{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:JMMFZVQZ4D43CFIENVCNEANSOR","short_pith_number":"pith:JMMFZVQZ","canonical_record":{"source":{"id":"2606.04642","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-03T09:11:37Z","cross_cats_sorted":[],"title_canon_sha256":"f14fb79c16198a7fa15a2711f6e923eb91cdf4bb5572b437bb07ef787a056e4f","abstract_canon_sha256":"e7347460bce94fd5e002e90e5d10a613add73cf0a592307203f33556f83c4ace"},"schema_version":"1.0"},"canonical_sha256":"4b185cd619e0f9b115046d44d201b27457c5fce5022929bd702796cb1a500ca9","source":{"kind":"arxiv","id":"2606.04642","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.04642","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"arxiv_version","alias_value":"2606.04642v1","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04642","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"pith_short_12","alias_value":"JMMFZVQZ4D43","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"pith_short_16","alias_value":"JMMFZVQZ4D43CFIE","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"pith_short_8","alias_value":"JMMFZVQZ","created_at":"2026-06-04T01:09:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:JMMFZVQZ4D43CFIENVCNEANSOR","target":"record","payload":{"canonical_record":{"source":{"id":"2606.04642","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-03T09:11:37Z","cross_cats_sorted":[],"title_canon_sha256":"f14fb79c16198a7fa15a2711f6e923eb91cdf4bb5572b437bb07ef787a056e4f","abstract_canon_sha256":"e7347460bce94fd5e002e90e5d10a613add73cf0a592307203f33556f83c4ace"},"schema_version":"1.0"},"canonical_sha256":"4b185cd619e0f9b115046d44d201b27457c5fce5022929bd702796cb1a500ca9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T01:09:22.749341Z","signature_b64":"UEGPr1bn4wcURxRh6Irx5Hgzfj23fxSLXPQf6+xeIkjzf1jwVQf7+qcuwkw9BSf75Xy/nQDt3PSQc04t5vUWCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b185cd619e0f9b115046d44d201b27457c5fce5022929bd702796cb1a500ca9","last_reissued_at":"2026-06-04T01:09:22.748470Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T01:09:22.748470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.04642","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":"2026-06-04T01:09:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v8FxYQp6OL/mhSQWDOJUQU65v0cf5o4rV0FfWXC6MnKYqoC3XF7hlvcqw6bF/Ph0kfn/0XkaGYPrp43LHYa3Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T06:06:59.721146Z"},"content_sha256":"9a70eaa1526f0d54bff9e49bfa556c0787b18023f56ec3e0429fd40627a7c489","schema_version":"1.0","event_id":"sha256:9a70eaa1526f0d54bff9e49bfa556c0787b18023f56ec3e0429fd40627a7c489"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:JMMFZVQZ4D43CFIENVCNEANSOR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Products of involutons in symplectic groups II","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Klaus Nielsen","submitted_at":"2026-06-03T09:11:37Z","abstract_excerpt":"We show that the symplectic groups Sp(2n, K) over a field $K$ are 5-reflectional if $2n \\ge 4$. If $n$ is even and $(2n, K) \\ne (4, {\\rm GF}(3))$ they are even 4-reflectional."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04642","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/2606.04642/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":"2026-06-04T01:09:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gYimYxBC2379bapXPecWn/sqh6U54fYc5xtIvF0b/RALZz4kxR3aFWyaB/ahrLvRl8j17PIvjyCy8kJJVGiXBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T06:06:59.721975Z"},"content_sha256":"5f7d272bc6e2bc2078903894794ff38b770ba0e377ea9a8477215ab972094cd5","schema_version":"1.0","event_id":"sha256:5f7d272bc6e2bc2078903894794ff38b770ba0e377ea9a8477215ab972094cd5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JMMFZVQZ4D43CFIENVCNEANSOR/bundle.json","state_url":"https://pith.science/pith/JMMFZVQZ4D43CFIENVCNEANSOR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JMMFZVQZ4D43CFIENVCNEANSOR/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":"2026-06-08T06:06:59Z","links":{"resolver":"https://pith.science/pith/JMMFZVQZ4D43CFIENVCNEANSOR","bundle":"https://pith.science/pith/JMMFZVQZ4D43CFIENVCNEANSOR/bundle.json","state":"https://pith.science/pith/JMMFZVQZ4D43CFIENVCNEANSOR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JMMFZVQZ4D43CFIENVCNEANSOR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:JMMFZVQZ4D43CFIENVCNEANSOR","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":"e7347460bce94fd5e002e90e5d10a613add73cf0a592307203f33556f83c4ace","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-03T09:11:37Z","title_canon_sha256":"f14fb79c16198a7fa15a2711f6e923eb91cdf4bb5572b437bb07ef787a056e4f"},"schema_version":"1.0","source":{"id":"2606.04642","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.04642","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"arxiv_version","alias_value":"2606.04642v1","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04642","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"pith_short_12","alias_value":"JMMFZVQZ4D43","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"pith_short_16","alias_value":"JMMFZVQZ4D43CFIE","created_at":"2026-06-04T01:09:22Z"},{"alias_kind":"pith_short_8","alias_value":"JMMFZVQZ","created_at":"2026-06-04T01:09:22Z"}],"graph_snapshots":[{"event_id":"sha256:5f7d272bc6e2bc2078903894794ff38b770ba0e377ea9a8477215ab972094cd5","target":"graph","created_at":"2026-06-04T01:09:22Z","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/2606.04642/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that the symplectic groups Sp(2n, K) over a field $K$ are 5-reflectional if $2n \\ge 4$. If $n$ is even and $(2n, K) \\ne (4, {\\rm GF}(3))$ they are even 4-reflectional.","authors_text":"Klaus Nielsen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-03T09:11:37Z","title":"Products of involutons in symplectic groups II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04642","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":"sha256:9a70eaa1526f0d54bff9e49bfa556c0787b18023f56ec3e0429fd40627a7c489","target":"record","created_at":"2026-06-04T01:09:22Z","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":"e7347460bce94fd5e002e90e5d10a613add73cf0a592307203f33556f83c4ace","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-03T09:11:37Z","title_canon_sha256":"f14fb79c16198a7fa15a2711f6e923eb91cdf4bb5572b437bb07ef787a056e4f"},"schema_version":"1.0","source":{"id":"2606.04642","kind":"arxiv","version":1}},"canonical_sha256":"4b185cd619e0f9b115046d44d201b27457c5fce5022929bd702796cb1a500ca9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b185cd619e0f9b115046d44d201b27457c5fce5022929bd702796cb1a500ca9","first_computed_at":"2026-06-04T01:09:22.748470Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T01:09:22.748470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UEGPr1bn4wcURxRh6Irx5Hgzfj23fxSLXPQf6+xeIkjzf1jwVQf7+qcuwkw9BSf75Xy/nQDt3PSQc04t5vUWCQ==","signature_status":"signed_v1","signed_at":"2026-06-04T01:09:22.749341Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.04642","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a70eaa1526f0d54bff9e49bfa556c0787b18023f56ec3e0429fd40627a7c489","sha256:5f7d272bc6e2bc2078903894794ff38b770ba0e377ea9a8477215ab972094cd5"],"state_sha256":"86a7788e4ad1151f483fc1c2db90f7d45d40c2b2d4bd7da91e9429624741aa88"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FanmE5u5GSLECR00vlWdW/ZQIYX+K7xPzcnZNUGbQOm8jfR7jlDhti65hXb+OJJf5/wI7syNW2RYWHoON3lzBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T06:06:59.727853Z","bundle_sha256":"a85c582c4ca6cdcbba0f2f7317d953830e987b9734dd85f48f5ac5eeee2a9684"}}