrity","json":"/paper/2506.05915/citation-record.json","paper":"/paper/2506.05915"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:49.109171Z","title":null,"venue":null,"work_id":"a2f52eed-f0ce-4a49-980d-e6307f44b62d","year":1963},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":1,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.420464Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:948d90f590a7cc54a0868888d5cf54c3d0c32a974a30dada185fdba1983d1b8d","observation_id":"61e2231a-0cf4-4da9-a94a-1029104fb755","resolution":{"observed_at":"2026-08-07T10:20:49.115719Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.427971Z","title":"A simple intrinsic proof of the gauss-bonnet formula for closed riemannian manifolds","venue":null,"work_id":null,"year":1944},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":2,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.427971Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:8aa40b2ca7b0610205c4beb946ec11373c45af3d71bc4a6c525b42f700df5ef3","observation_id":"d691c830-4d80-46cf-89b4-be86698476d1","resolution":{"observed_at":"2026-08-07T10:20:48.427971Z","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":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:49.072745Z","title":"Intersection Theory , volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete","venue":null,"work_id":"1bc795c8-cb0f-420e-8db6-a10500b26ecb","year":1998},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":3,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.433435Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:bbaf81f77af93f0cfeb46b67b399072f41a462416dbb13e98356459ac5a7f5f3","observation_id":"78e678bb-c844-4c98-b0f8-fafcd2ee7f64","resolution":{"observed_at":"2026-08-07T10:20:49.079077Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:49.045211Z","title":"Principles of Algebraic Geometry","venue":null,"work_id":"5d562ab7-73f8-46fc-adfc-020cacd80a63","year":1978},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":4,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.439512Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:ba54515b454e22d03411af709789d94c74257e149ee1e2455e054b2106b2c39e","observation_id":"59d9a939-9acd-4fb5-98ec-eb740cbd74de","resolution":{"observed_at":"2026-08-07T10:20:49.053574Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:49.020355Z","title":"Integrability criteria for systems of nonlinear partial differential equations","venue":null,"work_id":"4ff4fefd-9a0e-4a49-a332-49a329e17de4","year":1967},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":5,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.448111Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:a33397fec433b20e79603d6774a7dced8c0ed7f8838b96616669af358f825d2a","observation_id":"237f94cc-b08e-44f1-bb50-d31c5e96d123","resolution":{"observed_at":"2026-08-07T10:20:49.027636Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.997966Z","title":"Sur quelques points d'algèbre homologique","venue":null,"work_id":"e90c9b10-e36e-4e7f-b943-8fae1f34b5b3","year":1957},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":6,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.455065Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:df80a3b6407e2d1387563589a37ba2ffeae4dae763bd58063978ea9318886f53","observation_id":"c2a448ec-e6dd-4522-a9ac-475286332832","resolution":{"observed_at":"2026-08-07T10:20:49.004798Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.976453Z","title":"Topological Methods in Algebraic Geometry , volume 131 of Grundlehren der Mathematischen Wissenschaften","venue":null,"work_id":"dbbad5cf-6947-46f1-8b28-b0a15b56d4f2","year":1966},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":7,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.461996Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:ff7a55a4eb99c2f4f68cdc78cc9dccc14beeacf9b6a50ef93da51018027ccd48","observation_id":"43b57bbe-7c0a-4cc9-8277-1c4f3a45ecc5","resolution":{"observed_at":"2026-08-07T10:20:48.982936Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.952526Z","title":"Mirror Symmetry","venue":null,"work_id":"6af7fb1b-220f-46d1-b53b-4677d082f124","year":2003},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":8,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.468263Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:230493f2ab1eed0cc5737d633df5fda2f364c5a814ea25f8a2aa56f7f571d29e","observation_id":"a58a8b7a-bd8b-4f80-872b-ece246f71129","resolution":{"observed_at":"2026-08-07T10:20:48.958658Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.934048Z","title":"Humphreys","venue":null,"work_id":"5aab88b4-fa65-4cb3-aaf3-39ae6527dfcc","year":1972},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":9,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.474635Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:79387380f89f9c36b717cdf387e3c5d998edb2166cb987a2a30da3773106cf73","observation_id":"f5b50445-0b20-4640-ade6-7f7a04b1be6a","resolution":{"observed_at":"2026-08-07T10:20:48.940997Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.908305Z","title":"Macdonald","venue":null,"work_id":"48dc1365-b812-48d4-89b0-c085b262032a","year":1998},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":10,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.481213Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:9581b0086ba97b3360c54559ca347239d44dc5eba99951f8220044247dd8e16e","observation_id":"8f33f7eb-8c9c-462c-81da-608fe50fb202","resolution":{"observed_at":"2026-08-07T10:20:48.918054Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.882787Z","title":"Milnor and James D","venue":null,"work_id":"0e3c684b-fb99-4d2f-bf2b-fb923af0d8d8","year":1974},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":11,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.488782Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:5895c341dbb1c61077e4abbbd25b0fe0d77c6153917e79fed3030092be0b7f6f","observation_id":"dc5918a7-939d-4dd3-b310-65baa67fefe2","resolution":{"observed_at":"2026-08-07T10:20:48.890802Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.861562Z","title":"Formal properties of over-determined systems of linear partial differential equations","venue":null,"work_id":"7ca1e7b2-5335-4a86-baf0-9ef4b58eb400","year":1964},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":12,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.499820Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:493071b06435f9246b2a350e7eecf122d0c48c5b6c99e016d23492f71cf8a7f9","observation_id":"3ecc97d4-8ad9-40a7-a2e3-c74c5d1de37d","resolution":{"observed_at":"2026-08-07T10:20:48.868079Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.828673Z","title":"G \\'e om \\'e trie alg \\'e brique et g \\'e om \\'e trie analytique","venue":null,"work_id":"81b4d3df-46d5-4583-a486-79630cd37b22","year":1956},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":13,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.506233Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:b2ca7011dd55e6741c1b292382d925177fcf66ff44c01b3bb45381d74e0b18e9","observation_id":"c170fb17-3829-4089-a4e3-f69b16c7bdbd","resolution":{"observed_at":"2026-08-07T10:20:48.839611Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.801952Z","title":null,"venue":null,"work_id":"0b51def8-7687-4eef-b7ae-441289455ac3","year":1997},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":14,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.512762Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:ba0732b3803b71c1cacb36274bf9545e5ea7bf6ccec55ae7fc807b61b05547f4","observation_id":"3bf98670-10ad-40b8-abcd-346a9c7150d7","resolution":{"observed_at":"2026-08-07T10:20:48.809322Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.776977Z","title":"Deformation of structures on manifolds defined by transitive, continuous pseudogroups","venue":null,"work_id":"c77c5cbb-9af1-41e2-8b70-38f8d4763ba7","year":1962},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":15,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.518757Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:496edd89e95d894ee6378194d6e0509b72f67cf9cfb64a02cc734fbc815f8e3f","observation_id":"364b688a-a4d4-4a1f-8e7d-f6f1e385fb58","resolution":{"observed_at":"2026-08-07T10:20:48.784441Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.755043Z","title":"The invariants of algebraic loci","venue":null,"work_id":"9af3a76c-fa35-4c6f-bc28-f4022c24ad33","year":1937},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":16,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.526777Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:021ada4179fea6626cf2e564a7cc8d368368365834785e7c643b330449953e49","observation_id":"4779c039-183c-4a86-8de5-838a2ba7c75e","resolution":{"observed_at":"2026-08-07T10:20:48.761339Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.732344Z","title":null,"venue":null,"work_id":"2ce343c5-1435-4449-bdd8-706f910e845c","year":1983},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":17,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.533224Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:934a5ab17168a6aa12c2f8f68d438e464d431e7dbc4d5bff56403599e567c592","observation_id":"bd1f3086-81c7-4440-aaf2-4525ace541ff","resolution":{"observed_at":"2026-08-07T10:20:48.739384Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2506.00752","last_updated":"2025-06-11T05:41:47Z","snapshot_observed_at":"2026-08-13T07:13:53.507138Z","submitted_at":"2025-05-31T23:39:45Z","title":"Constructing Two Metrics for Spencer Cohomology: Hodge Decomposition of Constrained Bundles","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.00752","snapshot_observed_at":"2026-08-07T10:20:48.541092Z","title":"Constructing two metrics for spencer cohomology: Hodge decomposition of constrained bundles","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":18,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.541092Z"},"links":{"cited_paper":"/paper/2506.00752","citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:2f05ea9a0f73434cd9278615f5c78782eb3d2b9dd3b1c0ba5d925e25b0ea51ce","observation_id":"e8eb1648-f9fc-4f49-a9ea-bb529d01a430","resolution":{"observed_at":"2026-08-07T10:20:48.541092Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2505.16766","last_updated":"2025-05-25T01:46:02Z","snapshot_observed_at":"2026-08-07T14:53:24.559980Z","submitted_at":"2025-05-22T15:07:59Z","title":"Dynamical Geometric Theory of Principal Bundle Constrained Systems: Strong Transversality Conditions and Variational Framework for Gauge Field Coupling","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2505.16766","snapshot_observed_at":"2026-08-07T10:20:48.546906Z","title":"Dynamical geometric theory of principal bundle constrained systems: Strong transversality conditions and variational framework for gauge field coupling","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":19,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.546906Z"},"links":{"cited_paper":"/paper/2505.16766","citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:9a8c05c4231c1bf086e12275e0759d1a7e86a2561b1efab3bae83cd1c7d01b53","observation_id":"e18042d6-670f-4127-826d-95c5434d1eeb","resolution":{"observed_at":"2026-08-07T10:20:48.546906Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2506.00728","last_updated":"2025-06-11T05:25:22Z","snapshot_observed_at":"2026-08-14T02:11:12.837043Z","submitted_at":"2025-05-31T22:04:53Z","title":"Geometric Duality Between Constraints and Gauge Fields: Mirror Symmetry and Spencer Isomorphisms of Compatible Pairs on Principal Bundles","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2506.00728","snapshot_observed_at":"2026-08-07T10:20:48.552751Z","title":"Geometric duality between constraints and gauge fields: Mirror symmetry and spencer isomorphisms of compatible pairs on principal bundles","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":20,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.552751Z"},"links":{"cited_paper":"/paper/2506.00728","citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:252eb014515f595f2c38f527d37fb0ca59ae005eee17e76f44d433c6d7f72f3e","observation_id":"718be92f-e621-4346-ba38-55616272d2c2","resolution":{"observed_at":"2026-08-07T10:20:48.552751Z","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":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-07T10:20:48.697868Z","title":"Mirror symmetry of spencer-hodge decompositions in constrained geometric systems","venue":null,"work_id":"a551f051-16b1-4401-9b0c-aa8d75b9a72d","year":2025},"citing_paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry","version":2},"reference_index":21,"source":"arxiv_source","source_observed_at":"2026-08-07T10:20:48.559782Z"},"links":{"citing_paper":"/paper/2506.05915"},"observation_digest":"sha256:3fd7dea4219de76c217ad92a8837dc982169c9601381f2e57b734700c396b66d","observation_id":"82820585-44f3-4292-ba30-7f9fa9926fc8","resolution":{"observed_at":"2026-08-07T10:20:48.709622Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2506.05915","last_updated":"2025-06-11T06:16:15Z","latest_version":2,"primary_category":"math.GM","snapshot_observed_at":"2026-08-12T16:52:37.285973Z","submitted_at":"2025-06-06T09:36:54Z","title":"Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry"},"reference_resolution":{"displayed":21,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":7,"verified_exact":0,"verified_fuzzy":14},"total_outbound_references":21},"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":"2026-08-15T06:32:42.880941+00:00","source":"crossref"},{"observed_at":"2026-08-15T06:32:39.529945+00:00","source":"retraction_watch"}],"thesis":"As of 15 August 2026, this Paper Citation Record lists 21 of 21 outbound references and 4 inbound Pith citation observations for arXiv:2506.05915."}