{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:SDOY4JUBJRFOWXTZHRVYL2WQJT","short_pith_number":"pith:SDOY4JUB","canonical_record":{"source":{"id":"2506.04541","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-05T01:22:02Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"5432801cb605e955fa86feb04c067f139f6db0d916907df9102c792440647056","abstract_canon_sha256":"f145ce2e449ea589f6c3417c785ab12a8cba2fa99ff367379bce26d2d64a4ce7"},"schema_version":"1.0"},"canonical_sha256":"90dd8e26814c4aeb5e793c6b85ead04cdd834446501327ebc1ac22381e56bab0","source":{"kind":"arxiv","id":"2506.04541","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.04541","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"arxiv_version","alias_value":"2506.04541v1","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.04541","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"pith_short_12","alias_value":"SDOY4JUBJRFO","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"pith_short_16","alias_value":"SDOY4JUBJRFOWXTZ","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"pith_short_8","alias_value":"SDOY4JUB","created_at":"2026-07-05T11:16:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:SDOY4JUBJRFOWXTZHRVYL2WQJT","target":"record","payload":{"canonical_record":{"source":{"id":"2506.04541","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-05T01:22:02Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"5432801cb605e955fa86feb04c067f139f6db0d916907df9102c792440647056","abstract_canon_sha256":"f145ce2e449ea589f6c3417c785ab12a8cba2fa99ff367379bce26d2d64a4ce7"},"schema_version":"1.0"},"canonical_sha256":"90dd8e26814c4aeb5e793c6b85ead04cdd834446501327ebc1ac22381e56bab0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:16:16.336942Z","signature_b64":"gAe8pVwD1FrtpQW9XcZy3tclZ9Zcrd250ihkRMJgMtnE/Nb7Ps/uOcoTbsuSqUSjanP7r3+SDTD5b9TAdSZxCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90dd8e26814c4aeb5e793c6b85ead04cdd834446501327ebc1ac22381e56bab0","last_reissued_at":"2026-07-05T11:16:16.336448Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:16:16.336448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.04541","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":"2026-07-05T11:16:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SJxKnfKJa8OSmR0+KoUrhNeCtXnvtKfNUex653jHWVuesybmx/QukbdcZFkRQQv+TJO6WVWyuv6aQlmTkeTSDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T22:36:39.582188Z"},"content_sha256":"9c29e3cde59a15d7cd96d638b2b52469ee5cf278885520ea6afbbbbe1879eaf1","schema_version":"1.0","event_id":"sha256:9c29e3cde59a15d7cd96d638b2b52469ee5cf278885520ea6afbbbbe1879eaf1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:SDOY4JUBJRFOWXTZHRVYL2WQJT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.FA","authors_text":"Josu\\'e I. Rios-Cangas","submitted_at":"2025-06-05T01:22:02Z","abstract_excerpt":"This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\\mathcal H$, and provide a characterization of all inner products by means of positive operators in $\\mathcal {B(H)}$. Next, we establish necessary and sufficient conditions for an operator in $\\mathcal B(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04541","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/2506.04541/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":"2026-07-05T11:16:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yM6BEyoCgxp/du5KcaTgJKqne+aZYkYHRN+GjKzJsQ7UWyKihU33c2VtovMMf37XceIh/kg26u8Yd60MFiX5Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T22:36:39.583306Z"},"content_sha256":"acec9a4d7f14729bc2b2d64b7c79802718475d22dae8a0a98c80812baff4d934","schema_version":"1.0","event_id":"sha256:acec9a4d7f14729bc2b2d64b7c79802718475d22dae8a0a98c80812baff4d934"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SDOY4JUBJRFOWXTZHRVYL2WQJT/bundle.json","state_url":"https://pith.science/pith/SDOY4JUBJRFOWXTZHRVYL2WQJT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SDOY4JUBJRFOWXTZHRVYL2WQJT/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":"2026-08-07T22:36:39Z","links":{"resolver":"https://pith.science/pith/SDOY4JUBJRFOWXTZHRVYL2WQJT","bundle":"https://pith.science/pith/SDOY4JUBJRFOWXTZHRVYL2WQJT/bundle.json","state":"https://pith.science/pith/SDOY4JUBJRFOWXTZHRVYL2WQJT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SDOY4JUBJRFOWXTZHRVYL2WQJT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:SDOY4JUBJRFOWXTZHRVYL2WQJT","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":"f145ce2e449ea589f6c3417c785ab12a8cba2fa99ff367379bce26d2d64a4ce7","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-05T01:22:02Z","title_canon_sha256":"5432801cb605e955fa86feb04c067f139f6db0d916907df9102c792440647056"},"schema_version":"1.0","source":{"id":"2506.04541","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.04541","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"arxiv_version","alias_value":"2506.04541v1","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.04541","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"pith_short_12","alias_value":"SDOY4JUBJRFO","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"pith_short_16","alias_value":"SDOY4JUBJRFOWXTZ","created_at":"2026-07-05T11:16:16Z"},{"alias_kind":"pith_short_8","alias_value":"SDOY4JUB","created_at":"2026-07-05T11:16:16Z"}],"graph_snapshots":[{"event_id":"sha256:acec9a4d7f14729bc2b2d64b7c79802718475d22dae8a0a98c80812baff4d934","target":"graph","created_at":"2026-07-05T11:16:16Z","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/2506.04541/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\\mathcal H$, and provide a characterization of all inner products by means of positive operators in $\\mathcal {B(H)}$. Next, we establish necessary and sufficient conditions for an operator in $\\mathcal B(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$.","authors_text":"Josu\\'e I. Rios-Cangas","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-05T01:22:02Z","title":"Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04541","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":"sha256:9c29e3cde59a15d7cd96d638b2b52469ee5cf278885520ea6afbbbbe1879eaf1","target":"record","created_at":"2026-07-05T11:16:16Z","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":"f145ce2e449ea589f6c3417c785ab12a8cba2fa99ff367379bce26d2d64a4ce7","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-05T01:22:02Z","title_canon_sha256":"5432801cb605e955fa86feb04c067f139f6db0d916907df9102c792440647056"},"schema_version":"1.0","source":{"id":"2506.04541","kind":"arxiv","version":1}},"canonical_sha256":"90dd8e26814c4aeb5e793c6b85ead04cdd834446501327ebc1ac22381e56bab0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"90dd8e26814c4aeb5e793c6b85ead04cdd834446501327ebc1ac22381e56bab0","first_computed_at":"2026-07-05T11:16:16.336448Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:16:16.336448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gAe8pVwD1FrtpQW9XcZy3tclZ9Zcrd250ihkRMJgMtnE/Nb7Ps/uOcoTbsuSqUSjanP7r3+SDTD5b9TAdSZxCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:16:16.336942Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.04541","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9c29e3cde59a15d7cd96d638b2b52469ee5cf278885520ea6afbbbbe1879eaf1","sha256:acec9a4d7f14729bc2b2d64b7c79802718475d22dae8a0a98c80812baff4d934"],"state_sha256":"c4c329c65817a9e732b41f707b7a5aed6ea9e5cb8517555612e6265ef8555d04"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QN6HOpqf8jB51o6wZ23d6iuGVMT8snXJBNUNxrjcSYyQ5xBJ6kAcFRT7jbQRWTxPQtF6wPwqY9RK55bcno21DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T22:36:39.589619Z","bundle_sha256":"3735a9c5299a74beac631a4ff1f1ffe948f9166f8714752918e75376cccff823"}}