{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:UWZITQNP3XG5ZXHWIRBA4N3AUL","short_pith_number":"pith:UWZITQNP","canonical_record":{"source":{"id":"1603.00534","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-02T00:41:43Z","cross_cats_sorted":[],"title_canon_sha256":"43a43237c91ae98beff59127efeb54855114b5bd491b3dbe964d301003a3f8a6","abstract_canon_sha256":"8bc2829fbb575188cb8121b4e1d4b2af307aaf03f24d9b5dc649ce84581bcb75"},"schema_version":"1.0"},"canonical_sha256":"a5b289c1afddcddcdcf644420e3760a2c6b524d48fcb8a9947ea14178420fda3","source":{"kind":"arxiv","id":"1603.00534","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.00534","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"arxiv_version","alias_value":"1603.00534v1","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.00534","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"pith_short_12","alias_value":"UWZITQNP3XG5","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UWZITQNP3XG5ZXHW","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UWZITQNP","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:UWZITQNP3XG5ZXHWIRBA4N3AUL","target":"record","payload":{"canonical_record":{"source":{"id":"1603.00534","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-02T00:41:43Z","cross_cats_sorted":[],"title_canon_sha256":"43a43237c91ae98beff59127efeb54855114b5bd491b3dbe964d301003a3f8a6","abstract_canon_sha256":"8bc2829fbb575188cb8121b4e1d4b2af307aaf03f24d9b5dc649ce84581bcb75"},"schema_version":"1.0"},"canonical_sha256":"a5b289c1afddcddcdcf644420e3760a2c6b524d48fcb8a9947ea14178420fda3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:58.028212Z","signature_b64":"iuGjcGdFE9+MQ2CzFuIk+WgwPSqUzxkeMFrdUA+A3aWCEa1aBu5PUH43V+vao7lYIPOKWDTpAS2nWTE0XLysCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5b289c1afddcddcdcf644420e3760a2c6b524d48fcb8a9947ea14178420fda3","last_reissued_at":"2026-05-18T00:51:58.027334Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:58.027334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.00534","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":"2026-05-18T00:51:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DEkTnYRPxUpeFZIlKI1hph9n46Zmy6MfgnyrkMoi9Gv/QiOB/9fsZbxer7RTc4ot1wzTdJdr2+Fhkb1G97b+Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:18:33.873181Z"},"content_sha256":"e5a56f7c924a575ddc25579fd64df55ac0d475aeb6dfc0da5b6f21d5b4069ba9","schema_version":"1.0","event_id":"sha256:e5a56f7c924a575ddc25579fd64df55ac0d475aeb6dfc0da5b6f21d5b4069ba9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:UWZITQNP3XG5ZXHWIRBA4N3AUL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular periodicity of exponential sums of symmetric Boolean functions and some of its consequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Francis N. Castro, Luis A. Medina","submitted_at":"2016-03-02T00:41:43Z","abstract_excerpt":"This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Boolean functions. In particular, the periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean functions is considered. Periods modulo $p$, bounds for periods and relations between them are obtained for these exponential sums. The concept of avoiding primes is also introduced. This concept and the bounds presented in this work are used to show that some classes of symmetric Boolean functions are not balanced. In particular, every elementary symmetric Boo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00534","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":"2026-05-18T00:51:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y+h0xpJwEHoem9uiz6lOpHbnWLCE51PBCe7FTWwjTWRp3kSAGSr7CTCNOVhWY1yJYGv69790ZKSQvMKeY8dFDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:18:33.873853Z"},"content_sha256":"7d85b1c291c7d52e362ed59fe84f7f29a7402e3eae8f70fd9d684a6f0ef4345a","schema_version":"1.0","event_id":"sha256:7d85b1c291c7d52e362ed59fe84f7f29a7402e3eae8f70fd9d684a6f0ef4345a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UWZITQNP3XG5ZXHWIRBA4N3AUL/bundle.json","state_url":"https://pith.science/pith/UWZITQNP3XG5ZXHWIRBA4N3AUL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UWZITQNP3XG5ZXHWIRBA4N3AUL/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":"2026-05-31T18:18:33Z","links":{"resolver":"https://pith.science/pith/UWZITQNP3XG5ZXHWIRBA4N3AUL","bundle":"https://pith.science/pith/UWZITQNP3XG5ZXHWIRBA4N3AUL/bundle.json","state":"https://pith.science/pith/UWZITQNP3XG5ZXHWIRBA4N3AUL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UWZITQNP3XG5ZXHWIRBA4N3AUL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UWZITQNP3XG5ZXHWIRBA4N3AUL","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":"8bc2829fbb575188cb8121b4e1d4b2af307aaf03f24d9b5dc649ce84581bcb75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-02T00:41:43Z","title_canon_sha256":"43a43237c91ae98beff59127efeb54855114b5bd491b3dbe964d301003a3f8a6"},"schema_version":"1.0","source":{"id":"1603.00534","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.00534","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"arxiv_version","alias_value":"1603.00534v1","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.00534","created_at":"2026-05-18T00:51:58Z"},{"alias_kind":"pith_short_12","alias_value":"UWZITQNP3XG5","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UWZITQNP3XG5ZXHW","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UWZITQNP","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:7d85b1c291c7d52e362ed59fe84f7f29a7402e3eae8f70fd9d684a6f0ef4345a","target":"graph","created_at":"2026-05-18T00:51:58Z","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":"This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Boolean functions. In particular, the periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean functions is considered. Periods modulo $p$, bounds for periods and relations between them are obtained for these exponential sums. The concept of avoiding primes is also introduced. This concept and the bounds presented in this work are used to show that some classes of symmetric Boolean functions are not balanced. In particular, every elementary symmetric Boo","authors_text":"Francis N. Castro, Luis A. Medina","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-02T00:41:43Z","title":"Modular periodicity of exponential sums of symmetric Boolean functions and some of its consequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00534","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:e5a56f7c924a575ddc25579fd64df55ac0d475aeb6dfc0da5b6f21d5b4069ba9","target":"record","created_at":"2026-05-18T00:51:58Z","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":"8bc2829fbb575188cb8121b4e1d4b2af307aaf03f24d9b5dc649ce84581bcb75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-02T00:41:43Z","title_canon_sha256":"43a43237c91ae98beff59127efeb54855114b5bd491b3dbe964d301003a3f8a6"},"schema_version":"1.0","source":{"id":"1603.00534","kind":"arxiv","version":1}},"canonical_sha256":"a5b289c1afddcddcdcf644420e3760a2c6b524d48fcb8a9947ea14178420fda3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5b289c1afddcddcdcf644420e3760a2c6b524d48fcb8a9947ea14178420fda3","first_computed_at":"2026-05-18T00:51:58.027334Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:58.027334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iuGjcGdFE9+MQ2CzFuIk+WgwPSqUzxkeMFrdUA+A3aWCEa1aBu5PUH43V+vao7lYIPOKWDTpAS2nWTE0XLysCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:58.028212Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.00534","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e5a56f7c924a575ddc25579fd64df55ac0d475aeb6dfc0da5b6f21d5b4069ba9","sha256:7d85b1c291c7d52e362ed59fe84f7f29a7402e3eae8f70fd9d684a6f0ef4345a"],"state_sha256":"accdc5db5372a0fccb7252985c1304b94b342b54e8e705cdb4f9f60af152ca47"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eTqx+bJP3oas639WrJX22Ml8/y4RliDZRuy/xD0k8HYBusFFa49khQ0eyZ/169l+zGUbjDJsw0t6+UwvjXqVBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T18:18:33.877076Z","bundle_sha256":"88c39b979f2b37f36c7c97446b78cee68b2f9d16aaa6f41fdf7c909113d6ca63"}}