{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2","short_pith_number":"pith:OB4AJ3Y3","canonical_record":{"source":{"id":"1605.04480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-14T23:49:00Z","cross_cats_sorted":[],"title_canon_sha256":"6464152a0773c5363353414a4833b04fb0a66065f8d4175ece2328da67042ace","abstract_canon_sha256":"f28e3ab7880a5525a19556959497fad60bffc239cfb60f34ecc72a4831de447e"},"schema_version":"1.0"},"canonical_sha256":"707804ef1b5053eeaf3fcd58a59972968c14ffa569a14955eb56d11bba84892e","source":{"kind":"arxiv","id":"1605.04480","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.04480","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"arxiv_version","alias_value":"1605.04480v2","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.04480","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"pith_short_12","alias_value":"OB4AJ3Y3KBJ6","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OB4AJ3Y3KBJ65LZ7","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OB4AJ3Y3","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2","target":"record","payload":{"canonical_record":{"source":{"id":"1605.04480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-14T23:49:00Z","cross_cats_sorted":[],"title_canon_sha256":"6464152a0773c5363353414a4833b04fb0a66065f8d4175ece2328da67042ace","abstract_canon_sha256":"f28e3ab7880a5525a19556959497fad60bffc239cfb60f34ecc72a4831de447e"},"schema_version":"1.0"},"canonical_sha256":"707804ef1b5053eeaf3fcd58a59972968c14ffa569a14955eb56d11bba84892e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:37.961817Z","signature_b64":"mB9TYlr8PKXoZ/DZIx5Hxu6Jf62HWvFMOWKwWbxhhK7r4QKV8MNspxMS3Wy4ugNonOnYcDIkoxI74B6qI8RAAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"707804ef1b5053eeaf3fcd58a59972968c14ffa569a14955eb56d11bba84892e","last_reissued_at":"2026-05-18T00:49:37.961189Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:37.961189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.04480","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:49:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h4kDABHUrh7Q2eB6UK4jOopWvTHHEnQ8sO+kVeVMeOgepgOa+8maSgmqw/1exo9HARN8U9rEXshAhNNG6cX8BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T04:42:00.722451Z"},"content_sha256":"9ad88c970d2434d2f38839a9108da897a40f73534eaa7fc5e2a38b1eda6c4f53","schema_version":"1.0","event_id":"sha256:9ad88c970d2434d2f38839a9108da897a40f73534eaa7fc5e2a38b1eda6c4f53"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal Mock Jacobi Theta Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"John F. R. Duncan, Miranda C. N. Cheng","submitted_at":"2016-05-14T23:49:00Z","abstract_excerpt":"We classify the optimal mock Jacobi forms of weight one with rational coefficients. The space they span is thirty-four-dimensional, and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane. We show that their Fourier coefficients can be expressed explicitly in terms of singular moduli, and obtain positivity conditions which distinguish the optimal mock Jacobi forms that appear in umbral moonshine. We find that all of Ramanujan's mock theta functions can be expressed simply in terms of the optimal mock Jacobi forms with rational coefficients."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04480","kind":"arxiv","version":2},"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":"2026-05-18T00:49:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BaVqh0gVIVVBGGdl5A1jyIJmv5VEog+b9hjEsWru87qYNkcb3iP42W7znoQDJv/DW9CTueepdiQfTtsYVOFLDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T04:42:00.722797Z"},"content_sha256":"3d1216beba43037f72524812e70a28cfe8dbd4ccef468bf060bc9e58df23e3bb","schema_version":"1.0","event_id":"sha256:3d1216beba43037f72524812e70a28cfe8dbd4ccef468bf060bc9e58df23e3bb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2/bundle.json","state_url":"https://pith.science/pith/OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2/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":"2026-05-21T04:42:00Z","links":{"resolver":"https://pith.science/pith/OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2","bundle":"https://pith.science/pith/OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2/bundle.json","state":"https://pith.science/pith/OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OB4AJ3Y3KBJ65LZ7ZVMKLGLSS2","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":"f28e3ab7880a5525a19556959497fad60bffc239cfb60f34ecc72a4831de447e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-14T23:49:00Z","title_canon_sha256":"6464152a0773c5363353414a4833b04fb0a66065f8d4175ece2328da67042ace"},"schema_version":"1.0","source":{"id":"1605.04480","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.04480","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"arxiv_version","alias_value":"1605.04480v2","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.04480","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"pith_short_12","alias_value":"OB4AJ3Y3KBJ6","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OB4AJ3Y3KBJ65LZ7","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OB4AJ3Y3","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:3d1216beba43037f72524812e70a28cfe8dbd4ccef468bf060bc9e58df23e3bb","target":"graph","created_at":"2026-05-18T00:49:37Z","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 classify the optimal mock Jacobi forms of weight one with rational coefficients. The space they span is thirty-four-dimensional, and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane. We show that their Fourier coefficients can be expressed explicitly in terms of singular moduli, and obtain positivity conditions which distinguish the optimal mock Jacobi forms that appear in umbral moonshine. We find that all of Ramanujan's mock theta functions can be expressed simply in terms of the optimal mock Jacobi forms with rational coefficients.","authors_text":"John F. R. Duncan, Miranda C. N. Cheng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-14T23:49:00Z","title":"Optimal Mock Jacobi Theta Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04480","kind":"arxiv","version":2},"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":"sha256:9ad88c970d2434d2f38839a9108da897a40f73534eaa7fc5e2a38b1eda6c4f53","target":"record","created_at":"2026-05-18T00:49:37Z","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":"f28e3ab7880a5525a19556959497fad60bffc239cfb60f34ecc72a4831de447e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-14T23:49:00Z","title_canon_sha256":"6464152a0773c5363353414a4833b04fb0a66065f8d4175ece2328da67042ace"},"schema_version":"1.0","source":{"id":"1605.04480","kind":"arxiv","version":2}},"canonical_sha256":"707804ef1b5053eeaf3fcd58a59972968c14ffa569a14955eb56d11bba84892e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"707804ef1b5053eeaf3fcd58a59972968c14ffa569a14955eb56d11bba84892e","first_computed_at":"2026-05-18T00:49:37.961189Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:37.961189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mB9TYlr8PKXoZ/DZIx5Hxu6Jf62HWvFMOWKwWbxhhK7r4QKV8MNspxMS3Wy4ugNonOnYcDIkoxI74B6qI8RAAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:37.961817Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.04480","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9ad88c970d2434d2f38839a9108da897a40f73534eaa7fc5e2a38b1eda6c4f53","sha256:3d1216beba43037f72524812e70a28cfe8dbd4ccef468bf060bc9e58df23e3bb"],"state_sha256":"cc66cc54b8ece11c4289c81c0dd977e5822db905485d5fd4e04857150493211b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Lh2qDHJqdzo9FV4SX9bDFIvIkx2BRK1lUPsar/epS6FAvp0zVDTzZPiPHsrLLjdHomGIawgFlhEA2NsHGsnDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T04:42:00.724848Z","bundle_sha256":"8578a8481dee2119837dc353b17aafef7a85b07cf97f91db810c73a9dcb88932"}}