{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261112-r06-legendre-symbol-0091", "pilot_number": "R0046-23741", "number": 23741, "family_id": "qf-mod43-legendre-domain-method-boundary", "title": { "zh-Hant": "第 23741 題：Legendre符號", "en": "Question 23741: Domain Extensions and Premature Methods Are Different Legendre Boundaries" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261112-r06-legendre-symbol-0091", "batch_id": "batch-20261014-advance-20261112-r06-legendre-symbol-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-legendre-symbol" ], "title_zh": "Legendre符號", "stem_zh": "odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)", "explanation_zh": "Wave6第六層核對=(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "Legendre符號" ] }, "variant": { "family_id": "qf-mod43-legendre-domain-method-boundary", "seed": 202611126091, "parameters": { "mode": "even_prime", "modulus": 2, "a": 16, "domain": "Odd_prime_Legendre_symbol", "operation": "legendre-domain-method-boundary", "semantic_template": "qf-mod43-legendre-domain-method-boundary/{\"a\":16,\"mode\":\"even_prime\",\"modulus\":2}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261112-r06-legendre-symbol-0091", "batch_id": "batch-20261014-advance-20261112-r06-legendre-symbol-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1c61681b6ce17b49238c745778f5809091d3daa17c893d30499c1d0d3e5fd010", "family_id": "qf-mod43-legendre-domain-method-boundary", "seed": 202611126091, "semantic_template": "qf-mod43-legendre-domain-method-boundary/{\"a\":16,\"mode\":\"even_prime\",\"modulus\":2}" }, "fields": { "title": "Domain Extensions and Premature Methods Are Different Legendre Boundaries", "stem": "Use mode=even_prime, modulus=2, original a=16. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).", "answer": { "kind": "expression", "value": "(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)", "explanation": "Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)." }, "skills": [ "Domain Extensions and Premature Methods Are Different Legendre Boundaries" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=even_prime, modulus=2, original a=16. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=legendre-domain-method-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "保留原題介面快照，本頁直接列來源欄序，不需JavaScript。radius由原mode選6／11欄，不補欄；No與實際否定依來源責任分清。", "en": "Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Legendre ORIGINAL boundary class quotient times modulus := mul(8, 2)", "after": "Legendre ORIGINAL boundary class quotient times modulus = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Legendre ORIGINAL boundary class quotient times modulus，不以小數近似替代。", "en": "Recompute Legendre ORIGINAL boundary class quotient times modulus with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "2" ], "output": "16", "label": "Legendre ORIGINAL boundary class quotient times modulus" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple := sub(16, 16)", "after": "Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似，也不把有限測試冒充全稱極限。", "en": "Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple，不以小數近似替代。", "en": "Recompute Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "16", "16" ], "output": "0", "label": "Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Legendre ORIGINAL boundary class:ORIGINAL a=16,positive modulus=2;actual quotient=8", "after": "normalized class=0;a=quotient*modulus+class;ORIGINAL a retained in output where declared", "relation": "implies", "rule_id": "legendre_ORIGINAL_signed_class_Euclidean_normalization", "conditions": [], "explanation": { "zh-Hant": "負a亦保同一congruence class，以0到p−1表示；不把負餘數直接當標準類。", "en": "Retain the same class for negative a and normalize into0..p−1,rather than using a negative language-specific remainder." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "ORIGINAL mode=even_prime;modulus=2,a=16,normalized class=0", "after": "unsupported_domain_or_method_certificate fixed No;modulus outside declared odd-prime Legendre domain", "relation": "implies", "rule_id": "legendre_DOMAIN_or_METHOD_No_responsibilities_distinct", "conditions": [], "explanation": { "zh-Hant": "premature_euler的No不否定Euler criterion，也不把合法input說成域外；p2／composite則是真域限制。不偷偷改為Jacobi等其他符號。", "en": "For premature_euler,the No neither refutes Euler criterion nor puts a valid input out of domain.Modulus2/composite cases have a domain restriction.No Jacobi or other extension is silently adopted." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ "even_prime", 2, 0, false, "Legendre_symbol_domain_requires_an_odd_prime_modulus" ] }, "conclusion": { "canonical_value": "(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261112-r06-legendre-symbol-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261112-r06-legendre-symbol-foundations/questions.jsonl", "batch_sha256": "ece4cc3b3fcf3c5b1dd2d6a1e92fe38d880eb9a488ab04606bc610adb75aede4", "original_line": 91, "original_line_sha256": "0825efaa4c55033232964a0e56f0d27b1a4910001a6d7f781b66b34626ed4497", "question_compact_sha256": "1c61681b6ce17b49238c745778f5809091d3daa17c893d30499c1d0d3e5fd010", "english_row_compact_sha256": "44f664ac78df6f3f97ee1819a3aa5a84ef28eca4745c8a83af05b75bb855d2be", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261112-r06-legendre-symbol-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261112-r06-legendre-symbol-foundations.en.jsonl" }, "mko_bindings": { "mko-legendre-symbol": { "sha256": "56ce955775c56864a29a398a2adae655ad1f8d53a82d5d9970bc7d0e1a4cebde", "english_sha256": "c00926f8c5fb9b2c966956a02566cfdbfe62191fd6143c23260619d3a0da2cee", "original_path": "public/data/mko/legendre-symbol.json", "english_path": "public/data/i18n/editions/mko-legendre-symbol.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "04d5fe00084f382c9e6853d5f11c80b558d9cf857160ceb644fa20b64948d1fe", "original_candidate_sha256": "2e15e006c482383c886528e93d23056db05c7f77380c0170b25acc51c133bb55", "original_candidate_line_sha256": "2e15e006c482383c886528e93d23056db05c7f77380c0170b25acc51c133bb55" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-009-p47", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1", "question_first_witness_sha256": "c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "24b07c8f59fe73e6b97c0527a41ded080e61f01f68fdd832fa7da2b02b601889", "arithmetic_steps": 2 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }