{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:UWLQZX63CTXLEH54GW7KIKZ4UD","short_pith_number":"pith:UWLQZX63","canonical_record":{"source":{"id":"1509.01030","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-09-03T10:55:24Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"7aa8bb2040d8ceac98ebd0c178c5d391d095a0337b577fbda64a544c130b6e71","abstract_canon_sha256":"9947360a08ae240b7e16fba0825d3b6e340d395312de86fae936c1bec2ca7b26"},"schema_version":"1.0"},"canonical_sha256":"a5970cdfdb14eeb21fbc35bea42b3ca0c5250c94a2f8703347c3cf36233c3ef9","source":{"kind":"arxiv","id":"1509.01030","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01030","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01030v1","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01030","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"pith_short_12","alias_value":"UWLQZX63CTXL","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UWLQZX63CTXLEH54","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UWLQZX63","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:UWLQZX63CTXLEH54GW7KIKZ4UD","target":"record","payload":{"canonical_record":{"source":{"id":"1509.01030","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-09-03T10:55:24Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"7aa8bb2040d8ceac98ebd0c178c5d391d095a0337b577fbda64a544c130b6e71","abstract_canon_sha256":"9947360a08ae240b7e16fba0825d3b6e340d395312de86fae936c1bec2ca7b26"},"schema_version":"1.0"},"canonical_sha256":"a5970cdfdb14eeb21fbc35bea42b3ca0c5250c94a2f8703347c3cf36233c3ef9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:04.505735Z","signature_b64":"DXh2zExWJsz1VbQSQWY0TR+tZohp5r4ox2XholN85oGyuzpNOcIdSyyWpD+5BZFITBJtWMNejyCoTtCITs/gCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5970cdfdb14eeb21fbc35bea42b3ca0c5250c94a2f8703347c3cf36233c3ef9","last_reissued_at":"2026-05-18T01:34:04.505049Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:04.505049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.01030","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":"2026-05-18T01:34:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iRnNQBfBUCXuSU/vuVblHopXV1S0XtjlpsilfSQ6osZT9avZ03wXvuuZGv0JBNBEWuvcR5WbQK2n91iQr+4TCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T02:19:50.486171Z"},"content_sha256":"c8b715373fe030f554d910ebaab45cd59fdb88d5a22eec4b074f6fe4ed056884","schema_version":"1.0","event_id":"sha256:c8b715373fe030f554d910ebaab45cd59fdb88d5a22eec4b074f6fe4ed056884"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:UWLQZX63CTXLEH54GW7KIKZ4UD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gap Theorem for Separated Sequences without Pain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CV","authors_text":"Alexander Ulanovskii, Anton Baranov, Yurii Belov","submitted_at":"2015-09-03T10:55:24Z","abstract_excerpt":"We give a simple and straightforward proof of the Gap Theorem for separated sequences by A. Poltoratski and M. Mitkovski using the Beurling--Malliavin formula for the radius of completeness."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01030","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":"2026-05-18T01:34:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fezB372mkoCOB6Has3jpOC0rzj2G6/ljQAiMp1plyAnkjqaByRcKi9YZim/n00luv9g30GK17G8EOL1VU9L0AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T02:19:50.486534Z"},"content_sha256":"1a80d70ceb99dd33b833741680cbffd10397f5fdafdbef0573524a24eb4a3d66","schema_version":"1.0","event_id":"sha256:1a80d70ceb99dd33b833741680cbffd10397f5fdafdbef0573524a24eb4a3d66"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UWLQZX63CTXLEH54GW7KIKZ4UD/bundle.json","state_url":"https://pith.science/pith/UWLQZX63CTXLEH54GW7KIKZ4UD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UWLQZX63CTXLEH54GW7KIKZ4UD/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":"2026-05-28T02:19:50Z","links":{"resolver":"https://pith.science/pith/UWLQZX63CTXLEH54GW7KIKZ4UD","bundle":"https://pith.science/pith/UWLQZX63CTXLEH54GW7KIKZ4UD/bundle.json","state":"https://pith.science/pith/UWLQZX63CTXLEH54GW7KIKZ4UD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UWLQZX63CTXLEH54GW7KIKZ4UD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:UWLQZX63CTXLEH54GW7KIKZ4UD","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":"9947360a08ae240b7e16fba0825d3b6e340d395312de86fae936c1bec2ca7b26","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-09-03T10:55:24Z","title_canon_sha256":"7aa8bb2040d8ceac98ebd0c178c5d391d095a0337b577fbda64a544c130b6e71"},"schema_version":"1.0","source":{"id":"1509.01030","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01030","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01030v1","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01030","created_at":"2026-05-18T01:34:04Z"},{"alias_kind":"pith_short_12","alias_value":"UWLQZX63CTXL","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UWLQZX63CTXLEH54","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UWLQZX63","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:1a80d70ceb99dd33b833741680cbffd10397f5fdafdbef0573524a24eb4a3d66","target":"graph","created_at":"2026-05-18T01:34:04Z","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":"We give a simple and straightforward proof of the Gap Theorem for separated sequences by A. Poltoratski and M. Mitkovski using the Beurling--Malliavin formula for the radius of completeness.","authors_text":"Alexander Ulanovskii, Anton Baranov, Yurii Belov","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-09-03T10:55:24Z","title":"Gap Theorem for Separated Sequences without Pain"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01030","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:c8b715373fe030f554d910ebaab45cd59fdb88d5a22eec4b074f6fe4ed056884","target":"record","created_at":"2026-05-18T01:34:04Z","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":"9947360a08ae240b7e16fba0825d3b6e340d395312de86fae936c1bec2ca7b26","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-09-03T10:55:24Z","title_canon_sha256":"7aa8bb2040d8ceac98ebd0c178c5d391d095a0337b577fbda64a544c130b6e71"},"schema_version":"1.0","source":{"id":"1509.01030","kind":"arxiv","version":1}},"canonical_sha256":"a5970cdfdb14eeb21fbc35bea42b3ca0c5250c94a2f8703347c3cf36233c3ef9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5970cdfdb14eeb21fbc35bea42b3ca0c5250c94a2f8703347c3cf36233c3ef9","first_computed_at":"2026-05-18T01:34:04.505049Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:04.505049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DXh2zExWJsz1VbQSQWY0TR+tZohp5r4ox2XholN85oGyuzpNOcIdSyyWpD+5BZFITBJtWMNejyCoTtCITs/gCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:04.505735Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.01030","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8b715373fe030f554d910ebaab45cd59fdb88d5a22eec4b074f6fe4ed056884","sha256:1a80d70ceb99dd33b833741680cbffd10397f5fdafdbef0573524a24eb4a3d66"],"state_sha256":"8504f91ac7a5f852761b6d40f50d016ca741674268fc1a1ef1aa8edad0d11715"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hVG/KKectJ057f8T5UyLvLWPSAPvFiatS+D+YNS0YhEIkegUR503bv0c4HwB+5auEkdFgSPQt/1CzdRVRrNSCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T02:19:50.488636Z","bundle_sha256":"22c4d6ed4f02a3f032ce2d26b539d3643be564d9533d618b2a881ac1fe9ce692"}}