{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:WN5MVSC7EULTR7GDTJXZQUHQDV","short_pith_number":"pith:WN5MVSC7","canonical_record":{"source":{"id":"2509.01155","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-01T06:24:11Z","cross_cats_sorted":[],"title_canon_sha256":"8788a609cbe4e8680aa0aaf7894049245a4ede980bdd6bc237860e6728a56b6c","abstract_canon_sha256":"61a8b3736c5486e83b5b04538c49c6f19a23eb803ab72e4c019417c761b88e74"},"schema_version":"1.0"},"canonical_sha256":"b37acac85f251738fcc39a6f9850f01d5135e9fbc98bf0829e6e665fb079765d","source":{"kind":"arxiv","id":"2509.01155","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.01155","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"arxiv_version","alias_value":"2509.01155v2","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.01155","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"pith_short_12","alias_value":"WN5MVSC7EULT","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"pith_short_16","alias_value":"WN5MVSC7EULTR7GD","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"pith_short_8","alias_value":"WN5MVSC7","created_at":"2026-05-22T01:03:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:WN5MVSC7EULTR7GDTJXZQUHQDV","target":"record","payload":{"canonical_record":{"source":{"id":"2509.01155","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-01T06:24:11Z","cross_cats_sorted":[],"title_canon_sha256":"8788a609cbe4e8680aa0aaf7894049245a4ede980bdd6bc237860e6728a56b6c","abstract_canon_sha256":"61a8b3736c5486e83b5b04538c49c6f19a23eb803ab72e4c019417c761b88e74"},"schema_version":"1.0"},"canonical_sha256":"b37acac85f251738fcc39a6f9850f01d5135e9fbc98bf0829e6e665fb079765d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:45.714220Z","signature_b64":"pspWMU34ZDhhARTUeGkYo8NBdR8PaWoc1gHEpYLZBFG2g65n09ygkBkV5wkavget1neyvb9Bg65Je892UaJ9DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b37acac85f251738fcc39a6f9850f01d5135e9fbc98bf0829e6e665fb079765d","last_reissued_at":"2026-05-22T01:03:45.713298Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:45.713298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.01155","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":"2026-05-22T01:03:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jp9EHdu6O3ulYvu/WARs+ejf8Eiq+NfS/mSDBMmJXzIZk7B3WGHbjJWblDcAF4TyihWf3HkJXu9mFZKLqsj1AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:21:01.468064Z"},"content_sha256":"cc467821550713f1dae35ccbfaa16aee4ca4c8411e3b359a59100736321384b3","schema_version":"1.0","event_id":"sha256:cc467821550713f1dae35ccbfaa16aee4ca4c8411e3b359a59100736321384b3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:WN5MVSC7EULTR7GDTJXZQUHQDV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On finite-energy solutions of Kazan-Warner equations on the lattice graph","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bobo Hua, Huyuan Chen","submitted_at":"2025-09-01T06:24:11Z","abstract_excerpt":"We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$ - \\Delta u= \\varepsilon e^{\\kappa u} +\\beta\\delta_0\\quad {\\rm in}\\ \\mathbb{Z}^2,$$ where $\\varepsilon=\\pm1$, $\\kappa>0$ and $\\beta\\in\\mathbb{R}$.\n  When $\\varepsilon=1$, we prove the existence of a continuous family of finite-energy solutions for some parameter $\\kappa$. This provides a partial resolution of the open problem on the existence of finite-energy solutions to the Liouville equation.\n  When $\\varepsilon=-1$ and $\\beta>\\frac{4\\pi}{\\kappa}$, we prove that the set of finite-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.01155","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.01155/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-22T01:03:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mOJzwRCShI523wrgxNmNcIuSwxoKmCBm3UITvzlvu3oIpHQ7V6Ydgl4kQSH69d7kvL6e2NIJzcUQE/9waEMODg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T18:21:01.468790Z"},"content_sha256":"3f8a73715555c01d909216f988035ba9dc4dab8dacfd07c298934af8f71ec123","schema_version":"1.0","event_id":"sha256:3f8a73715555c01d909216f988035ba9dc4dab8dacfd07c298934af8f71ec123"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WN5MVSC7EULTR7GDTJXZQUHQDV/bundle.json","state_url":"https://pith.science/pith/WN5MVSC7EULTR7GDTJXZQUHQDV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WN5MVSC7EULTR7GDTJXZQUHQDV/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":"2026-05-26T18:21:01Z","links":{"resolver":"https://pith.science/pith/WN5MVSC7EULTR7GDTJXZQUHQDV","bundle":"https://pith.science/pith/WN5MVSC7EULTR7GDTJXZQUHQDV/bundle.json","state":"https://pith.science/pith/WN5MVSC7EULTR7GDTJXZQUHQDV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WN5MVSC7EULTR7GDTJXZQUHQDV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:WN5MVSC7EULTR7GDTJXZQUHQDV","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":"61a8b3736c5486e83b5b04538c49c6f19a23eb803ab72e4c019417c761b88e74","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-01T06:24:11Z","title_canon_sha256":"8788a609cbe4e8680aa0aaf7894049245a4ede980bdd6bc237860e6728a56b6c"},"schema_version":"1.0","source":{"id":"2509.01155","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.01155","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"arxiv_version","alias_value":"2509.01155v2","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.01155","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"pith_short_12","alias_value":"WN5MVSC7EULT","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"pith_short_16","alias_value":"WN5MVSC7EULTR7GD","created_at":"2026-05-22T01:03:45Z"},{"alias_kind":"pith_short_8","alias_value":"WN5MVSC7","created_at":"2026-05-22T01:03:45Z"}],"graph_snapshots":[{"event_id":"sha256:3f8a73715555c01d909216f988035ba9dc4dab8dacfd07c298934af8f71ec123","target":"graph","created_at":"2026-05-22T01:03:45Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2509.01155/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$ - \\Delta u= \\varepsilon e^{\\kappa u} +\\beta\\delta_0\\quad {\\rm in}\\ \\mathbb{Z}^2,$$ where $\\varepsilon=\\pm1$, $\\kappa>0$ and $\\beta\\in\\mathbb{R}$.\n  When $\\varepsilon=1$, we prove the existence of a continuous family of finite-energy solutions for some parameter $\\kappa$. This provides a partial resolution of the open problem on the existence of finite-energy solutions to the Liouville equation.\n  When $\\varepsilon=-1$ and $\\beta>\\frac{4\\pi}{\\kappa}$, we prove that the set of finite-","authors_text":"Bobo Hua, Huyuan Chen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-01T06:24:11Z","title":"On finite-energy solutions of Kazan-Warner equations on the lattice graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.01155","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:cc467821550713f1dae35ccbfaa16aee4ca4c8411e3b359a59100736321384b3","target":"record","created_at":"2026-05-22T01:03:45Z","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":"61a8b3736c5486e83b5b04538c49c6f19a23eb803ab72e4c019417c761b88e74","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2025-09-01T06:24:11Z","title_canon_sha256":"8788a609cbe4e8680aa0aaf7894049245a4ede980bdd6bc237860e6728a56b6c"},"schema_version":"1.0","source":{"id":"2509.01155","kind":"arxiv","version":2}},"canonical_sha256":"b37acac85f251738fcc39a6f9850f01d5135e9fbc98bf0829e6e665fb079765d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b37acac85f251738fcc39a6f9850f01d5135e9fbc98bf0829e6e665fb079765d","first_computed_at":"2026-05-22T01:03:45.713298Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:03:45.713298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pspWMU34ZDhhARTUeGkYo8NBdR8PaWoc1gHEpYLZBFG2g65n09ygkBkV5wkavget1neyvb9Bg65Je892UaJ9DQ==","signature_status":"signed_v1","signed_at":"2026-05-22T01:03:45.714220Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.01155","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cc467821550713f1dae35ccbfaa16aee4ca4c8411e3b359a59100736321384b3","sha256:3f8a73715555c01d909216f988035ba9dc4dab8dacfd07c298934af8f71ec123"],"state_sha256":"216f9a74da4bda1f9108a7855eacb44fa9e24f504a92fa6f73dcd97f1e972688"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KgDbWCuA2Tq+yE/GfeEqD3FRQY3PPogCYwid87qbO7JdwY71vnwgZHSv4DIaX7Ppfy9c71Dc6jy1ON4Z+Tc7BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T18:21:01.472397Z","bundle_sha256":"5da1dde5101b87bb257bbd222e881bfd609fc83c77d2d9768bb409c2215ba71f"}}