{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261022-r03-complex-number-0075", "pilot_number": "R0057-29050", "number": 29050, "family_id": "qf-str22-complex-multiplication", "title": { "zh-Hant": "第 29050 題：複數有序對運算", "en": "Question 29050: The Negative Real Contribution from the Product of Imaginary Parts" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261022-r03-complex-number-0075", "batch_id": "batch-20261013-advance-20261022-r03-complex-number-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-complex-number" ], "title_zh": "複數有序對運算", "stem_zh": "複數以整數有序對表示。操作=complex-multiplication；資料={\"left\":[12,0],\"right\":[2,39]}。請依 [left,right,result] 欄序回答。", "answer": { "kind": "expression", "value": "([12, 0], [2, 39], [24, 468])", "explanation_zh": "結構基礎核對=([12, 0], [2, 39], [24, 468])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；有序對運算依i平方等於-1；不由複平面外觀補值。" }, "difficulty": { "band": "upper_secondary", "level": 3, "skills": [ "複數有序對運算" ] }, "variant": { "family_id": "qf-str22-complex-multiplication", "seed": 202610223075, "parameters": { "left": [ 12, 0 ], "right": [ 2, 39 ], "domain": "Integer_pair_complex_arithmetic", "operation": "complex-multiplication", "semantic_template": "qf-str22-complex-multiplication/{\"left\":[12,0],\"right\":[2,39]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261022-r03-complex-number-0075", "batch_id": "batch-20261013-advance-20261022-r03-complex-number-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "38a8cebd607ca5ae33b8fc74f11c37ebe161693e8fa1ac5383727250b7680fc3", "family_id": "qf-str22-complex-multiplication", "seed": 202610223075, "semantic_template": "qf-str22-complex-multiplication/{\"left\":[12,0],\"right\":[2,39]}" }, "fields": { "title": "The Negative Real Contribution from the Product of Imaginary Parts", "stem": "Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[12, 0], right=[c,d]=[2, 39]. Give (left,right,[a*c−b*d,a*d+b*c]).", "answer": { "kind": "expression", "value": "([12, 0], [2, 39], [24, 468])", "explanation": "Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 0], [2, 39], [24, 468])." }, "skills": [ "The Negative Real Contribution from the Product of Imaginary Parts" ], "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": "複數以整數有序對表示。操作=complex-multiplication；資料={\"left\":[12,0],\"right\":[2,39]}。請依 [left,right,result] 欄序回答。", "en": "Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[12, 0], right=[c,d]=[2, 39]. Give (left,right,[a*c−b*d,a*d+b*c])." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-str22-complex-multiplication, complex-multiplication]", "after": "derived = [[left, right, result]]", "relation": "implies", "rule_id": "STR22_exact_interface_and_source_ORDERED_tuple_roles", "conditions": [], "explanation": { "zh-Hant": "此銀行依原tuple欄序與hyphen operation。fraction約分、正分母且整數省略/1，與前輪舊契約不同；不改原中英快照或題意。", "en": "Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Complex ORIGINAL real-real product := mul(12, 2)", "after": "Complex ORIGINAL real-real product = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Complex ORIGINAL real-real product，不以小數近似替代。", "en": "Recompute Complex ORIGINAL real-real product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "2" ], "output": "24", "label": "Complex ORIGINAL real-real product" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Complex ORIGINAL imaginary-imaginary product := mul(0, 39)", "after": "Complex ORIGINAL imaginary-imaginary product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Complex ORIGINAL imaginary-imaginary product，不以小數近似替代。", "en": "Recompute Complex ORIGINAL imaginary-imaginary product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "39" ], "output": "0", "label": "Complex ORIGINAL imaginary-imaginary product" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Complex ORIGINAL real-imaginary product := mul(12, 39)", "after": "Complex ORIGINAL real-imaginary product = 468", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Complex ORIGINAL real-imaginary product，不以小數近似替代。", "en": "Recompute Complex ORIGINAL real-imaginary product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "39" ], "output": "468", "label": "Complex ORIGINAL real-imaginary product" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Complex ORIGINAL imaginary-real product := mul(0, 2)", "after": "Complex ORIGINAL imaginary-real product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Complex ORIGINAL imaginary-real product，不以小數近似替代。", "en": "Recompute Complex ORIGINAL imaginary-real product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "2" ], "output": "0", "label": "Complex ORIGINAL imaginary-real product" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Complex ORIGINAL real part ac minus bd := sub(24, 0)", "after": "Complex ORIGINAL real part ac minus bd = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Complex ORIGINAL real part ac minus bd，不以小數近似替代。", "en": "Recompute Complex ORIGINAL real part ac minus bd with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "24", "0" ], "output": "24", "label": "Complex ORIGINAL real part ac minus bd" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Complex ORIGINAL imaginary coefficient ad plus bc := sum(468, 0)", "after": "Complex ORIGINAL imaginary coefficient ad plus bc = 468", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Complex ORIGINAL imaginary coefficient ad plus bc，不以小數近似替代。", "en": "Recompute Complex ORIGINAL imaginary coefficient ad plus bc with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "468", "0" ], "output": "468", "label": "Complex ORIGINAL imaginary coefficient ad plus bc" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [[12, 0], [2, 39]]", "after": "derived = [24, 0, 468, 0, [24, 468]]", "relation": "implies", "rule_id": "Complex_distribute_i_squared_MINUSone_not_Hadamard", "conditions": [], "explanation": { "zh-Hant": "完整分配得ac+(ad+bc)i+bd*i²，代i²=-1得(ac-bd,ad+bc)。derived可超raw1000或default1e6，無新cap；不是逐coordinate相乘。", "en": "Distribute to ac+(ad+bc)i+bd*i²,and use i²=-1 to obtain(ac-bd,ad+bc).Derived components may exceed raw1000/default1e6 bounds;do not impose a new cap or use a Hadamard product." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null } ], "math_model": { "result": [ [ 12, 0 ], [ 2, 39 ], [ 24, 468 ] ] }, "conclusion": { "canonical_value": "([12, 0], [2, 39], [24, 468])", "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-20261013-advance-20261022-r03-complex-number-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261022-r03-complex-number-foundations/questions.jsonl", "batch_sha256": "9328a7410834a1b930f6821174cd6458453cbac067d8be94a87f81f1b6fb2a32", "original_line": 75, "original_line_sha256": "8f0afdfa1cfeb0d5852d8ae131dc338720224e84a7ab32d70a57e6ed09e438b4", "question_compact_sha256": "38a8cebd607ca5ae33b8fc74f11c37ebe161693e8fa1ac5383727250b7680fc3", "english_row_compact_sha256": "b4dda700f9e19703a3da958c4e7ef70e26b689aff6f32021bff6f90815f6c55d", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261022-r03-complex-number-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261022-r03-complex-number-foundations.en.jsonl" }, "mko_bindings": { "mko-complex-number": { "sha256": "a0d0b2223ccef6be06ad99f3ba6f05b76e1570d35ae43f7caac2509712c0431a", "english_sha256": "b38702b90e90f16fa680c51c92fc226a64a08dabcfed32e524b7b9465b4e9a75", "original_path": "public/data/mko/complex-number.json", "english_path": "public/data/i18n/editions/mko-complex-number.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "cd714b09e116fbc0088007895c4d956bc0f4fad5a61507e4a16f09649d05a3b0", "original_candidate_sha256": "435aaff171c5792f565140165e64e2cd0741491e797ab559fad56acfac45e7ba", "original_candidate_line_sha256": "435aaff171c5792f565140165e64e2cd0741491e797ab559fad56acfac45e7ba" }, "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-20261026-011-p58", "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": "db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6", "question_first_witness_sha256": "58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 8, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "382b9527dda98705a623350ea65ff2847c5062625eebb1dd14f609e451834122", "arithmetic_steps": 6 }, "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." } }