{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KFBO2YQLYI3EIDXZPP4YR6MJHV","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":"d0fb83f7c57008264b45bca7bc92e3fddf23f47634fe701754dedb61549ef6cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-06T05:26:26Z","title_canon_sha256":"a604626d9df7e93f9430049eb5b76d65ea37a633189cd7b6a09c9cb5f5641cb4"},"schema_version":"1.0","source":{"id":"1311.1289","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.1289","created_at":"2026-05-18T03:07:33Z"},{"alias_kind":"arxiv_version","alias_value":"1311.1289v2","created_at":"2026-05-18T03:07:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.1289","created_at":"2026-05-18T03:07:33Z"},{"alias_kind":"pith_short_12","alias_value":"KFBO2YQLYI3E","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KFBO2YQLYI3EIDXZ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KFBO2YQL","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:732e0b2ef6e6fd1f561b53e3599c5d22098a73cc018d5f06eab8c765707f1025","target":"graph","created_at":"2026-05-18T03:07:33Z","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":"In this paper, we introduce the 4-th multiple residue symbol $[p_1, p_2, p_3, p_4]$ for certain four prime numbers $p_1, p_2, p_3, p_4$, which extends the Legendre symbol and the R\\'{e}dei triple symbol in a natural manner. For this we construct concretely a certain nilpotent extension K over Q of degree 64, where ramified prime numbers are $p_1$, $p_2$ and $p_3$, such that the symbol $[p_1, p_2, p_3, p_4]$ describes the decomposition law of $p_4$ in the extension K/Q. We then establish the relation of our symbol and the 4-th arithmetic Milnor invariant (an arithmetic analogue of the 4-th orde","authors_text":"Fumiya Amano","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-06T05:26:26Z","title":"On a certain nilpotent extension over Q of degree 64 and the 4-th multiple residue symbol"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.1289","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:c9e797c8fec9bac92e036f69dbe4adcb223e892bb32549475eff5888d3938210","target":"record","created_at":"2026-05-18T03:07:33Z","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":"d0fb83f7c57008264b45bca7bc92e3fddf23f47634fe701754dedb61549ef6cc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-06T05:26:26Z","title_canon_sha256":"a604626d9df7e93f9430049eb5b76d65ea37a633189cd7b6a09c9cb5f5641cb4"},"schema_version":"1.0","source":{"id":"1311.1289","kind":"arxiv","version":2}},"canonical_sha256":"5142ed620bc236440ef97bf988f9893d4823c8fbe945435e2ee3d958c2c07b2f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5142ed620bc236440ef97bf988f9893d4823c8fbe945435e2ee3d958c2c07b2f","first_computed_at":"2026-05-18T03:07:33.645105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:33.645105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ee8syO1Wr5DLfHDiMaDsRNaWJCvMeNLlxgQUJKHScUNQhj6jrDh5GZ2RukgxG1gEM4eqKsOGhW7fuFCd5k3nDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:33.645749Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.1289","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c9e797c8fec9bac92e036f69dbe4adcb223e892bb32549475eff5888d3938210","sha256:732e0b2ef6e6fd1f561b53e3599c5d22098a73cc018d5f06eab8c765707f1025"],"state_sha256":"7e19437befff6c10053f7671399980d2af7a729fc993c810f7ee1585936f3dc6"}