{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:4W5ADCM6PMMPHXNSZHBP3A35NH","short_pith_number":"pith:4W5ADCM6","canonical_record":{"source":{"id":"2507.16394","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-22T09:38:29Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"e81170da34bedf980b8b571b906c02cd13583f87472bdcc69bf45b144e48d393","abstract_canon_sha256":"903ebc946076b3856edf4738e52b172c8a288c9e26bc29f7a25f26ee3fdbce72"},"schema_version":"1.0"},"canonical_sha256":"e5ba01899e7b18f3ddb2c9c2fd837d69c6036dec6b05cc2fa7f2b4038d17208f","source":{"kind":"arxiv","id":"2507.16394","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.16394","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"arxiv_version","alias_value":"2507.16394v1","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.16394","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"pith_short_12","alias_value":"4W5ADCM6PMMP","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"pith_short_16","alias_value":"4W5ADCM6PMMPHXNS","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"pith_short_8","alias_value":"4W5ADCM6","created_at":"2026-07-05T11:41:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:4W5ADCM6PMMPHXNSZHBP3A35NH","target":"record","payload":{"canonical_record":{"source":{"id":"2507.16394","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-22T09:38:29Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"e81170da34bedf980b8b571b906c02cd13583f87472bdcc69bf45b144e48d393","abstract_canon_sha256":"903ebc946076b3856edf4738e52b172c8a288c9e26bc29f7a25f26ee3fdbce72"},"schema_version":"1.0"},"canonical_sha256":"e5ba01899e7b18f3ddb2c9c2fd837d69c6036dec6b05cc2fa7f2b4038d17208f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:41:08.702139Z","signature_b64":"OPf87zVN1eziRZyNwRuaAyw/2De8OJ2ciaAgDLq807qeLYVeaDGaVGMPF+I9Ie8eSfVvEW676O9a89WnwNAiBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5ba01899e7b18f3ddb2c9c2fd837d69c6036dec6b05cc2fa7f2b4038d17208f","last_reissued_at":"2026-07-05T11:41:08.701762Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:41:08.701762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.16394","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":"2026-07-05T11:41:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OnT95zdtz9Xt5ybf/beM5n5LGSG8nsz4iSDyHv7sR/r67en2szydyaILu0hjIYsq9wid7Ra6Bnyx9Ql9++OTCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T13:37:03.944483Z"},"content_sha256":"1128d836d16ecc2bcd236a1c04dfd41cb63f116c21215099a2b230d87154a5fd","schema_version":"1.0","event_id":"sha256:1128d836d16ecc2bcd236a1c04dfd41cb63f116c21215099a2b230d87154a5fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:4W5ADCM6PMMPHXNSZHBP3A35NH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The $\\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\\frac{n}{2}$ and beyond","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Jonah A. J. Duncan, Luc Nguyen","submitted_at":"2025-07-22T09:38:29Z","abstract_excerpt":"Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\\geq 3$ with smooth non-empty boundary $\\partial M$. Let $\\Gamma\\subset\\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $\\Gamma$ satisfying standard assumptions. Denoting by $A_{g_u}$ the Schouten tensor of a conformal metric $g_u = u^{-2}g_0$, we show that the associated fully nonlinear Loewner-Nirenberg problem\n  \\begin{align*}\n  \\begin{cases}\n  f(\\lambda(-g_u^{-1}A_{g_u})) = \\frac{1}{2}, \\quad \\lambda(-g_u^{-1}A_{g_u})\\in\\Gamma & \\text{on }M\\backslash \\partial M \\newline\n  u = 0 & \\text{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.16394","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.16394/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":"2026-07-05T11:41:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LMlS+Pc6clbkbUdyd5i1q8PS46RUjUbdWWdWtsGK3RYl+IZFM0OCtwk2W/PS9zRS0AI4w0wSfdWFcdkZReqDAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T13:37:03.945009Z"},"content_sha256":"22877e62907fe13c2604b8c0cc7b2d65705a4f4f372a5260fa2d2cf6c4770050","schema_version":"1.0","event_id":"sha256:22877e62907fe13c2604b8c0cc7b2d65705a4f4f372a5260fa2d2cf6c4770050"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4W5ADCM6PMMPHXNSZHBP3A35NH/bundle.json","state_url":"https://pith.science/pith/4W5ADCM6PMMPHXNSZHBP3A35NH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4W5ADCM6PMMPHXNSZHBP3A35NH/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":"2026-08-08T13:37:03Z","links":{"resolver":"https://pith.science/pith/4W5ADCM6PMMPHXNSZHBP3A35NH","bundle":"https://pith.science/pith/4W5ADCM6PMMPHXNSZHBP3A35NH/bundle.json","state":"https://pith.science/pith/4W5ADCM6PMMPHXNSZHBP3A35NH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4W5ADCM6PMMPHXNSZHBP3A35NH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:4W5ADCM6PMMPHXNSZHBP3A35NH","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":"903ebc946076b3856edf4738e52b172c8a288c9e26bc29f7a25f26ee3fdbce72","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-22T09:38:29Z","title_canon_sha256":"e81170da34bedf980b8b571b906c02cd13583f87472bdcc69bf45b144e48d393"},"schema_version":"1.0","source":{"id":"2507.16394","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.16394","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"arxiv_version","alias_value":"2507.16394v1","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.16394","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"pith_short_12","alias_value":"4W5ADCM6PMMP","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"pith_short_16","alias_value":"4W5ADCM6PMMPHXNS","created_at":"2026-07-05T11:41:08Z"},{"alias_kind":"pith_short_8","alias_value":"4W5ADCM6","created_at":"2026-07-05T11:41:08Z"}],"graph_snapshots":[{"event_id":"sha256:22877e62907fe13c2604b8c0cc7b2d65705a4f4f372a5260fa2d2cf6c4770050","target":"graph","created_at":"2026-07-05T11:41:08Z","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/2507.16394/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\\geq 3$ with smooth non-empty boundary $\\partial M$. Let $\\Gamma\\subset\\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $\\Gamma$ satisfying standard assumptions. Denoting by $A_{g_u}$ the Schouten tensor of a conformal metric $g_u = u^{-2}g_0$, we show that the associated fully nonlinear Loewner-Nirenberg problem\n  \\begin{align*}\n  \\begin{cases}\n  f(\\lambda(-g_u^{-1}A_{g_u})) = \\frac{1}{2}, \\quad \\lambda(-g_u^{-1}A_{g_u})\\in\\Gamma & \\text{on }M\\backslash \\partial M \\newline\n  u = 0 & \\text{","authors_text":"Jonah A. J. Duncan, Luc Nguyen","cross_cats":["math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-22T09:38:29Z","title":"The $\\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\\frac{n}{2}$ and beyond"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.16394","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":"sha256:1128d836d16ecc2bcd236a1c04dfd41cb63f116c21215099a2b230d87154a5fd","target":"record","created_at":"2026-07-05T11:41:08Z","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":"903ebc946076b3856edf4738e52b172c8a288c9e26bc29f7a25f26ee3fdbce72","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-22T09:38:29Z","title_canon_sha256":"e81170da34bedf980b8b571b906c02cd13583f87472bdcc69bf45b144e48d393"},"schema_version":"1.0","source":{"id":"2507.16394","kind":"arxiv","version":1}},"canonical_sha256":"e5ba01899e7b18f3ddb2c9c2fd837d69c6036dec6b05cc2fa7f2b4038d17208f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5ba01899e7b18f3ddb2c9c2fd837d69c6036dec6b05cc2fa7f2b4038d17208f","first_computed_at":"2026-07-05T11:41:08.701762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:41:08.701762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OPf87zVN1eziRZyNwRuaAyw/2De8OJ2ciaAgDLq807qeLYVeaDGaVGMPF+I9Ie8eSfVvEW676O9a89WnwNAiBg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:41:08.702139Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.16394","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1128d836d16ecc2bcd236a1c04dfd41cb63f116c21215099a2b230d87154a5fd","sha256:22877e62907fe13c2604b8c0cc7b2d65705a4f4f372a5260fa2d2cf6c4770050"],"state_sha256":"9af72bbbaf2c9465b2bcd7689e0b59340022230c08940b57fdefdbfe89452f26"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cNh0lHwB+22n15dXKSKK2P/LPCjJ8aF0Se39rkZiwQ1fWhogkG1oDtX9PQaPsaigraTI1FgKiVpSPEQWHkeAAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T13:37:03.949255Z","bundle_sha256":"795e892d0dc53acfb2fb5e736254d9f2b405f26c3912c06cd2a65cb8e37dbce5"}}