{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0096", "pilot_number": "P1000-0996", "number": 996, "family_id": "qf-mod47-morder-boundary", "title": { "zh-Hant": "第 996 題：模n乘法階", "en": "Question 996: Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0096", "batch_id": "batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-multiplicative-order-mod-n" ], "title_zh": "模n乘法階", "stem_zh": "n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":48,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(noncoprime, 48, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)", "explanation_zh": "Wave6第十層核對=(noncoprime, 48, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "模n乘法階" ] }, "variant": { "family_id": "qf-mod47-morder-boundary", "seed": 202611167096, "parameters": { "mode": "noncoprime", "n": 48, "a": 2, "domain": "Exact_multiplicative_order_mod_n", "operation": "morder-boundary", "semantic_template": "qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":48}" } }, "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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0096", "batch_id": "batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "670ef2cda62eb6cc3a47ed40346e0a094563be8976f9f801348b7a7e4b38a76d", "family_id": "qf-mod47-morder-boundary", "seed": 202611167096, "semantic_template": "qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":48}" }, "fields": { "title": "Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues", "stem": "Use exact modulus n=48 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).", "answer": { "kind": "expression", "value": "(noncoprime, 48, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)", "explanation": "For invalid modulus keep residue/gcd null; for a noncoprime valid modulus retain the actual normalized residue and gcd. Both have null power table and undefined order. order_one requires residue1; negative_normalization requires a negative representative and a qualified unit, with the actual normalized power table. The exact result, retaining the requested order, is (noncoprime, 48, 2, 2, 2, No, null, undefined, mode_specific_multiplicative_order_boundary)." }, "skills": [ "Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues" ], "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": "n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":48,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use exact modulus n=48 and integer a=2. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Mode=noncoprime. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "n=48; a=2", "after": "order is defined here only for n≥2 and units modulo n", "relation": "implies", "rule_id": "order_domain", "conditions": [], "explanation": { "zh-Hant": "先固定本節點的定義域；加法階數與乘法階數不能混用。", "en": "Fix this node's domain first; do not confuse additive and multiplicative order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。", "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "a=2", "after": "normalized residue=2; gcd(2,48)=2", "relation": "implies", "rule_id": "normalize_and_qualify", "conditions": [], "explanation": { "zh-Hant": "先把負代表元也正規化到[0,n)，再檢查互質。", "en": "Normalize even a negative representative into[0,n), then test coprimality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。", "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "g=2>1 divides both residue and n", "after": "no positive power can be congruent to1", "relation": "implies", "rule_id": "nonunit_power_obstruction", "conditions": [], "explanation": { "zh-Hant": "每個正冪及模數的共同因數仍整除其整數線性組合，不可能得到1。", "en": "The common divisor divides each positive power and the modulus, so a congruence to1 is impossible." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。", "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "not a unit", "after": "power table=null; order=undefined", "relation": "implies", "rule_id": "nonunit_order_boundary", "conditions": [], "explanation": { "zh-Hant": "這是否定定義域資格，不是把未定義階數設成1。", "en": "This rejects the domain qualification; it does not set an undefined order to1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "階數必為第一個正指數回到1；先檢查 n≥2 及互質，不宣稱階數等於 phi。", "en": "Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi." } }, "arithmetic": null } ], "math_model": { "result": [ "noncoprime", 48, 2, 2, 2, false, null, "undefined", "mode_specific_multiplicative_order_boundary" ] }, "conclusion": { "canonical_value": "(noncoprime, 48, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)", "explanation": { "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。", "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings." } }, "definition_note": { "zh-Hant": "n<2 或不互質時階數未定義；1 的階數為 1；負代表元先正規化到 [0,n)。", "en": "For n<2 or a noncoprime residue the order is undefined; residue 1 has order 1; normalize a negative representative into [0,n) first." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations/questions.jsonl", "batch_sha256": "c245b764f43fe987ce3c9b7683ff9d31d6ebd86803fc9add4a5b4951ce49ec5c", "original_line": 96, "original_line_sha256": "4afef94ad819fcb199b077a62ec5669fe40a32ea2e8918f21a3096b5e21c3245", "question_compact_sha256": "670ef2cda62eb6cc3a47ed40346e0a094563be8976f9f801348b7a7e4b38a76d", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations.en.jsonl", "english_row_compact_sha256": "7052a4f5fcea694e134042b96d8812dd8b42c68ca0725f409fb022ff0b5975e7" }, "mko_bindings": { "mko-multiplicative-order-mod-n": { "sha256": "70fb02dd0cfb34ce7cc4a848496bc4599fe6b0d22028bb31d567c1c29e501ec3", "english_sha256": "f483395f63a625c33cb8d24ecdc29750ab5bf5a3547024df4c65a99269a1616e", "original_path": "public/data/mko/multiplicative-order-mod-n.json", "english_path": "public/data/i18n/editions/mko-multiplicative-order-mod-n.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e", "original_candidate_sha256": "d338fe590ac2749212640c6d95598be9cd8123dac4e938a7bc2bb153049ec0ed", "original_candidate_line_sha256": "d338fe590ac2749212640c6d95598be9cd8123dac4e938a7bc2bb153049ec0ed", "mko_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c", "previous_english_sha256": { "mko-multiplicative-order-mod-n": "1bb68b21117fcfd1f0a12761eb7a57dfbedfa6d9c7f950810031ce61acb6201e" } } }, "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": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。", "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI 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-20261026-003-04", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results 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": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772", "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "ca0ed008cef5a8cfe1430dffd643634e629ab77f104a25ea98ff6399a129bc8b", "arithmetic_steps": 0 }, "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." } }