{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:57XUYTPJLKPN6C5BWTSM4EK6OP","short_pith_number":"pith:57XUYTPJ","canonical_record":{"source":{"id":"2508.09047","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-12T16:09:08Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"1f820c010ca96ca1d547b3ef96fe0c650e8d531f318939460dd1800ffdc03d35","abstract_canon_sha256":"4db47e1355dde28663de1c7de63a6d30a0a548c8132218b4c1f00b0344acc598"},"schema_version":"1.0"},"canonical_sha256":"efef4c4de95a9edf0ba1b4e4ce115e73dd42e00f0c5447beaebe27f502cb6e06","source":{"kind":"arxiv","id":"2508.09047","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.09047","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"arxiv_version","alias_value":"2508.09047v2","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.09047","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"pith_short_12","alias_value":"57XUYTPJLKPN","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"pith_short_16","alias_value":"57XUYTPJLKPN6C5B","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"pith_short_8","alias_value":"57XUYTPJ","created_at":"2026-07-05T11:56:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:57XUYTPJLKPN6C5BWTSM4EK6OP","target":"record","payload":{"canonical_record":{"source":{"id":"2508.09047","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-12T16:09:08Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"1f820c010ca96ca1d547b3ef96fe0c650e8d531f318939460dd1800ffdc03d35","abstract_canon_sha256":"4db47e1355dde28663de1c7de63a6d30a0a548c8132218b4c1f00b0344acc598"},"schema_version":"1.0"},"canonical_sha256":"efef4c4de95a9edf0ba1b4e4ce115e73dd42e00f0c5447beaebe27f502cb6e06","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:56:22.470926Z","signature_b64":"IMFcyFcuIfKjYqV2zDCDmI09oeE0ux8tWh4f9uyU6hkhxHKzqkYwA4ZqT4W7plIt7uopjJTXEuThfIhyNzhLBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"efef4c4de95a9edf0ba1b4e4ce115e73dd42e00f0c5447beaebe27f502cb6e06","last_reissued_at":"2026-07-05T11:56:22.470512Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:56:22.470512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2508.09047","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":"2026-07-05T11:56:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zEXDERklhNHI+UWAMyMXkkOFWZg1DMTQJu6U5SVgagukEsU+Mx27KduBSlctV9fOOqvaHdgfTz+HtCbzFfYfAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T08:20:05.958516Z"},"content_sha256":"0446c0be51a3054205712fc2618beee944cd131323b00e29150df89b5787d1f7","schema_version":"1.0","event_id":"sha256:0446c0be51a3054205712fc2618beee944cd131323b00e29150df89b5787d1f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:57XUYTPJLKPN6C5BWTSM4EK6OP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stability of spinorial Sobolev inequalities on $\\mathbb{S}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.DG","authors_text":"Guofang Wang, Mingwei Zhang","submitted_at":"2025-08-12T16:09:08Z","abstract_excerpt":"The spinorial Sobolev inequality on the unit sphere states \\begin{equation*}\n  \\Big(\\int| D\\psi|^{\\frac{2n}{n+1}}\\Big)^{\\frac{n+1}{n}}-\\frac{n}{2}\\omega_{n}^{1/n}\\int\\langle D\\psi,\\psi\\rangle\n  \\geq 0, \\end{equation*} with equality if and only if $\\psi \\in {\\mathcal M}$, the set of all $-\\frac 12$-Killing spinors and their conformal transformations.\n  Our main result in this paper is to refine this inequality by establishing a stability inequality\n  \\begin{equation*} \\Big(\\int| D\\psi|^{\\frac{2n}{n+1}}\\Big)^{\\frac{n+1}{n}}-\\frac{n}{2}\\omega_{n}^{1/n}\\int\\langle D\\psi,\\psi\\rangle\n  \\geq {\\bf c}_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.09047","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/2508.09047/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":"2026-07-05T11:56:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zsiCK6NNooRgjy+k9jTfmEljnV3NbSWkH1T5jMxs16emP/lohKtxfSrZL7jAKPXVRNTemuesQMeKH42dJ97XCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T08:20:05.959018Z"},"content_sha256":"5634ce7aa5149f1272a487b2b36dc1aa4f9f1b83ce5cf244d3c0b02279d9257f","schema_version":"1.0","event_id":"sha256:5634ce7aa5149f1272a487b2b36dc1aa4f9f1b83ce5cf244d3c0b02279d9257f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/57XUYTPJLKPN6C5BWTSM4EK6OP/bundle.json","state_url":"https://pith.science/pith/57XUYTPJLKPN6C5BWTSM4EK6OP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/57XUYTPJLKPN6C5BWTSM4EK6OP/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":"2026-08-09T08:20:05Z","links":{"resolver":"https://pith.science/pith/57XUYTPJLKPN6C5BWTSM4EK6OP","bundle":"https://pith.science/pith/57XUYTPJLKPN6C5BWTSM4EK6OP/bundle.json","state":"https://pith.science/pith/57XUYTPJLKPN6C5BWTSM4EK6OP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/57XUYTPJLKPN6C5BWTSM4EK6OP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:57XUYTPJLKPN6C5BWTSM4EK6OP","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":"4db47e1355dde28663de1c7de63a6d30a0a548c8132218b4c1f00b0344acc598","cross_cats_sorted":["math-ph","math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-12T16:09:08Z","title_canon_sha256":"1f820c010ca96ca1d547b3ef96fe0c650e8d531f318939460dd1800ffdc03d35"},"schema_version":"1.0","source":{"id":"2508.09047","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.09047","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"arxiv_version","alias_value":"2508.09047v2","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.09047","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"pith_short_12","alias_value":"57XUYTPJLKPN","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"pith_short_16","alias_value":"57XUYTPJLKPN6C5B","created_at":"2026-07-05T11:56:22Z"},{"alias_kind":"pith_short_8","alias_value":"57XUYTPJ","created_at":"2026-07-05T11:56:22Z"}],"graph_snapshots":[{"event_id":"sha256:5634ce7aa5149f1272a487b2b36dc1aa4f9f1b83ce5cf244d3c0b02279d9257f","target":"graph","created_at":"2026-07-05T11:56:22Z","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/2508.09047/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The spinorial Sobolev inequality on the unit sphere states \\begin{equation*}\n  \\Big(\\int| D\\psi|^{\\frac{2n}{n+1}}\\Big)^{\\frac{n+1}{n}}-\\frac{n}{2}\\omega_{n}^{1/n}\\int\\langle D\\psi,\\psi\\rangle\n  \\geq 0, \\end{equation*} with equality if and only if $\\psi \\in {\\mathcal M}$, the set of all $-\\frac 12$-Killing spinors and their conformal transformations.\n  Our main result in this paper is to refine this inequality by establishing a stability inequality\n  \\begin{equation*} \\Big(\\int| D\\psi|^{\\frac{2n}{n+1}}\\Big)^{\\frac{n+1}{n}}-\\frac{n}{2}\\omega_{n}^{1/n}\\int\\langle D\\psi,\\psi\\rangle\n  \\geq {\\bf c}_","authors_text":"Guofang Wang, Mingwei Zhang","cross_cats":["math-ph","math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-12T16:09:08Z","title":"Stability of spinorial Sobolev inequalities on $\\mathbb{S}^n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.09047","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":"sha256:0446c0be51a3054205712fc2618beee944cd131323b00e29150df89b5787d1f7","target":"record","created_at":"2026-07-05T11:56:22Z","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":"4db47e1355dde28663de1c7de63a6d30a0a548c8132218b4c1f00b0344acc598","cross_cats_sorted":["math-ph","math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-12T16:09:08Z","title_canon_sha256":"1f820c010ca96ca1d547b3ef96fe0c650e8d531f318939460dd1800ffdc03d35"},"schema_version":"1.0","source":{"id":"2508.09047","kind":"arxiv","version":2}},"canonical_sha256":"efef4c4de95a9edf0ba1b4e4ce115e73dd42e00f0c5447beaebe27f502cb6e06","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"efef4c4de95a9edf0ba1b4e4ce115e73dd42e00f0c5447beaebe27f502cb6e06","first_computed_at":"2026-07-05T11:56:22.470512Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:56:22.470512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IMFcyFcuIfKjYqV2zDCDmI09oeE0ux8tWh4f9uyU6hkhxHKzqkYwA4ZqT4W7plIt7uopjJTXEuThfIhyNzhLBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:56:22.470926Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.09047","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0446c0be51a3054205712fc2618beee944cd131323b00e29150df89b5787d1f7","sha256:5634ce7aa5149f1272a487b2b36dc1aa4f9f1b83ce5cf244d3c0b02279d9257f"],"state_sha256":"3b7404977bb71e862b47c458cf7bbf63f0a17f9290766e041f0b7c0943021e2f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bCGcoigGM5FMH91a/KB9dxx2wEptdCr1kITw9r1SFcOsjinZFq94+o3UapNCbRjbI3xE6nwpAKXlwy/Z9J7kBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T08:20:05.962990Z","bundle_sha256":"c48e2de7006368eb247d6fe22c99d5c2b333afc46696ba82b70c380893592dfd"}}