{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:FNMRS4FNKNN4VNF5QGWRUVX2LZ","short_pith_number":"pith:FNMRS4FN","canonical_record":{"source":{"id":"2410.12970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-10-16T19:04:26Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"519f52cc4eb508f6e7d048751d1593f33d2518699761dad2d82ef0f4b287aed7","abstract_canon_sha256":"0bfd358fcd596c876e98ac73c6a8e87f4f5a77a631d1bf0997773d68773fa073"},"schema_version":"1.0"},"canonical_sha256":"2b591970ad535bcab4bd81ad1a56fa5e5cfaef5059e31fd97f8627a9a96c6434","source":{"kind":"arxiv","id":"2410.12970","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.12970","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"arxiv_version","alias_value":"2410.12970v1","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.12970","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"pith_short_12","alias_value":"FNMRS4FNKNN4","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"pith_short_16","alias_value":"FNMRS4FNKNN4VNF5","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"pith_short_8","alias_value":"FNMRS4FN","created_at":"2026-06-10T01:09:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:FNMRS4FNKNN4VNF5QGWRUVX2LZ","target":"record","payload":{"canonical_record":{"source":{"id":"2410.12970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-10-16T19:04:26Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"519f52cc4eb508f6e7d048751d1593f33d2518699761dad2d82ef0f4b287aed7","abstract_canon_sha256":"0bfd358fcd596c876e98ac73c6a8e87f4f5a77a631d1bf0997773d68773fa073"},"schema_version":"1.0"},"canonical_sha256":"2b591970ad535bcab4bd81ad1a56fa5e5cfaef5059e31fd97f8627a9a96c6434","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:09:13.464417Z","signature_b64":"7SW/c0iN+qilYojISWlVz8xhzyahUJE0nizcfsJrETDw3RAFVQL/s2SrgikGxV7Wxe2jQDI+4rHJkcMGoDhhBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b591970ad535bcab4bd81ad1a56fa5e5cfaef5059e31fd97f8627a9a96c6434","last_reissued_at":"2026-06-10T01:09:13.463816Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:09:13.463816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2410.12970","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":"2026-06-10T01:09:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4YIwt9yj6dvaxelJPQEi39CRwQwb101Ym1WKOP+i2usNVcC8PTQWiSNK3zIcNXlBoAgtc5TZ0sCOckkYUCmhCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T21:55:58.911123Z"},"content_sha256":"b8f76792558dea660758fd37802383096d25ac4ecb14439886140116dd582a6c","schema_version":"1.0","event_id":"sha256:b8f76792558dea660758fd37802383096d25ac4ecb14439886140116dd582a6c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:FNMRS4FNKNN4VNF5QGWRUVX2LZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weakly Divisible Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Gaurav Digambar Patil","submitted_at":"2024-10-16T19:04:26Z","abstract_excerpt":"We define a new class of rings parameterized by binary forms of a certain type, and give an effective lower bound for the number of such rings whose discriminant is less than a bound $X$. We also obtain a lower bound for the number of number fields whose ring of integers lies in the above class and whose discriminant is less than a bound $X$. Our results improve an estimate of Bhargava-Shankar-Wang in \\cite{bhargava2022squarefree}. In particular we show the following:\n  $\\bullet$ When $n\\ge 4,$ the number of rings of rank $n$ over $\\mathbb{Z}$ with discriminant less than or equal to $X$ is $$\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.12970","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.12970/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":"2026-06-10T01:09:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DuUR/rQis1qMWQ0tYiuzftI48DvkM9oJ9EBCtRJQqRxkY/Bn5EZYLqjzFFhTm30+O3ssO+35JRRT7BfZvgg8Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T21:55:58.911506Z"},"content_sha256":"cf5838151af9f36aed2c8b3f8e055ed63e665a865f27aff4c52b02cfc3818d2a","schema_version":"1.0","event_id":"sha256:cf5838151af9f36aed2c8b3f8e055ed63e665a865f27aff4c52b02cfc3818d2a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FNMRS4FNKNN4VNF5QGWRUVX2LZ/bundle.json","state_url":"https://pith.science/pith/FNMRS4FNKNN4VNF5QGWRUVX2LZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FNMRS4FNKNN4VNF5QGWRUVX2LZ/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":"2026-06-29T21:55:58Z","links":{"resolver":"https://pith.science/pith/FNMRS4FNKNN4VNF5QGWRUVX2LZ","bundle":"https://pith.science/pith/FNMRS4FNKNN4VNF5QGWRUVX2LZ/bundle.json","state":"https://pith.science/pith/FNMRS4FNKNN4VNF5QGWRUVX2LZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FNMRS4FNKNN4VNF5QGWRUVX2LZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:FNMRS4FNKNN4VNF5QGWRUVX2LZ","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":"0bfd358fcd596c876e98ac73c6a8e87f4f5a77a631d1bf0997773d68773fa073","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-10-16T19:04:26Z","title_canon_sha256":"519f52cc4eb508f6e7d048751d1593f33d2518699761dad2d82ef0f4b287aed7"},"schema_version":"1.0","source":{"id":"2410.12970","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.12970","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"arxiv_version","alias_value":"2410.12970v1","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.12970","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"pith_short_12","alias_value":"FNMRS4FNKNN4","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"pith_short_16","alias_value":"FNMRS4FNKNN4VNF5","created_at":"2026-06-10T01:09:13Z"},{"alias_kind":"pith_short_8","alias_value":"FNMRS4FN","created_at":"2026-06-10T01:09:13Z"}],"graph_snapshots":[{"event_id":"sha256:cf5838151af9f36aed2c8b3f8e055ed63e665a865f27aff4c52b02cfc3818d2a","target":"graph","created_at":"2026-06-10T01:09:13Z","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/2410.12970/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We define a new class of rings parameterized by binary forms of a certain type, and give an effective lower bound for the number of such rings whose discriminant is less than a bound $X$. We also obtain a lower bound for the number of number fields whose ring of integers lies in the above class and whose discriminant is less than a bound $X$. Our results improve an estimate of Bhargava-Shankar-Wang in \\cite{bhargava2022squarefree}. In particular we show the following:\n  $\\bullet$ When $n\\ge 4,$ the number of rings of rank $n$ over $\\mathbb{Z}$ with discriminant less than or equal to $X$ is $$\\","authors_text":"Gaurav Digambar Patil","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-10-16T19:04:26Z","title":"Weakly Divisible Rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.12970","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:b8f76792558dea660758fd37802383096d25ac4ecb14439886140116dd582a6c","target":"record","created_at":"2026-06-10T01:09:13Z","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":"0bfd358fcd596c876e98ac73c6a8e87f4f5a77a631d1bf0997773d68773fa073","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-10-16T19:04:26Z","title_canon_sha256":"519f52cc4eb508f6e7d048751d1593f33d2518699761dad2d82ef0f4b287aed7"},"schema_version":"1.0","source":{"id":"2410.12970","kind":"arxiv","version":1}},"canonical_sha256":"2b591970ad535bcab4bd81ad1a56fa5e5cfaef5059e31fd97f8627a9a96c6434","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b591970ad535bcab4bd81ad1a56fa5e5cfaef5059e31fd97f8627a9a96c6434","first_computed_at":"2026-06-10T01:09:13.463816Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:09:13.463816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7SW/c0iN+qilYojISWlVz8xhzyahUJE0nizcfsJrETDw3RAFVQL/s2SrgikGxV7Wxe2jQDI+4rHJkcMGoDhhBA==","signature_status":"signed_v1","signed_at":"2026-06-10T01:09:13.464417Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.12970","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8f76792558dea660758fd37802383096d25ac4ecb14439886140116dd582a6c","sha256:cf5838151af9f36aed2c8b3f8e055ed63e665a865f27aff4c52b02cfc3818d2a"],"state_sha256":"84494be5df02e9bfe5bbd0e6dc3f1532a0a887d31c99c2ac10b558c76685cdee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1mxw1wS/WbcTm3vPB6QERZiScmZR0hLcNcdytgKpemvp4p8z8TxTDvLuJAEFV7GWl0eK8bGILho7Mn+/vkGFCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T21:55:58.913448Z","bundle_sha256":"dd63a528e40d114c471045d954b09880c3db024f6d301f3f49d4da23329e4135"}}