{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:32IPRTLVLRICKNAYIGOKXSXK6O","short_pith_number":"pith:32IPRTLV","canonical_record":{"source":{"id":"0709.2878","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2007-09-18T16:46:21Z","cross_cats_sorted":[],"title_canon_sha256":"549ea5fccc4d180fd0dc9a6666c86ba5d6ec99167fd2b795d772dcd676d552d1","abstract_canon_sha256":"f5e6601f8b94121368c994c2922f1fe9633b128b8aebaf58400d366c9682b006"},"schema_version":"1.0"},"canonical_sha256":"de90f8cd755c50253418419cabcaeaf3a9d7adc6c2c23f2c4b985a8764420645","source":{"kind":"arxiv","id":"0709.2878","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0709.2878","created_at":"2026-05-18T02:16:11Z"},{"alias_kind":"arxiv_version","alias_value":"0709.2878v1","created_at":"2026-05-18T02:16:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.2878","created_at":"2026-05-18T02:16:11Z"},{"alias_kind":"pith_short_12","alias_value":"32IPRTLVLRIC","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"32IPRTLVLRICKNAY","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"32IPRTLV","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:32IPRTLVLRICKNAYIGOKXSXK6O","target":"record","payload":{"canonical_record":{"source":{"id":"0709.2878","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2007-09-18T16:46:21Z","cross_cats_sorted":[],"title_canon_sha256":"549ea5fccc4d180fd0dc9a6666c86ba5d6ec99167fd2b795d772dcd676d552d1","abstract_canon_sha256":"f5e6601f8b94121368c994c2922f1fe9633b128b8aebaf58400d366c9682b006"},"schema_version":"1.0"},"canonical_sha256":"de90f8cd755c50253418419cabcaeaf3a9d7adc6c2c23f2c4b985a8764420645","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:11.165818Z","signature_b64":"0eIRVwUrA/kEKieGZm2AYbDNRK/9zVDwHycGF0jiysT8btXQRrQnTsIHK2R6pmP24TmiPzoxbS8ZtcwNX9OFBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de90f8cd755c50253418419cabcaeaf3a9d7adc6c2c23f2c4b985a8764420645","last_reissued_at":"2026-05-18T02:16:11.165232Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:11.165232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0709.2878","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:16:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sMPh0Lt8xmedsyJ4GBYkQ6fFq+3MOdBQCcI8+TRc2RZjN17FoGOj+zpXyUe/RCtFBSK6VT7Jc70gA82yXqWeBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:51:31.513476Z"},"content_sha256":"e4b046a9aedf81faf4cce2ba429207e644bd7de4d972aef8196191534e08e818","schema_version":"1.0","event_id":"sha256:e4b046a9aedf81faf4cce2ba429207e644bd7de4d972aef8196191534e08e818"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:32IPRTLVLRICKNAYIGOKXSXK6O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singular limits for the bi-laplacian operator with exponential nonlinearity in $\\R^4$","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Claudio Mu\\~noz, M\\'onica Clapp, Monica Musso","submitted_at":"2007-09-18T16:46:21Z","abstract_excerpt":"Let $\\Omega$ be a bounded smooth domain in $\\mathbb{R}^{4}$ such that for some integer $d\\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem\n  {\\Delta^{2}u-\\rho^{4}k(x)e^{u}=0 & \\hbox{in}\\Omega,\n  u=\\Delta u=0 & \\hbox{on}\\partial\\Omega,\n  has a solution blowing-up, as $\\rho\\to0$, at $m$ points of $\\Omega$, for any given number $m$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.2878","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:16:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0Ycpv/zCplGyU1ogLxzxdDsgJmitNfW/3D2SjofceAK9YTFrShq49MrgzXv+aCtH/GmRDGTaf1tCbzsh/G8HCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:51:31.514209Z"},"content_sha256":"b2acc815ed514a9318ab04f90fe527d8f9b23931a5fa6433fbcf9eac437116b1","schema_version":"1.0","event_id":"sha256:b2acc815ed514a9318ab04f90fe527d8f9b23931a5fa6433fbcf9eac437116b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/32IPRTLVLRICKNAYIGOKXSXK6O/bundle.json","state_url":"https://pith.science/pith/32IPRTLVLRICKNAYIGOKXSXK6O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/32IPRTLVLRICKNAYIGOKXSXK6O/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":"2026-05-27T05:51:31Z","links":{"resolver":"https://pith.science/pith/32IPRTLVLRICKNAYIGOKXSXK6O","bundle":"https://pith.science/pith/32IPRTLVLRICKNAYIGOKXSXK6O/bundle.json","state":"https://pith.science/pith/32IPRTLVLRICKNAYIGOKXSXK6O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/32IPRTLVLRICKNAYIGOKXSXK6O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:32IPRTLVLRICKNAYIGOKXSXK6O","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":"f5e6601f8b94121368c994c2922f1fe9633b128b8aebaf58400d366c9682b006","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2007-09-18T16:46:21Z","title_canon_sha256":"549ea5fccc4d180fd0dc9a6666c86ba5d6ec99167fd2b795d772dcd676d552d1"},"schema_version":"1.0","source":{"id":"0709.2878","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0709.2878","created_at":"2026-05-18T02:16:11Z"},{"alias_kind":"arxiv_version","alias_value":"0709.2878v1","created_at":"2026-05-18T02:16:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.2878","created_at":"2026-05-18T02:16:11Z"},{"alias_kind":"pith_short_12","alias_value":"32IPRTLVLRIC","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"32IPRTLVLRICKNAY","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"32IPRTLV","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:b2acc815ed514a9318ab04f90fe527d8f9b23931a5fa6433fbcf9eac437116b1","target":"graph","created_at":"2026-05-18T02:16:11Z","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 $\\Omega$ be a bounded smooth domain in $\\mathbb{R}^{4}$ such that for some integer $d\\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem\n  {\\Delta^{2}u-\\rho^{4}k(x)e^{u}=0 & \\hbox{in}\\Omega,\n  u=\\Delta u=0 & \\hbox{on}\\partial\\Omega,\n  has a solution blowing-up, as $\\rho\\to0$, at $m$ points of $\\Omega$, for any given number $m$.","authors_text":"Claudio Mu\\~noz, M\\'onica Clapp, Monica Musso","cross_cats":[],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2007-09-18T16:46:21Z","title":"Singular limits for the bi-laplacian operator with exponential nonlinearity in $\\R^4$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.2878","kind":"arxiv","version":1},"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:e4b046a9aedf81faf4cce2ba429207e644bd7de4d972aef8196191534e08e818","target":"record","created_at":"2026-05-18T02:16:11Z","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":"f5e6601f8b94121368c994c2922f1fe9633b128b8aebaf58400d366c9682b006","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2007-09-18T16:46:21Z","title_canon_sha256":"549ea5fccc4d180fd0dc9a6666c86ba5d6ec99167fd2b795d772dcd676d552d1"},"schema_version":"1.0","source":{"id":"0709.2878","kind":"arxiv","version":1}},"canonical_sha256":"de90f8cd755c50253418419cabcaeaf3a9d7adc6c2c23f2c4b985a8764420645","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de90f8cd755c50253418419cabcaeaf3a9d7adc6c2c23f2c4b985a8764420645","first_computed_at":"2026-05-18T02:16:11.165232Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:16:11.165232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0eIRVwUrA/kEKieGZm2AYbDNRK/9zVDwHycGF0jiysT8btXQRrQnTsIHK2R6pmP24TmiPzoxbS8ZtcwNX9OFBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:16:11.165818Z","signed_message":"canonical_sha256_bytes"},"source_id":"0709.2878","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e4b046a9aedf81faf4cce2ba429207e644bd7de4d972aef8196191534e08e818","sha256:b2acc815ed514a9318ab04f90fe527d8f9b23931a5fa6433fbcf9eac437116b1"],"state_sha256":"ccf80ad569decaa85cb4c973b3a6fdda47e07c5834f195ebb507b2ac91d9c5df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JY+KyImLANw7X1tz1yrEXWWtLz7hJPr2dWx65a5FjP+AKTPaeS5qDUTPWNm8BjJ/NHWeyslKu1Hn37r9FvzJCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:51:31.517559Z","bundle_sha256":"0df1d8940d000c1283d50cac687bf48573b734b59d2e9457520a7f034c0f5738"}}