{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:H2NKEU4EY6BCVFDLOYJ4BLX5C4","short_pith_number":"pith:H2NKEU4E","canonical_record":{"source":{"id":"1512.03703","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-06T17:56:21Z","cross_cats_sorted":["math-ph","math.FA","math.MP","math.SP"],"title_canon_sha256":"343d0639aa71064c32fbc2e9139e99d014d7733642dcddc09824f8957e63872b","abstract_canon_sha256":"4d41d2a50496934306e040ee9cc529a2b01d141c5d77ce74414a879bb9b57c68"},"schema_version":"1.0"},"canonical_sha256":"3e9aa25384c7822a946b7613c0aefd171ea851d29ab7fdbe4b452e18732c673c","source":{"kind":"arxiv","id":"1512.03703","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.03703","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"arxiv_version","alias_value":"1512.03703v2","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03703","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"pith_short_12","alias_value":"H2NKEU4EY6BC","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"H2NKEU4EY6BCVFDL","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"H2NKEU4E","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:H2NKEU4EY6BCVFDLOYJ4BLX5C4","target":"record","payload":{"canonical_record":{"source":{"id":"1512.03703","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-06T17:56:21Z","cross_cats_sorted":["math-ph","math.FA","math.MP","math.SP"],"title_canon_sha256":"343d0639aa71064c32fbc2e9139e99d014d7733642dcddc09824f8957e63872b","abstract_canon_sha256":"4d41d2a50496934306e040ee9cc529a2b01d141c5d77ce74414a879bb9b57c68"},"schema_version":"1.0"},"canonical_sha256":"3e9aa25384c7822a946b7613c0aefd171ea851d29ab7fdbe4b452e18732c673c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:31.470255Z","signature_b64":"0wl6BZUTYUDvuxbLtbewPlpH4rsy8+pCBIx00RRQFbsXy8q/Lcdu6zGcW8JmmiTbzopqXptmIP7ZbXlCiQ1CBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e9aa25384c7822a946b7613c0aefd171ea851d29ab7fdbe4b452e18732c673c","last_reissued_at":"2026-05-18T00:38:31.469682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:31.469682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.03703","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":"2026-05-18T00:38:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uNUX3xGegQYdEjTFwTKx2K37Drn8NBMGTGAP0YqnOniTEMCl9j2UTbk8UDiVzh5d0sM0lnOMMyXlhmOpOL96AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:37:25.036658Z"},"content_sha256":"d68f2b3a44e19b70f7299a6afaba59a354976f463792ba79a717a92fb23ee9f8","schema_version":"1.0","event_id":"sha256:d68f2b3a44e19b70f7299a6afaba59a354976f463792ba79a717a92fb23ee9f8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:H2NKEU4EY6BCVFDLOYJ4BLX5C4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singularities of solutions to quadratic vector equations on complex upper half-plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP","math.SP"],"primary_cat":"math.PR","authors_text":"Laszlo Erdos, Oskari Ajanki, Torben Kr\\\"uger","submitted_at":"2015-12-06T17:56:21Z","abstract_excerpt":"Let $ S $ be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation $ -\\frac{1}{m}=z+Sm $ with a parameter $ z $ in the complex upper half-plane $ \\mathbb{H} $ has a unique solution $ m $ with values in $ \\mathbb{H} $. We show that the $ z $-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures $ v $ on $ \\mathbb{R} $. Under suitable conditions on $ S $, we show that $ v $ has a real analytic density apart from finitely many algebraic singularities of degree at most three.\n  Our motivation c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03703","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":"2026-05-18T00:38:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5DckffHF5/Dk6bdB+wowgjTEcrfXUD57XuASoGL9YAkJVxnD80wAFIB3//8tWgy9wOsTlpKdmogLfMg77i5rBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:37:25.037012Z"},"content_sha256":"3491c3f5f94c1b8d6be18297f7d05c4ac883b84f0fd44d19428c6de47551c279","schema_version":"1.0","event_id":"sha256:3491c3f5f94c1b8d6be18297f7d05c4ac883b84f0fd44d19428c6de47551c279"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H2NKEU4EY6BCVFDLOYJ4BLX5C4/bundle.json","state_url":"https://pith.science/pith/H2NKEU4EY6BCVFDLOYJ4BLX5C4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H2NKEU4EY6BCVFDLOYJ4BLX5C4/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":"2026-06-02T18:37:25Z","links":{"resolver":"https://pith.science/pith/H2NKEU4EY6BCVFDLOYJ4BLX5C4","bundle":"https://pith.science/pith/H2NKEU4EY6BCVFDLOYJ4BLX5C4/bundle.json","state":"https://pith.science/pith/H2NKEU4EY6BCVFDLOYJ4BLX5C4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H2NKEU4EY6BCVFDLOYJ4BLX5C4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:H2NKEU4EY6BCVFDLOYJ4BLX5C4","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":"4d41d2a50496934306e040ee9cc529a2b01d141c5d77ce74414a879bb9b57c68","cross_cats_sorted":["math-ph","math.FA","math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-06T17:56:21Z","title_canon_sha256":"343d0639aa71064c32fbc2e9139e99d014d7733642dcddc09824f8957e63872b"},"schema_version":"1.0","source":{"id":"1512.03703","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.03703","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"arxiv_version","alias_value":"1512.03703v2","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03703","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"pith_short_12","alias_value":"H2NKEU4EY6BC","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"H2NKEU4EY6BCVFDL","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"H2NKEU4E","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:3491c3f5f94c1b8d6be18297f7d05c4ac883b84f0fd44d19428c6de47551c279","target":"graph","created_at":"2026-05-18T00:38:31Z","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":"Let $ S $ be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation $ -\\frac{1}{m}=z+Sm $ with a parameter $ z $ in the complex upper half-plane $ \\mathbb{H} $ has a unique solution $ m $ with values in $ \\mathbb{H} $. We show that the $ z $-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures $ v $ on $ \\mathbb{R} $. Under suitable conditions on $ S $, we show that $ v $ has a real analytic density apart from finitely many algebraic singularities of degree at most three.\n  Our motivation c","authors_text":"Laszlo Erdos, Oskari Ajanki, Torben Kr\\\"uger","cross_cats":["math-ph","math.FA","math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-06T17:56:21Z","title":"Singularities of solutions to quadratic vector equations on complex upper half-plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03703","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:d68f2b3a44e19b70f7299a6afaba59a354976f463792ba79a717a92fb23ee9f8","target":"record","created_at":"2026-05-18T00:38:31Z","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":"4d41d2a50496934306e040ee9cc529a2b01d141c5d77ce74414a879bb9b57c68","cross_cats_sorted":["math-ph","math.FA","math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-06T17:56:21Z","title_canon_sha256":"343d0639aa71064c32fbc2e9139e99d014d7733642dcddc09824f8957e63872b"},"schema_version":"1.0","source":{"id":"1512.03703","kind":"arxiv","version":2}},"canonical_sha256":"3e9aa25384c7822a946b7613c0aefd171ea851d29ab7fdbe4b452e18732c673c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e9aa25384c7822a946b7613c0aefd171ea851d29ab7fdbe4b452e18732c673c","first_computed_at":"2026-05-18T00:38:31.469682Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:31.469682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0wl6BZUTYUDvuxbLtbewPlpH4rsy8+pCBIx00RRQFbsXy8q/Lcdu6zGcW8JmmiTbzopqXptmIP7ZbXlCiQ1CBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:31.470255Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.03703","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d68f2b3a44e19b70f7299a6afaba59a354976f463792ba79a717a92fb23ee9f8","sha256:3491c3f5f94c1b8d6be18297f7d05c4ac883b84f0fd44d19428c6de47551c279"],"state_sha256":"e2d2162a151aff9e4c7229d3234edc52b32d293e1d5d6e4eaf7ebe6da925c0b3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SgAvrB5WiCAcv5C1esG+PjW3N2QebqvF/nPGe9gLKqak6i06LjOmu3UgEUmdR9Em9A6Y5KrMDxU7dybP37zxAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T18:37:25.038977Z","bundle_sha256":"180f3c3afd526d8f61ed9ebfc7024b59c298531f45ac76819f950a7b0c0288d2"}}