{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:PNWMFH3JEEGV3K5T7JU7UKRMY2","short_pith_number":"pith:PNWMFH3J","canonical_record":{"source":{"id":"2507.01529","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-07-02T09:35:01Z","cross_cats_sorted":[],"title_canon_sha256":"99a4ca940bdf527b725c5771cf06f0721edfe99169559df054e13806a16bb2ff","abstract_canon_sha256":"b0b241eae4b6eefe7acfb1994df00744bedc8b272eb7d96328f2c9ef5b1ac165"},"schema_version":"1.0"},"canonical_sha256":"7b6cc29f69210d5dabb3fa69fa2a2cc684e3ec428387fef3e67c8b06be69ff5c","source":{"kind":"arxiv","id":"2507.01529","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.01529","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"arxiv_version","alias_value":"2507.01529v2","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.01529","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"pith_short_12","alias_value":"PNWMFH3JEEGV","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"pith_short_16","alias_value":"PNWMFH3JEEGV3K5T","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"pith_short_8","alias_value":"PNWMFH3J","created_at":"2026-07-05T11:48:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:PNWMFH3JEEGV3K5T7JU7UKRMY2","target":"record","payload":{"canonical_record":{"source":{"id":"2507.01529","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-07-02T09:35:01Z","cross_cats_sorted":[],"title_canon_sha256":"99a4ca940bdf527b725c5771cf06f0721edfe99169559df054e13806a16bb2ff","abstract_canon_sha256":"b0b241eae4b6eefe7acfb1994df00744bedc8b272eb7d96328f2c9ef5b1ac165"},"schema_version":"1.0"},"canonical_sha256":"7b6cc29f69210d5dabb3fa69fa2a2cc684e3ec428387fef3e67c8b06be69ff5c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:48:51.007696Z","signature_b64":"JKbI32RlYd3XnrjzOqnSPvQ9Z5rYVU6V/7OTRNabURXsJkjvdc7vdUOvaZj3uyEZVxbwo6KW9Lm/VcaiyFlWAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b6cc29f69210d5dabb3fa69fa2a2cc684e3ec428387fef3e67c8b06be69ff5c","last_reissued_at":"2026-07-05T11:48:51.007181Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:48:51.007181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.01529","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":"2026-07-05T11:48:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k8ZW8Fit97Ef/rhyR9BQpRzQ8rD0RcY5wppo4poS1bO8yBBSb7aXbEJrOoKP5ELV21+ngnDZQ4MFgFk17vT3BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T05:29:18.798005Z"},"content_sha256":"fcc166d6a6d5ee08615724f69ca0b7eae8498c02b85bc4b7218450b92abef2de","schema_version":"1.0","event_id":"sha256:fcc166d6a6d5ee08615724f69ca0b7eae8498c02b85bc4b7218450b92abef2de"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:PNWMFH3JEEGV3K5T7JU7UKRMY2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some new congruences on biregular overpartitions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"N. K. Meher","submitted_at":"2025-07-02T09:35:01Z","abstract_excerpt":"Recently, Nadji, Ahmia and Ram\\'{i}rez \\cite{Nadji2025} investigate the arithmetic properties of ${\\bar B}_{\\ell_1,\\ell_2}(n)$, the number of overpartitions where no part is divisible by $\\ell_1$ or $\\ell_2$ with $\\gcd(\\ell_1,\\ell_2)$$=1$ and $\\ell_1$,$\\ell_2$$>1$. Specifically, they established congruences modulo $3$ and powers of $2$ for the pairs of $(\\ell_1,\\ell_2)$$\\in$$\\{(4,3),(4,9),(8,3),(8,9)\\}$, using the concept of generating functions, dissection formulas and Smoot's implementation of Radu's Ramanujan-Kolberg algorithm. After that, Alanazi, Munagi and Saikia \\cite{Alanazi2024} studi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.01529","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.01529/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":"2026-07-05T11:48:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gDWJFJth9vNMlQ2dpR5BPKKrva2Jfu8fVaLjFAB+jW+d7ULjp7h1XLjZLzTIJbF2Ykg3WEPU2soJTTZkxc03Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T05:29:18.798933Z"},"content_sha256":"d1577823e40b4b264bc324fb2ea6c062df28e55a52e02c92edb8a8066b6dc09e","schema_version":"1.0","event_id":"sha256:d1577823e40b4b264bc324fb2ea6c062df28e55a52e02c92edb8a8066b6dc09e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PNWMFH3JEEGV3K5T7JU7UKRMY2/bundle.json","state_url":"https://pith.science/pith/PNWMFH3JEEGV3K5T7JU7UKRMY2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PNWMFH3JEEGV3K5T7JU7UKRMY2/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":"2026-08-09T05:29:18Z","links":{"resolver":"https://pith.science/pith/PNWMFH3JEEGV3K5T7JU7UKRMY2","bundle":"https://pith.science/pith/PNWMFH3JEEGV3K5T7JU7UKRMY2/bundle.json","state":"https://pith.science/pith/PNWMFH3JEEGV3K5T7JU7UKRMY2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PNWMFH3JEEGV3K5T7JU7UKRMY2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:PNWMFH3JEEGV3K5T7JU7UKRMY2","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":"b0b241eae4b6eefe7acfb1994df00744bedc8b272eb7d96328f2c9ef5b1ac165","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-07-02T09:35:01Z","title_canon_sha256":"99a4ca940bdf527b725c5771cf06f0721edfe99169559df054e13806a16bb2ff"},"schema_version":"1.0","source":{"id":"2507.01529","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.01529","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"arxiv_version","alias_value":"2507.01529v2","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.01529","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"pith_short_12","alias_value":"PNWMFH3JEEGV","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"pith_short_16","alias_value":"PNWMFH3JEEGV3K5T","created_at":"2026-07-05T11:48:51Z"},{"alias_kind":"pith_short_8","alias_value":"PNWMFH3J","created_at":"2026-07-05T11:48:51Z"}],"graph_snapshots":[{"event_id":"sha256:d1577823e40b4b264bc324fb2ea6c062df28e55a52e02c92edb8a8066b6dc09e","target":"graph","created_at":"2026-07-05T11:48:51Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2507.01529/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Recently, Nadji, Ahmia and Ram\\'{i}rez \\cite{Nadji2025} investigate the arithmetic properties of ${\\bar B}_{\\ell_1,\\ell_2}(n)$, the number of overpartitions where no part is divisible by $\\ell_1$ or $\\ell_2$ with $\\gcd(\\ell_1,\\ell_2)$$=1$ and $\\ell_1$,$\\ell_2$$>1$. Specifically, they established congruences modulo $3$ and powers of $2$ for the pairs of $(\\ell_1,\\ell_2)$$\\in$$\\{(4,3),(4,9),(8,3),(8,9)\\}$, using the concept of generating functions, dissection formulas and Smoot's implementation of Radu's Ramanujan-Kolberg algorithm. After that, Alanazi, Munagi and Saikia \\cite{Alanazi2024} studi","authors_text":"N. K. Meher","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-07-02T09:35:01Z","title":"Some new congruences on biregular overpartitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.01529","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:fcc166d6a6d5ee08615724f69ca0b7eae8498c02b85bc4b7218450b92abef2de","target":"record","created_at":"2026-07-05T11:48:51Z","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":"b0b241eae4b6eefe7acfb1994df00744bedc8b272eb7d96328f2c9ef5b1ac165","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-07-02T09:35:01Z","title_canon_sha256":"99a4ca940bdf527b725c5771cf06f0721edfe99169559df054e13806a16bb2ff"},"schema_version":"1.0","source":{"id":"2507.01529","kind":"arxiv","version":2}},"canonical_sha256":"7b6cc29f69210d5dabb3fa69fa2a2cc684e3ec428387fef3e67c8b06be69ff5c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b6cc29f69210d5dabb3fa69fa2a2cc684e3ec428387fef3e67c8b06be69ff5c","first_computed_at":"2026-07-05T11:48:51.007181Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:48:51.007181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JKbI32RlYd3XnrjzOqnSPvQ9Z5rYVU6V/7OTRNabURXsJkjvdc7vdUOvaZj3uyEZVxbwo6KW9Lm/VcaiyFlWAw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:48:51.007696Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.01529","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fcc166d6a6d5ee08615724f69ca0b7eae8498c02b85bc4b7218450b92abef2de","sha256:d1577823e40b4b264bc324fb2ea6c062df28e55a52e02c92edb8a8066b6dc09e"],"state_sha256":"ee09007237361d22e5339aa8937bfef4423615419d301bd608be1e6889532bda"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2acsCY46I0UukruYzrabRJmNPXwxJOYH32Bw+lm7q0Zo8PLnZXGH3dW6Mf+7J0Xqj5eACjmO3i6r5LkDUAY4AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T05:29:18.804652Z","bundle_sha256":"8c0482f20bd6089954a9dedf61a4e0b5ce4f73c44bb70f9b765fbb30e048ef7b"}}