{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:E64P4S6PKZT2WVYT5AFNPI7XEY","short_pith_number":"pith:E64P4S6P","canonical_record":{"source":{"id":"1503.08888","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-03-31T02:31:51Z","cross_cats_sorted":[],"title_canon_sha256":"4d2cf48b896fe4820c12b3cabf529eb74751dc74ef281bad035ebcfa89e1cfa4","abstract_canon_sha256":"86cd05dddf8688d2575516f30a795d7d9ccd7d1f4719bc4031a38e4c2ac22434"},"schema_version":"1.0"},"canonical_sha256":"27b8fe4bcf5667ab5713e80ad7a3f726348195a16b7373d131f9f2d4eaec890f","source":{"kind":"arxiv","id":"1503.08888","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.08888","created_at":"2026-05-18T02:19:51Z"},{"alias_kind":"arxiv_version","alias_value":"1503.08888v1","created_at":"2026-05-18T02:19:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08888","created_at":"2026-05-18T02:19:51Z"},{"alias_kind":"pith_short_12","alias_value":"E64P4S6PKZT2","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"E64P4S6PKZT2WVYT","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"E64P4S6P","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:E64P4S6PKZT2WVYT5AFNPI7XEY","target":"record","payload":{"canonical_record":{"source":{"id":"1503.08888","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-03-31T02:31:51Z","cross_cats_sorted":[],"title_canon_sha256":"4d2cf48b896fe4820c12b3cabf529eb74751dc74ef281bad035ebcfa89e1cfa4","abstract_canon_sha256":"86cd05dddf8688d2575516f30a795d7d9ccd7d1f4719bc4031a38e4c2ac22434"},"schema_version":"1.0"},"canonical_sha256":"27b8fe4bcf5667ab5713e80ad7a3f726348195a16b7373d131f9f2d4eaec890f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:51.928537Z","signature_b64":"x0/aQX4uG339iZOptunSDXnRzDL42HZlsnASIBCmPoUahAj2Ui2d8Xx32Twiqh6H6XRdTL+Vb3EhnZJm1smUCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27b8fe4bcf5667ab5713e80ad7a3f726348195a16b7373d131f9f2d4eaec890f","last_reissued_at":"2026-05-18T02:19:51.927999Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:51.927999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.08888","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":"2026-05-18T02:19:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kj82CZCwR7WOCozX/0wpefdgOXy3v1EGebOHYdbiXrg06WyH0fxZT9kj8/TaXQXFHI777T5KncL361kGSyy8Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T09:51:22.670585Z"},"content_sha256":"424af7b0464813beda76c91c0ab82af081a9a4d841f89f6cbe600df1eae79cb8","schema_version":"1.0","event_id":"sha256:424af7b0464813beda76c91c0ab82af081a9a4d841f89f6cbe600df1eae79cb8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:E64P4S6PKZT2WVYT5AFNPI7XEY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Caustics of world hyper-sheets in the Minkowski space-time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Shyuichi Izumiya","submitted_at":"2015-03-31T02:31:51Z","abstract_excerpt":"In the Minkowski space-time, a world hyper-sheet is a timelike hypersurface consisting of a one-parameter family of spacelike submanifolds. Recently, Bousso and Randall introduced the notion of caustics of world hyper-sheets in order to define the notion of holographic domains in space-time. Here, we give a mathematical framework for describing the caustics of world hyper-sheets in the Minkowski space-time. As a consequence, we investigate the singularities of the caustics of world hyper-sheets and whose geometrical meanings. Although the Minkowski space-time has zero gravity, this framework g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08888","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":"2026-05-18T02:19:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uXPCHOZWarYuz8S/jfqIiXm58gXwKiBwdv+wQPegwR73rPPDZKSf9xalHmgQT1emYbuAie+XdQM4C3jaLdtLDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T09:51:22.670933Z"},"content_sha256":"8fc5255ca4df07f5554a88180e501b327d4bc27fd062a3f81a7fc89a4584b712","schema_version":"1.0","event_id":"sha256:8fc5255ca4df07f5554a88180e501b327d4bc27fd062a3f81a7fc89a4584b712"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E64P4S6PKZT2WVYT5AFNPI7XEY/bundle.json","state_url":"https://pith.science/pith/E64P4S6PKZT2WVYT5AFNPI7XEY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E64P4S6PKZT2WVYT5AFNPI7XEY/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":"2026-06-03T09:51:22Z","links":{"resolver":"https://pith.science/pith/E64P4S6PKZT2WVYT5AFNPI7XEY","bundle":"https://pith.science/pith/E64P4S6PKZT2WVYT5AFNPI7XEY/bundle.json","state":"https://pith.science/pith/E64P4S6PKZT2WVYT5AFNPI7XEY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E64P4S6PKZT2WVYT5AFNPI7XEY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:E64P4S6PKZT2WVYT5AFNPI7XEY","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":"86cd05dddf8688d2575516f30a795d7d9ccd7d1f4719bc4031a38e4c2ac22434","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-03-31T02:31:51Z","title_canon_sha256":"4d2cf48b896fe4820c12b3cabf529eb74751dc74ef281bad035ebcfa89e1cfa4"},"schema_version":"1.0","source":{"id":"1503.08888","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.08888","created_at":"2026-05-18T02:19:51Z"},{"alias_kind":"arxiv_version","alias_value":"1503.08888v1","created_at":"2026-05-18T02:19:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08888","created_at":"2026-05-18T02:19:51Z"},{"alias_kind":"pith_short_12","alias_value":"E64P4S6PKZT2","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"E64P4S6PKZT2WVYT","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"E64P4S6P","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:8fc5255ca4df07f5554a88180e501b327d4bc27fd062a3f81a7fc89a4584b712","target":"graph","created_at":"2026-05-18T02:19:51Z","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":"In the Minkowski space-time, a world hyper-sheet is a timelike hypersurface consisting of a one-parameter family of spacelike submanifolds. Recently, Bousso and Randall introduced the notion of caustics of world hyper-sheets in order to define the notion of holographic domains in space-time. Here, we give a mathematical framework for describing the caustics of world hyper-sheets in the Minkowski space-time. As a consequence, we investigate the singularities of the caustics of world hyper-sheets and whose geometrical meanings. Although the Minkowski space-time has zero gravity, this framework g","authors_text":"Shyuichi Izumiya","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-03-31T02:31:51Z","title":"Caustics of world hyper-sheets in the Minkowski space-time"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08888","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":"sha256:424af7b0464813beda76c91c0ab82af081a9a4d841f89f6cbe600df1eae79cb8","target":"record","created_at":"2026-05-18T02:19:51Z","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":"86cd05dddf8688d2575516f30a795d7d9ccd7d1f4719bc4031a38e4c2ac22434","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-03-31T02:31:51Z","title_canon_sha256":"4d2cf48b896fe4820c12b3cabf529eb74751dc74ef281bad035ebcfa89e1cfa4"},"schema_version":"1.0","source":{"id":"1503.08888","kind":"arxiv","version":1}},"canonical_sha256":"27b8fe4bcf5667ab5713e80ad7a3f726348195a16b7373d131f9f2d4eaec890f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"27b8fe4bcf5667ab5713e80ad7a3f726348195a16b7373d131f9f2d4eaec890f","first_computed_at":"2026-05-18T02:19:51.927999Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:51.927999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x0/aQX4uG339iZOptunSDXnRzDL42HZlsnASIBCmPoUahAj2Ui2d8Xx32Twiqh6H6XRdTL+Vb3EhnZJm1smUCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:51.928537Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.08888","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:424af7b0464813beda76c91c0ab82af081a9a4d841f89f6cbe600df1eae79cb8","sha256:8fc5255ca4df07f5554a88180e501b327d4bc27fd062a3f81a7fc89a4584b712"],"state_sha256":"8ce6da694849b32a2e84134579cb3f2482017cfb0ea04177abb84b90051f27b2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h2fMYiweaHlkeqUW82odl3JvBcsn/FrULNMKfbel/oEoL1uy60kJgs+v/DzcdfwX3VQCHvi9jTd+unxQPZOJDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T09:51:22.672911Z","bundle_sha256":"0c34fef0f349f947d033250abbb2ece34fb55412b3edbedea5f600fc43f22b31"}}