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Let $q$ be an odd prime power, let $\\mathbb{F}_q$ denote the finite field of order $q$, and let $\\mathcal{R}_q$ denote the set of non-zero squares in $\\mathbb{F}_q$. Let $\\{a, b\\} \\subseteq \\mathbb{F}_q$ be such that $\\{ab, (a-1)(b-1)\\} \\subseteq \\mathcal{R}_q$. Let $\\mathcal{Q}_{a, b}$ denote the quadratic quasigroup $(\\mathbb{F}_q, *_{a, b})$ where $*_{a, b}$ is defined by\n "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.02446","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-03T05:08:28Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6e4151480020ab8130cb9ad4a7dbeb51726af8d7dbaadff5e43f55857ac08086","abstract_canon_sha256":"e2acf7e1fc83419f218384477ae12d58d7a6bb79e02628e2adeb35ec26116abb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:15:02.095846Z","signature_b64":"jW7w/cNzt6lVGTwoMu7KuwQd4wdZwS0pdggSr0uhWTIL2ksM0G/gEpw6A1G2L45yyat594E1ndBjLLjIZf+KBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78fa7fe0c285523bd9e6b3d97118ddf5b9d2229a26f34be6e84960ea8423408c","last_reissued_at":"2026-07-05T11:15:02.095448Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:15:02.095448Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Isotopisms of quadratic quasigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Jack Allsop","submitted_at":"2025-06-03T05:08:28Z","abstract_excerpt":"A quasigroup is a pair $(Q, \\cdot)$ where $Q$ is a non-empty set and $\\cdot$ is a binary operation on $Q$ such that for every $(u, v) \\in Q^2$ there exists a unique $(x, y) \\in Q^2$ such that $u \\cdot x = v = y \\cdot u$. Let $q$ be an odd prime power, let $\\mathbb{F}_q$ denote the finite field of order $q$, and let $\\mathcal{R}_q$ denote the set of non-zero squares in $\\mathbb{F}_q$. Let $\\{a, b\\} \\subseteq \\mathbb{F}_q$ be such that $\\{ab, (a-1)(b-1)\\} \\subseteq \\mathcal{R}_q$. Let $\\mathcal{Q}_{a, b}$ denote the quadratic quasigroup $(\\mathbb{F}_q, *_{a, b})$ where $*_{a, b}$ is defined by\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.02446","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/2506.02446/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.02446","created_at":"2026-07-05T11:15:02.095507+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.02446v1","created_at":"2026-07-05T11:15:02.095507+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.02446","created_at":"2026-07-05T11:15:02.095507+00:00"},{"alias_kind":"pith_short_12","alias_value":"PD5H7YGCQVJD","created_at":"2026-07-05T11:15:02.095507+00:00"},{"alias_kind":"pith_short_16","alias_value":"PD5H7YGCQVJDXWPG","created_at":"2026-07-05T11:15:02.095507+00:00"},{"alias_kind":"pith_short_8","alias_value":"PD5H7YGC","created_at":"2026-07-05T11:15:02.095507+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W","json":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W.json","graph_json":"https://pith.science/api/pith-number/PD5H7YGCQVJDXWPGWPMXCGG56W/graph.json","events_json":"https://pith.science/api/pith-number/PD5H7YGCQVJDXWPGWPMXCGG56W/events.json","paper":"https://pith.science/paper/PD5H7YGC"},"agent_actions":{"view_html":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W","download_json":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W.json","view_paper":"https://pith.science/paper/PD5H7YGC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.02446&json=true","fetch_graph":"https://pith.science/api/pith-number/PD5H7YGCQVJDXWPGWPMXCGG56W/graph.json","fetch_events":"https://pith.science/api/pith-number/PD5H7YGCQVJDXWPGWPMXCGG56W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W/action/storage_attestation","attest_author":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W/action/author_attestation","sign_citation":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W/action/citation_signature","submit_replication":"https://pith.science/pith/PD5H7YGCQVJDXWPGWPMXCGG56W/action/replication_record"}},"created_at":"2026-07-05T11:15:02.095507+00:00","updated_at":"2026-07-05T11:15:02.095507+00:00"}