{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261006-advance-20261007-r02-linear-transformation-0086", "pilot_number": "R0064-32861", "number": 32861, "family_id": "qf-linear-map-homogeneity-values", "title": { "zh-Hant": "第 32861 題：矩陣模型純量兩側值", "en": "Question 32861: Both Values in a Matrix Scalar Identity" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261006-advance-20261007-r02-linear-transformation-0086", "batch_id": "batch-20261006-advance-20261007-r02-linear-transformation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-linear-transformation" ], "title_zh": "矩陣模型純量兩側值", "stem_zh": "實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-2, 3], [-2, 11]]，T(x)=Mx。u=[-2, 11]、c=1，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。", "answer": { "kind": "expression", "value": "[37, 125] = [37, 125]", "explanation_zh": "兩側值=[37, 125] = [37, 125]。零純量、負純量及零輸入均按宣告模型計算；這不是任意函數的抽樣證明。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "矩陣模型純量兩側值" ] }, "variant": { "family_id": "qf-linear-map-homogeneity-values", "seed": 202610072086, "parameters": { "M": [ [ -2, 3 ], [ -2, 11 ] ], "u": [ -2, 11 ], "c": 1, "operation": "homogeneity_values", "domain": "reals", "semantic_template": "qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],[\"-2\",\"11\"],\"1\"],\"operation\":\"homogeneity_values\"}" } }, "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-r02-linear-transformation-0086", "batch_id": "batch-20261006-advance-20261007-r02-linear-transformation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "ff815199d46d5a1e1b706737534d79b1ebddf94384a3e42e5e9cb479b4e9dfcc", "family_id": "qf-linear-map-homogeneity-values", "seed": 202610072086, "semantic_template": "qf-linear-map-homogeneity-values/{\"inputs\":[[[\"-2\",\"3\"],[\"-2\",\"11\"]],[\"-2\",\"11\"],\"1\"],\"operation\":\"homogeneity_values\"}" }, "fields": { "title": "Both Values in a Matrix Scalar Identity", "stem": "Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-2, 11] and c=1, compute T(cu) and cT(u). Answer as left vector = right vector.", "answer": { "kind": "expression", "value": "[37, 125] = [37, 125]", "explanation": "Evaluate both sides in the declared model to obtain [37, 125] = [37, 125], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function." }, "skills": [ "Both Values in a Matrix Scalar Identity" ], "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": "實平面線性模型，固定標準基底，以直行向量記輸入，M=[[-2, 3], [-2, 11]]，T(x)=Mx。u=[-2, 11]、c=1，分別求 T(cu) 與 cT(u)，答「左向量 = 右向量」。", "en": "Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-2, 11] and c=1, compute T(cu) and cT(u). Answer as left vector = right vector." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-linear-map-homogeneity-values, homogeneity_values, reals, []]", "after": "derived = [[VECTOR_EQUALS_VECTOR_STRING]]", "relation": "implies", "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles", "conditions": [], "explanation": { "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。", "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Homogeneity ORIGINAL c times u ORIGINAL component0 := mul(1, -2)", "after": "Homogeneity ORIGINAL c times u ORIGINAL component0 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity ORIGINAL c times u ORIGINAL component0，不以小數近似替代。", "en": "Recompute Homogeneity ORIGINAL c times u ORIGINAL component0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-2" ], "output": "-2", "label": "Homogeneity ORIGINAL c times u ORIGINAL component0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Homogeneity ORIGINAL c times u ORIGINAL component1 := mul(1, 11)", "after": "Homogeneity ORIGINAL c times u ORIGINAL component1 = 11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity ORIGINAL c times u ORIGINAL component1，不以小數近似替代。", "en": "Recompute Homogeneity ORIGINAL c times u ORIGINAL component1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "11" ], "output": "11", "label": "Homogeneity ORIGINAL c times u ORIGINAL component1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product := mul(-2, -2)", "after": "Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product，不以小數近似替代。", "en": "Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "Homogeneity LEFT M of scaled input ORIGINAL row0 column0 product" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product := mul(3, 11)", "after": "Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product = 33", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product，不以小數近似替代。", "en": "Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "11" ], "output": "33", "label": "Homogeneity LEFT M of scaled input ORIGINAL row0 column1 product" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum := sum(4, 33)", "after": "Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum = 37", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum，不以小數近似替代。", "en": "Recompute Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "33" ], "output": "37", "label": "Homogeneity LEFT M of scaled input ORIGINAL row0 two products sum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Homogeneity LEFT M of scaled input, 0, [-2, 3], [-2, 11]]", "after": "derived = [[4, 33], 37]", "relation": "implies", "rule_id": "LinearMap_EACH_row_by_column_component_ALL_products", "conditions": [], "explanation": { "zh-Hant": "固定標準基底用row-by-column，兩個product都加；derived向量分量可超SAFE，不能再當raw vector重驗。", "en": "In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product := mul(-2, -2)", "after": "Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product，不以小數近似替代。", "en": "Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "Homogeneity LEFT M of scaled input ORIGINAL row1 column0 product" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product := mul(11, 11)", "after": "Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product = 121", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product，不以小數近似替代。", "en": "Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "11" ], "output": "121", "label": "Homogeneity LEFT M of scaled input ORIGINAL row1 column1 product" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum := sum(4, 121)", "after": "Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum = 125", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum，不以小數近似替代。", "en": "Recompute Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "121" ], "output": "125", "label": "Homogeneity LEFT M of scaled input ORIGINAL row1 two products sum" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [Homogeneity LEFT M of scaled input, 1, [-2, 11], [-2, 11]]", "after": "derived = [[4, 121], 125]", "relation": "implies", "rule_id": "LinearMap_EACH_row_by_column_component_ALL_products", "conditions": [], "explanation": { "zh-Hant": "固定標準基底用row-by-column，兩個product都加；derived向量分量可超SAFE，不能再當raw vector重驗。", "en": "In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Homogeneity RIGHT M of u ORIGINAL row0 column0 product := mul(-2, -2)", "after": "Homogeneity RIGHT M of u ORIGINAL row0 column0 product = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity RIGHT M of u ORIGINAL row0 column0 product，不以小數近似替代。", "en": "Recompute Homogeneity RIGHT M of u ORIGINAL row0 column0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "Homogeneity RIGHT M of u ORIGINAL row0 column0 product" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Homogeneity RIGHT M of u ORIGINAL row0 column1 product := mul(3, 11)", "after": "Homogeneity RIGHT M of u ORIGINAL row0 column1 product = 33", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity RIGHT M of u ORIGINAL row0 column1 product，不以小數近似替代。", "en": "Recompute Homogeneity RIGHT M of u ORIGINAL row0 column1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "11" ], "output": "33", "label": "Homogeneity RIGHT M of u ORIGINAL row0 column1 product" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Homogeneity RIGHT M of u ORIGINAL row0 two products sum := sum(4, 33)", "after": "Homogeneity RIGHT M of u ORIGINAL row0 two products sum = 37", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity RIGHT M of u ORIGINAL row0 two products sum，不以小數近似替代。", "en": "Recompute Homogeneity RIGHT M of u ORIGINAL row0 two products sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "33" ], "output": "37", "label": "Homogeneity RIGHT M of u ORIGINAL row0 two products sum" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "source roles = [Homogeneity RIGHT M of u, 0, [-2, 3], [-2, 11]]", "after": "derived = [[4, 33], 37]", "relation": "implies", "rule_id": "LinearMap_EACH_row_by_column_component_ALL_products", "conditions": [], "explanation": { "zh-Hant": "固定標準基底用row-by-column，兩個product都加；derived向量分量可超SAFE，不能再當raw vector重驗。", "en": "In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Homogeneity RIGHT M of u ORIGINAL row1 column0 product := mul(-2, -2)", "after": "Homogeneity RIGHT M of u ORIGINAL row1 column0 product = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity RIGHT M of u ORIGINAL row1 column0 product，不以小數近似替代。", "en": "Recompute Homogeneity RIGHT M of u ORIGINAL row1 column0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "Homogeneity RIGHT M of u ORIGINAL row1 column0 product" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Homogeneity RIGHT M of u ORIGINAL row1 column1 product := mul(11, 11)", "after": "Homogeneity RIGHT M of u ORIGINAL row1 column1 product = 121", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity RIGHT M of u ORIGINAL row1 column1 product，不以小數近似替代。", "en": "Recompute Homogeneity RIGHT M of u ORIGINAL row1 column1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "11" ], "output": "121", "label": "Homogeneity RIGHT M of u ORIGINAL row1 column1 product" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Homogeneity RIGHT M of u ORIGINAL row1 two products sum := sum(4, 121)", "after": "Homogeneity RIGHT M of u ORIGINAL row1 two products sum = 125", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity RIGHT M of u ORIGINAL row1 two products sum，不以小數近似替代。", "en": "Recompute Homogeneity RIGHT M of u ORIGINAL row1 two products sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "121" ], "output": "125", "label": "Homogeneity RIGHT M of u ORIGINAL row1 two products sum" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "source roles = [Homogeneity RIGHT M of u, 1, [-2, 11], [-2, 11]]", "after": "derived = [[4, 121], 125]", "relation": "implies", "rule_id": "LinearMap_EACH_row_by_column_component_ALL_products", "conditions": [], "explanation": { "zh-Hant": "固定標準基底用row-by-column，兩個product都加；derived向量分量可超SAFE，不能再當raw vector重驗。", "en": "In the fixed standard basis,use row-by-column and add BOTH products.Derived vector components may exceed SAFE;do not reparse them as RAW vectors." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Homogeneity RIGHT c times image ORIGINAL component0 := mul(1, 37)", "after": "Homogeneity RIGHT c times image ORIGINAL component0 = 37", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity RIGHT c times image ORIGINAL component0，不以小數近似替代。", "en": "Recompute Homogeneity RIGHT c times image ORIGINAL component0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "37" ], "output": "37", "label": "Homogeneity RIGHT c times image ORIGINAL component0" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Homogeneity RIGHT c times image ORIGINAL component1 := mul(1, 125)", "after": "Homogeneity RIGHT c times image ORIGINAL component1 = 125", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Homogeneity RIGHT c times image ORIGINAL component1，不以小數近似替代。", "en": "Recompute Homogeneity RIGHT c times image ORIGINAL component1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "125" ], "output": "125", "label": "Homogeneity RIGHT c times image ORIGINAL component1" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "source roles = [homogeneity_values, [[-2, 3], [-2, 11]], [-2, 11], 1]", "after": "derived = [[37, 125], [37, 125], [37, 125] = [37, 125]]", "relation": "implies", "rule_id": "LinearMap_BOTH_actual_vectors_equality_STRING_NOT_sample_linearity_proof", "conditions": [], "explanation": { "zh-Hant": "完整各算兩側向量再比相等，輸出[left]=[right]，不是YesNo／tuple／point。零或負c、零向量合法；這些是明示矩陣模型的值，不是任意函數的有限抽樣線性證明。", "en": "Compute BOTH actual vectors completely,then compare.Output[left]=[right],not YesNo/a tuple/a point.Zero/negative scalars and zero vectors are legal.These are values in the explicit matrix model,not finite-sample proof of linearity for an arbitrary function." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": "[37, 125] = [37, 125]" }, "conclusion": { "canonical_value": "[37, 125] = [37, 125]", "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-r02-linear-transformation-foundations", "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r02-linear-transformation-foundations/questions.jsonl", "batch_sha256": "8aa7124b4f367aa227816eaa77df8620070a5f7ef0a9cf180a0decb92ac73796", "original_line": 86, "original_line_sha256": "baa7fd11b34f7913ca5d6a16a620eaa6aef928f8ce615791fd6ee5eb00f5de9a", "question_compact_sha256": "ff815199d46d5a1e1b706737534d79b1ebddf94384a3e42e5e9cb479b4e9dfcc", "english_row_compact_sha256": "4dc6d4eea63b272f3765942e303a211732580b53dfeab8181e574e6adb300350", "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r02-linear-transformation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r02-linear-transformation-foundations.en.jsonl" }, "mko_bindings": { "mko-linear-transformation": { "sha256": "59ce4d8fac03871ea16eb62a76c9fae255470dc391ff7312ad4da120bafbf132", "english_sha256": "d95d3ac4155053ddc0d186089ae6ec2d7e4e20468aa14880d371ac260c3cd0a3", "original_path": "public/data/mko/linear-transformation.json", "english_path": "public/data/i18n/editions/mko-linear-transformation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "528186aa825c3054db15f4ce0a03ad58f72be7e7eb5bb236c4cc7ef34ff5a470", "original_candidate_sha256": "7ed79d81d33950968fc9bd5ff79e397a5ad0d578c41b61eb53c0d8f053ae3371", "original_candidate_line_sha256": "7ed79d81d33950968fc9bd5ff79e397a5ad0d578c41b61eb53c0d8f053ae3371" }, "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-p65", "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": "8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b", "question_first_witness_sha256": "066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 22, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "5d6e394a139894c8d4efd78302afbf9950292e08784c1e179362e8a66bda2bd1", "arithmetic_steps": 16 }, "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." } }