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We study the minimal obstruction modulus $\\kappa_Q := \\min \\{k \\in \\mathbb{Z}_{\\geq 1} \\mid \\text{there exists } l \\text{ such that } Q \\not\\equiv l \\pmod k \\}$, and we determine $\\kappa_Q$ completely, treating the cases $\\Delta \\equiv 0 \\pmod 4$ and $\\Delta \\equiv 1 \\pmod 4$ separately. We also determine the analogous invariants for primitive t"},"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":"2608.09063","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-08-10T03:11:33Z","cross_cats_sorted":[],"title_canon_sha256":"dbb40e0fda501dd2d0e9d895fe2b0431f36396538acf02c090b4e6b5cc92ba2e","abstract_canon_sha256":"9ac09233241ac30af108118ebce1bf93f00c8bcbf08521006bea12b2d0f4a113"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T02:21:42.819270Z","signature_b64":"OKdiR9mIhCGdnKITwjZzlmlrzRQgVpqpNRJuJMxe4dr1PGGo9f1pca/asvYWpZp5sTyK85saCpk31uCS5Ga3Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4dcd9ba740a2a70dfd1563f633ef9bd73ee6f81359f470a554012207be0a7ec","last_reissued_at":"2026-08-11T02:21:42.817705Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T02:21:42.817705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The minimal obstruction modulus for quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Naoki Ochi","submitted_at":"2026-08-10T03:11:33Z","abstract_excerpt":"Let $Q = ax^2+bxy+cy^2$ be a primitive positive definite integral binary quadratic form with discriminant $\\Delta = b^2-4ac$. It is known that $Q$ admits a local obstruction; that is, there exist $k,l \\in \\mathbb{Z}$ such that $Q \\not\\equiv l \\pmod k$. We study the minimal obstruction modulus $\\kappa_Q := \\min \\{k \\in \\mathbb{Z}_{\\geq 1} \\mid \\text{there exists } l \\text{ such that } Q \\not\\equiv l \\pmod k \\}$, and we determine $\\kappa_Q$ completely, treating the cases $\\Delta \\equiv 0 \\pmod 4$ and $\\Delta \\equiv 1 \\pmod 4$ separately. 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