{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:XYHOCIQKNYMD332XKRFRFDD532","short_pith_number":"pith:XYHOCIQK","canonical_record":{"source":{"id":"1804.03467","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-10T11:58:51Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"d9450e2fd833d7acf9d252f06d50eb7ce5b5657579de3243ff09af6dee000365","abstract_canon_sha256":"51dadb88de580292028cc6b0070cf27a8e6165ad8be5d14574585edd03c513a5"},"schema_version":"1.0"},"canonical_sha256":"be0ee1220a6e183def57544b128c7dde9882ff066fde9ed6cd62a5eb820f0ef8","source":{"kind":"arxiv","id":"1804.03467","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.03467","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"arxiv_version","alias_value":"1804.03467v1","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.03467","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"pith_short_12","alias_value":"XYHOCIQKNYMD","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"XYHOCIQKNYMD332X","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"XYHOCIQK","created_at":"2026-05-18T12:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:XYHOCIQKNYMD332XKRFRFDD532","target":"record","payload":{"canonical_record":{"source":{"id":"1804.03467","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-10T11:58:51Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"d9450e2fd833d7acf9d252f06d50eb7ce5b5657579de3243ff09af6dee000365","abstract_canon_sha256":"51dadb88de580292028cc6b0070cf27a8e6165ad8be5d14574585edd03c513a5"},"schema_version":"1.0"},"canonical_sha256":"be0ee1220a6e183def57544b128c7dde9882ff066fde9ed6cd62a5eb820f0ef8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:48.101715Z","signature_b64":"CA/ocsmFePAKqpUNEDqI7fMUTA/HZfl7UcW15GJB/KuoSnWZQC8rc9q7Fz3kuDKorqUXkVJf4duQ+2GsH0lKCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be0ee1220a6e183def57544b128c7dde9882ff066fde9ed6cd62a5eb820f0ef8","last_reissued_at":"2026-05-18T00:18:48.100944Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:48.100944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.03467","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":"2026-05-18T00:18:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8W0TEdxema90WwvZrnXbiFswrO3NN7vln2D2YPgzKvI2GNWZtkxOSvgdJN55r7aqcfQHvtVnXajzXJ2YnPJnBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:31:08.705798Z"},"content_sha256":"66f83d8fbe8a76085d750b51763531f0d0e02ccf2d0c2889c10923e512d705d6","schema_version":"1.0","event_id":"sha256:66f83d8fbe8a76085d750b51763531f0d0e02ccf2d0c2889c10923e512d705d6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:XYHOCIQKNYMD332XKRFRFDD532","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exact asymptotic volume and volume ratio of Schatten unit balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.FA","authors_text":"Christoph Thaele, Joscha Prochno, Zakhar Kabluchko","submitted_at":"2018-04-10T11:58:51Z","abstract_excerpt":"The unit ball $B_p^n(\\mathbb{R})$ of the finite-dimensional Schatten trace class $\\mathcal S_p^n$ consists of all real $n\\times n$ matrices $A$ whose singular values $s_1(A),\\ldots,s_n(A)$ satisfy $s_1^p(A)+\\ldots+s_n^p(A)\\leq 1$, where $p>0$. Saint Raymond [Studia Math.\\ 80, 63--75, 1984] showed that the limit $$ \\lim_{n\\to\\infty} n^{1/2 + 1/p} \\big(\\text{Vol}\\, B_p^n(\\mathbb{R})\\big)^{1/n^2} $$ exists in $(0,\\infty)$ and provided both lower and upper bounds. In this paper we determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03467","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":"2026-05-18T00:18:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"769bio5lQtbPgupezhHwkUvmhZwsDmAwMXlP9aVkT+1uJKcjxCqSYtNiu1WQ3oko3usLU/f4xVIFQo2XoKB9DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:31:08.706193Z"},"content_sha256":"64e8941ce83ca4c3ada23a1d6650ec67b998bb7c09cbb9d4de8126a358228374","schema_version":"1.0","event_id":"sha256:64e8941ce83ca4c3ada23a1d6650ec67b998bb7c09cbb9d4de8126a358228374"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XYHOCIQKNYMD332XKRFRFDD532/bundle.json","state_url":"https://pith.science/pith/XYHOCIQKNYMD332XKRFRFDD532/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XYHOCIQKNYMD332XKRFRFDD532/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":"2026-05-27T04:31:08Z","links":{"resolver":"https://pith.science/pith/XYHOCIQKNYMD332XKRFRFDD532","bundle":"https://pith.science/pith/XYHOCIQKNYMD332XKRFRFDD532/bundle.json","state":"https://pith.science/pith/XYHOCIQKNYMD332XKRFRFDD532/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XYHOCIQKNYMD332XKRFRFDD532/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:XYHOCIQKNYMD332XKRFRFDD532","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":"51dadb88de580292028cc6b0070cf27a8e6165ad8be5d14574585edd03c513a5","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-10T11:58:51Z","title_canon_sha256":"d9450e2fd833d7acf9d252f06d50eb7ce5b5657579de3243ff09af6dee000365"},"schema_version":"1.0","source":{"id":"1804.03467","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.03467","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"arxiv_version","alias_value":"1804.03467v1","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.03467","created_at":"2026-05-18T00:18:48Z"},{"alias_kind":"pith_short_12","alias_value":"XYHOCIQKNYMD","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"XYHOCIQKNYMD332X","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"XYHOCIQK","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:64e8941ce83ca4c3ada23a1d6650ec67b998bb7c09cbb9d4de8126a358228374","target":"graph","created_at":"2026-05-18T00:18:48Z","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":"The unit ball $B_p^n(\\mathbb{R})$ of the finite-dimensional Schatten trace class $\\mathcal S_p^n$ consists of all real $n\\times n$ matrices $A$ whose singular values $s_1(A),\\ldots,s_n(A)$ satisfy $s_1^p(A)+\\ldots+s_n^p(A)\\leq 1$, where $p>0$. Saint Raymond [Studia Math.\\ 80, 63--75, 1984] showed that the limit $$ \\lim_{n\\to\\infty} n^{1/2 + 1/p} \\big(\\text{Vol}\\, B_p^n(\\mathbb{R})\\big)^{1/n^2} $$ exists in $(0,\\infty)$ and provided both lower and upper bounds. In this paper we determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields.","authors_text":"Christoph Thaele, Joscha Prochno, Zakhar Kabluchko","cross_cats":["math.MG","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-10T11:58:51Z","title":"Exact asymptotic volume and volume ratio of Schatten unit balls"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03467","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":"sha256:66f83d8fbe8a76085d750b51763531f0d0e02ccf2d0c2889c10923e512d705d6","target":"record","created_at":"2026-05-18T00:18:48Z","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":"51dadb88de580292028cc6b0070cf27a8e6165ad8be5d14574585edd03c513a5","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-04-10T11:58:51Z","title_canon_sha256":"d9450e2fd833d7acf9d252f06d50eb7ce5b5657579de3243ff09af6dee000365"},"schema_version":"1.0","source":{"id":"1804.03467","kind":"arxiv","version":1}},"canonical_sha256":"be0ee1220a6e183def57544b128c7dde9882ff066fde9ed6cd62a5eb820f0ef8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"be0ee1220a6e183def57544b128c7dde9882ff066fde9ed6cd62a5eb820f0ef8","first_computed_at":"2026-05-18T00:18:48.100944Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:18:48.100944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CA/ocsmFePAKqpUNEDqI7fMUTA/HZfl7UcW15GJB/KuoSnWZQC8rc9q7Fz3kuDKorqUXkVJf4duQ+2GsH0lKCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:18:48.101715Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.03467","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:66f83d8fbe8a76085d750b51763531f0d0e02ccf2d0c2889c10923e512d705d6","sha256:64e8941ce83ca4c3ada23a1d6650ec67b998bb7c09cbb9d4de8126a358228374"],"state_sha256":"2f0c13d8d854b1e68c66104f27573afd88265a5e9a4ba576494fba27a568d76f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SEJGJddHIwlHRSTPIgpPS+zGa+p4Nw53SlK5x6UAOJQ9f/D+P12XgyKYyeY9RHOxM8Idp8yUogiGp3R5w9rFCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T04:31:08.709031Z","bundle_sha256":"01013b9349b7b805be7cf4240b6cb638933b4d8a981ea3ccfa3bf78013afcb6b"}}