{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261006-advance-20261007-r06-similarity-0081", "pilot_number": "R0056-28831", "number": 28831, "family_id": "qf-similarity-scaled-area", "title": { "zh-Hant": "第 28831 題：已給正面積的條件縮放", "en": "Question 28831: Conditional Scaling of a Given Positive Area" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261006-advance-20261007-r06-similarity-0081", "batch_id": "batch-20261006-advance-20261007-r06-similarity-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-similarity" ], "title_zh": "已給正面積的條件縮放", "stem_zh": "等單位正交座標系，兩軸採同一長度單位。已給某非退化三角形面積 11/2>0 平方單位，對應長度倍率 k=2/3>0。求縮放後條件面積，採最簡正分母分數。", "answer": { "kind": "expression", "value": "22/9", "explanation_zh": "條件面積=22/9。給定面積乘 k²；不從座標另計三角形面積，不繼承矩形面積 Evidence，也不宣稱一般測度理論。" }, "difficulty": { "band": "lower_secondary", "level": 1, "skills": [ "已給正面積的條件縮放" ] }, "variant": { "family_id": "qf-similarity-scaled-area", "seed": 202610076081, "parameters": { "area_n": 11, "area_d": 2, "p": 2, "q": 3, "operation": "scaled_area", "frame": "euclidean_orthonormal_equal_units", "semantic_template": "qf-similarity-scaled-area/{\"inputs\":[\"11/2\",\"2/3\"],\"operation\":\"scaled_area\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-06", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261006-advance-20261007-r06-similarity-0081", "batch_id": "batch-20261006-advance-20261007-r06-similarity-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4ef2c291393d4ac8dfe1a6f8f07ff49fc4d1fe7ed58e3d474e0b957c98ce5187", "family_id": "qf-similarity-scaled-area", "seed": 202610076081, "semantic_template": "qf-similarity-scaled-area/{\"inputs\":[\"11/2\",\"2/3\"],\"operation\":\"scaled_area\"}" }, "fields": { "title": "Conditional Scaling of a Given Positive Area", "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 11/2>0 squared units. At corresponding positive length scale k=2/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers.", "answer": { "kind": "expression", "value": "22/9", "explanation": "Multiply the given positive area by k² to obtain 22/9 squared units. Do not recompute a triangle area from coordinates, inherit rectangle-area Evidence, or claim general measure theory." }, "skills": [ "Conditional Scaling of a Given Positive Area" ], "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": "等單位正交座標系，兩軸採同一長度單位。已給某非退化三角形面積 11/2>0 平方單位，對應長度倍率 k=2/3>0。求縮放後條件面積，採最簡正分母分數。", "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. A nondegenerate triangle has given area 11/2>0 squared units. At corresponding positive length scale k=2/3, find the conditional scaled area. Give a reduced fraction with positive denominator, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-similarity-scaled-area, scaled_area]", "after": "derived = [given_positive_area_times_squared_scale]", "relation": "implies", "rule_id": "R56_source_answer_role_and_EXACT_family_interface", "conditions": [], "explanation": { "zh-Hant": "本題輸出依原scalar／set／是非／fraction角色；有理數整數保留/1。不加未宣告domain，不改中英來源、公式或題目。", "en": "Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Similarity ORIGINAL scale numerator squared := mul(2, 2)", "after": "Similarity ORIGINAL scale numerator squared = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Similarity ORIGINAL scale numerator squared，不以小數近似替代。", "en": "Recompute Similarity ORIGINAL scale numerator squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Similarity ORIGINAL scale numerator squared" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Similarity ORIGINAL scale denominator squared := mul(3, 3)", "after": "Similarity ORIGINAL scale denominator squared = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Similarity ORIGINAL scale denominator squared，不以小數近似替代。", "en": "Recompute Similarity ORIGINAL scale denominator squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Similarity ORIGINAL scale denominator squared" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Similarity ORIGINAL given area numerator times p squared := mul(11, 4)", "after": "Similarity ORIGINAL given area numerator times p squared = 44", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Similarity ORIGINAL given area numerator times p squared，不以小數近似替代。", "en": "Recompute Similarity ORIGINAL given area numerator times p squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "4" ], "output": "44", "label": "Similarity ORIGINAL given area numerator times p squared" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Similarity ORIGINAL given area denominator times q squared := mul(2, 9)", "after": "Similarity ORIGINAL given area denominator times q squared = 18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Similarity ORIGINAL given area denominator times q squared，不以小數近似替代。", "en": "Recompute Similarity ORIGINAL given area denominator times q squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "9" ], "output": "18", "label": "Similarity ORIGINAL given area denominator times q squared" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Similarity ORIGINAL scaled area fraction := div(44, 18)", "after": "Similarity ORIGINAL scaled area fraction = 22/9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Similarity ORIGINAL scaled area fraction，不以小數近似替代。", "en": "Recompute Similarity ORIGINAL scaled area fraction with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "44", "18" ], "output": "22/9", "label": "Similarity ORIGINAL scaled area fraction" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [scaled_area, [11, 2, 2, 3]]", "after": "derived = [4, 9, 22/9]", "relation": "implies", "rule_id": "Similarity_GIVEN_positive_square_length_or_area_times_SCALE_SQUARED", "conditions": [], "explanation": { "zh-Hant": "長度乘k=p/q，故平方邊長與面積乘k²，不乘k。面積由兩個獨立長度方向的行列式縮放得k²；本題是給定正area的條件式換算，不冒充由坐標另驗面積或一般measure定理。", "en": "Lengths scale by k=p/q,so squared lengths and area scale by k²,not k.Two independent length directions give determinant factor k².This is conditional conversion of the GIVEN positive area,not a coordinate-area recomputation or a general measure theorem." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null } ], "math_model": { "result": "22/9" }, "conclusion": { "canonical_value": "22/9", "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-20261006-advance-20261007-r06-similarity-foundations", "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r06-similarity-foundations/questions.jsonl", "batch_sha256": "85dde18b8aa8a90c777016e6a02bf75e3032d389b8b3a625c6c5ec208949a4f6", "original_line": 81, "original_line_sha256": "f350e8e70e0ab7b63a80e411709d3233c9157acd96a989449ce43ebed5026cd6", "question_compact_sha256": "4ef2c291393d4ac8dfe1a6f8f07ff49fc4d1fe7ed58e3d474e0b957c98ce5187", "english_row_compact_sha256": "a2157aa535431e0f1255a8cbcf9a2e382b7a09731522d98258633f423583b071", "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r06-similarity-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r06-similarity-foundations.en.jsonl" }, "mko_bindings": { "mko-similarity": { "sha256": "d60b59d396063a61d7b04c2039c88873846f8e3486fe0fa326adf55123c006f4", "english_sha256": "a5cdadef7a5b49238b9ced0b42ab794c51fdffe9bd83f46239b1c56d962af72e", "original_path": "public/data/mko/similarity.json", "english_path": "public/data/i18n/editions/mko-similarity.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "e340023047850507ee3fe2ae1466d606cfcbd26849aa3bb168d53248f38097e5", "original_candidate_sha256": "037d8defe20ffc971cc9babb05679927917dd022dcfd8d822e05d7171947677f", "original_candidate_line_sha256": "037d8defe20ffc971cc9babb05679927917dd022dcfd8d822e05d7171947677f" }, "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-20261026-010-p57", "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": "8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8", "question_first_witness_sha256": "3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 7, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "9b680a9729b7760ecdf6c2e6ed74b7983473e69de7fda277c3586257e683dd94", "arithmetic_steps": 5 }, "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." } }