{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:WBEXKAQFYQFTJBDLYCHLGKTF67","short_pith_number":"pith:WBEXKAQF","canonical_record":{"source":{"id":"1908.08359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-22T13:27:39Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"2503ae209eec54cfbbbf67ab61410b883ca1b14813e1ae062ee051ffd6c86190","abstract_canon_sha256":"868883247efb4881042dc829a0ecc4a052d6d149321e147136c672f1b8074c50"},"schema_version":"1.0"},"canonical_sha256":"b049750205c40b34846bc08eb32a65f7e2f41101f0ee939368d960fce7c1191b","source":{"kind":"arxiv","id":"1908.08359","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08359","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08359v1","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08359","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"pith_short_12","alias_value":"WBEXKAQFYQFT","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"pith_short_16","alias_value":"WBEXKAQFYQFTJBDL","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"pith_short_8","alias_value":"WBEXKAQF","created_at":"2026-07-05T00:48:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:WBEXKAQFYQFTJBDLYCHLGKTF67","target":"record","payload":{"canonical_record":{"source":{"id":"1908.08359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-22T13:27:39Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"2503ae209eec54cfbbbf67ab61410b883ca1b14813e1ae062ee051ffd6c86190","abstract_canon_sha256":"868883247efb4881042dc829a0ecc4a052d6d149321e147136c672f1b8074c50"},"schema_version":"1.0"},"canonical_sha256":"b049750205c40b34846bc08eb32a65f7e2f41101f0ee939368d960fce7c1191b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:48:17.918702Z","signature_b64":"TUo+mgwZNwcNJ2wxGOMEpAS+xzLRFGkVzYhw/5YfN4sWFhlHhSpBWw0GNUnBIssCWCzCjh58X5ailwonvhRWAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b049750205c40b34846bc08eb32a65f7e2f41101f0ee939368d960fce7c1191b","last_reissued_at":"2026-07-05T00:48:17.918275Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:48:17.918275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.08359","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":"2026-07-05T00:48:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IH/NwaxwIXZdZTRLHF9WNIO8RcX0hmpVbiwqbEMpIPQkiESP4+QOIYNF3morTDwllMHj95Kul9iSsbdWqyADBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:24:08.636563Z"},"content_sha256":"806bee97b7bcce015936c340a00bd5f7342daea92ac837484210fb94eceb6185","schema_version":"1.0","event_id":"sha256:806bee97b7bcce015936c340a00bd5f7342daea92ac837484210fb94eceb6185"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:WBEXKAQFYQFTJBDLYCHLGKTF67","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two variations on the periscope theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.DG","authors_text":"Serge Tabachnikov","submitted_at":"2019-08-22T13:27:39Z","abstract_excerpt":"A spherical periscope in multi-dimensional space is a system of two ideal mirrors that reflect the rays emanating from a fixed point to the rays coming back to the same point, and a reversed periscope is a system of two mirrors that reflect the rays having a fixed direction to the rays having the opposite direction. We describe the local diffeomorphisms of the wave fronts (spherical, in the former, and flat, in the latter cases), induced by these 2-mirror systems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08359","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/1908.08359/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":"2026-07-05T00:48:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qMX0ntPU+F1p8b9gUIH8t+fLlWlMhNGbXnXe43NaR2cy/FJYQfPfJ2jrXjyEZN64k5xuFelUOBAv6CJl7X+rCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:24:08.637058Z"},"content_sha256":"02dece0c292cf93739aaca986b528acb6c402a5de4ca40e3c7a84746b9a62b92","schema_version":"1.0","event_id":"sha256:02dece0c292cf93739aaca986b528acb6c402a5de4ca40e3c7a84746b9a62b92"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WBEXKAQFYQFTJBDLYCHLGKTF67/bundle.json","state_url":"https://pith.science/pith/WBEXKAQFYQFTJBDLYCHLGKTF67/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WBEXKAQFYQFTJBDLYCHLGKTF67/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":"2026-08-16T05:24:08Z","links":{"resolver":"https://pith.science/pith/WBEXKAQFYQFTJBDLYCHLGKTF67","bundle":"https://pith.science/pith/WBEXKAQFYQFTJBDLYCHLGKTF67/bundle.json","state":"https://pith.science/pith/WBEXKAQFYQFTJBDLYCHLGKTF67/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WBEXKAQFYQFTJBDLYCHLGKTF67/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:WBEXKAQFYQFTJBDLYCHLGKTF67","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":"868883247efb4881042dc829a0ecc4a052d6d149321e147136c672f1b8074c50","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-22T13:27:39Z","title_canon_sha256":"2503ae209eec54cfbbbf67ab61410b883ca1b14813e1ae062ee051ffd6c86190"},"schema_version":"1.0","source":{"id":"1908.08359","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08359","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08359v1","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08359","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"pith_short_12","alias_value":"WBEXKAQFYQFT","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"pith_short_16","alias_value":"WBEXKAQFYQFTJBDL","created_at":"2026-07-05T00:48:17Z"},{"alias_kind":"pith_short_8","alias_value":"WBEXKAQF","created_at":"2026-07-05T00:48:17Z"}],"graph_snapshots":[{"event_id":"sha256:02dece0c292cf93739aaca986b528acb6c402a5de4ca40e3c7a84746b9a62b92","target":"graph","created_at":"2026-07-05T00:48:17Z","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/1908.08359/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A spherical periscope in multi-dimensional space is a system of two ideal mirrors that reflect the rays emanating from a fixed point to the rays coming back to the same point, and a reversed periscope is a system of two mirrors that reflect the rays having a fixed direction to the rays having the opposite direction. We describe the local diffeomorphisms of the wave fronts (spherical, in the former, and flat, in the latter cases), induced by these 2-mirror systems.","authors_text":"Serge Tabachnikov","cross_cats":["math.SG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-22T13:27:39Z","title":"Two variations on the periscope theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08359","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":"sha256:806bee97b7bcce015936c340a00bd5f7342daea92ac837484210fb94eceb6185","target":"record","created_at":"2026-07-05T00:48:17Z","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":"868883247efb4881042dc829a0ecc4a052d6d149321e147136c672f1b8074c50","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-08-22T13:27:39Z","title_canon_sha256":"2503ae209eec54cfbbbf67ab61410b883ca1b14813e1ae062ee051ffd6c86190"},"schema_version":"1.0","source":{"id":"1908.08359","kind":"arxiv","version":1}},"canonical_sha256":"b049750205c40b34846bc08eb32a65f7e2f41101f0ee939368d960fce7c1191b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b049750205c40b34846bc08eb32a65f7e2f41101f0ee939368d960fce7c1191b","first_computed_at":"2026-07-05T00:48:17.918275Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:48:17.918275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TUo+mgwZNwcNJ2wxGOMEpAS+xzLRFGkVzYhw/5YfN4sWFhlHhSpBWw0GNUnBIssCWCzCjh58X5ailwonvhRWAA==","signature_status":"signed_v1","signed_at":"2026-07-05T00:48:17.918702Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.08359","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:806bee97b7bcce015936c340a00bd5f7342daea92ac837484210fb94eceb6185","sha256:02dece0c292cf93739aaca986b528acb6c402a5de4ca40e3c7a84746b9a62b92"],"state_sha256":"c750616178b43d5fb926d02de4747652b0d1e25b9e968e4a6095b4ce31362321"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c1HjkyFNNIWab7lSE+XIqF0sVOCmDcW4c+zMptGRNf0n4WJB5UplMgAJKdXOC6/mAZaPkZcvH0v0BBfucCwWAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T05:24:08.640444Z","bundle_sha256":"11cb33429914b7afe12fed1483a395b52ad0cdd2106a68fa2c21ca88557a7a24"}}