{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261026-r05-covariance-0061", "pilot_number": "R0023-12211", "number": 12211, "family_id": "qf-mod26-covariance-affine", "title": { "zh-Hant": "第 12211 題：協方差", "en": "Question 12211: Affine Shifts Drop Out of Covariance" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261026-r05-covariance-0061", "batch_id": "batch-20261013-advance-20261026-r05-covariance-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-covariance" ], "title_zh": "協方差", "stem_zh": "協方差必須綁定同一完整有限joint law。操作=covariance-affine；資料={\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"a\":0,\"b\":-2,\"c\":-2,\"d\":2}。請依 [a,b,c,d,covariance,transformed_covariance,identity] 欄序回答。", "answer": { "kind": "expression", "value": "(0, -2, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))", "explanation_zh": "結構深度核對=(0, -2, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；非零協方差可否證獨立；零協方差通常不推出獨立。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "協方差" ] }, "variant": { "family_id": "qf-mod26-covariance-affine", "seed": 202610265061, "parameters": { "support": [ [ -2, -1 ], [ -1, 0 ], [ 0, 1 ], [ 1, 2 ] ], "masses": [ [ 1, 8 ], [ 1, 4 ], [ 1, 8 ], [ 1, 2 ] ], "a": 0, "b": -2, "c": -2, "d": 2, "domain": "Exact_finite_joint_covariance", "operation": "covariance-affine", "semantic_template": "qf-mod26-covariance-affine/{\"a\":0,\"b\":-2,\"c\":-2,\"d\":2,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r05-covariance-0061", "batch_id": "batch-20261013-advance-20261026-r05-covariance-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "19bfae2c0461d8346dc0bb96cb8ba82e3be4e57307030ee7125c1d69e420baeb", "family_id": "qf-mod26-covariance-affine", "seed": 202610265061, "semantic_template": "qf-mod26-covariance-affine/{\"a\":0,\"b\":-2,\"c\":-2,\"d\":2,\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"support\":[[-2,-1],[-1,0],[0,1],[1,2]]}" }, "fields": { "title": "Affine Shifts Drop Out of Covariance", "stem": "Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[0, -2, -2, 2]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y)).", "answer": { "kind": "expression", "value": "(0, -2, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))", "explanation": "The shifts cancel when the product of means is subtracted; the two scale factors multiply the original covariance. Keep the scale signs. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -2, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))." }, "skills": [ "Affine Shifts Drop Out of Covariance" ], "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": "協方差必須綁定同一完整有限joint law。操作=covariance-affine；資料={\"support\":[[-2,-1],[-1,0],[0,1],[1,2]],\"masses\":[[1,8],[1,4],[1,8],[1,2]],\"a\":0,\"b\":-2,\"c\":-2,\"d\":2}。請依 [a,b,c,d,covariance,transformed_covariance,identity] 欄序回答。", "en": "Use the complete list of distinct integer pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]] and their aligned exact masses=[1/8, 1/4, 1/8, 1/2] on one common finite probability space. Qualification requires every mass to be nonnegative and their exact sum to be 1; zero masses are allowed. Preserve pair-mass alignment. Marginals alone do not specify this joint law or imply independence. No continuous or infinite-support conclusion is claimed. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require a qualified joint table. Transform to aX+b and cY+d with [a,b,c,d]=[0, -2, -2, 2]. Give (a,b,c,d,covariance,transformed_covariance,Cov(aX+b,cY+d)=ac Cov(X,Y))." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "joint law total mass := sum(1/8, 1/4, 1/8, 1/2)", "after": "joint law total mass = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 joint law total mass，不以小數近似替代。", "en": "Recompute joint law total mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/8", "1/4", "1/8", "1/2" ], "output": "1", "label": "joint law total mass" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "original distinct joint pairs=[[-2, -1], [-1, 0], [0, 1], [1, 2]];raw aligned masses=[[1, 8], [1, 4], [1, 8], [1, 2]]", "after": "normalized aligned masses=[1/8, 1/4, 1/8, 1/2];nonnegative=是;total=1", "relation": "implies", "rule_id": "qualify_whole_joint_law_preserve_alignment", "conditions": [], "explanation": { "zh-Hant": "只用同一共同機率空間的完整原始聯合表，保留順序、配對與零質量；每個質量非負且精確總和1才可求協方差。", "en": "Use the complete original joint table on one common probability space,preserving order,pairing and zero masses.Covariance requires nonnegative masses with exact total1." }, "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": "S003", "uses": [ "G001", "S002" ], "before": "original atom0 weighted X := mul(1/8, -2)", "after": "original atom0 weighted X = -1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom0 weighted X，不以小數近似替代。", "en": "Recompute original atom0 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "-2" ], "output": "-1/4", "label": "original atom0 weighted X" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "original atom0 weighted Y := mul(1/8, -1)", "after": "original atom0 weighted Y = -1/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom0 weighted Y，不以小數近似替代。", "en": "Recompute original atom0 weighted Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "-1" ], "output": "-1/8", "label": "original atom0 weighted Y" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "original atom0 XY := mul(-2, -1)", "after": "original atom0 XY = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom0 XY，不以小數近似替代。", "en": "Recompute original atom0 XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-1" ], "output": "2", "label": "original atom0 XY" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "original atom0 weighted XY := mul(1/8, 2)", "after": "original atom0 weighted XY = 1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom0 weighted XY，不以小數近似替代。", "en": "Recompute original atom0 weighted XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "2" ], "output": "1/4", "label": "original atom0 weighted XY" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "original atom1 weighted X := mul(1/4, -1)", "after": "original atom1 weighted X = -1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom1 weighted X，不以小數近似替代。", "en": "Recompute original atom1 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "-1" ], "output": "-1/4", "label": "original atom1 weighted X" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "original atom1 weighted Y := mul(1/4, 0)", "after": "original atom1 weighted Y = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom1 weighted Y，不以小數近似替代。", "en": "Recompute original atom1 weighted Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "0" ], "output": "0", "label": "original atom1 weighted Y" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "original atom1 XY := mul(-1, 0)", "after": "original atom1 XY = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom1 XY，不以小數近似替代。", "en": "Recompute original atom1 XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "0" ], "output": "0", "label": "original atom1 XY" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "original atom1 weighted XY := mul(1/4, 0)", "after": "original atom1 weighted XY = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom1 weighted XY，不以小數近似替代。", "en": "Recompute original atom1 weighted XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "0" ], "output": "0", "label": "original atom1 weighted XY" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "original atom2 weighted X := mul(1/8, 0)", "after": "original atom2 weighted X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom2 weighted X，不以小數近似替代。", "en": "Recompute original atom2 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "0" ], "output": "0", "label": "original atom2 weighted X" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "original atom2 weighted Y := mul(1/8, 1)", "after": "original atom2 weighted Y = 1/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom2 weighted Y，不以小數近似替代。", "en": "Recompute original atom2 weighted Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "1" ], "output": "1/8", "label": "original atom2 weighted Y" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "original atom2 XY := mul(0, 1)", "after": "original atom2 XY = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom2 XY，不以小數近似替代。", "en": "Recompute original atom2 XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "original atom2 XY" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "original atom2 weighted XY := mul(1/8, 0)", "after": "original atom2 weighted XY = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom2 weighted XY，不以小數近似替代。", "en": "Recompute original atom2 weighted XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "0" ], "output": "0", "label": "original atom2 weighted XY" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "original atom3 weighted X := mul(1/2, 1)", "after": "original atom3 weighted X = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom3 weighted X，不以小數近似替代。", "en": "Recompute original atom3 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "original atom3 weighted X" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "original atom3 weighted Y := mul(1/2, 2)", "after": "original atom3 weighted Y = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom3 weighted Y，不以小數近似替代。", "en": "Recompute original atom3 weighted Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "2" ], "output": "1", "label": "original atom3 weighted Y" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "original atom3 XY := mul(1, 2)", "after": "original atom3 XY = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom3 XY，不以小數近似替代。", "en": "Recompute original atom3 XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "original atom3 XY" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "original atom3 weighted XY := mul(1/2, 2)", "after": "original atom3 weighted XY = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original atom3 weighted XY，不以小數近似替代。", "en": "Recompute original atom3 weighted XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "2" ], "output": "1", "label": "original atom3 weighted XY" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "original E_X := sum(-1/4, -1/4, 0, 1/2)", "after": "original E_X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original E_X，不以小數近似替代。", "en": "Recompute original E_X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1/4", "-1/4", "0", "1/2" ], "output": "0", "label": "original E_X" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "original E_Y := sum(-1/8, 0, 1/8, 1)", "after": "original E_Y = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original E_Y，不以小數近似替代。", "en": "Recompute original E_Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1/8", "0", "1/8", "1" ], "output": "1", "label": "original E_Y" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "original E_XY := sum(1/4, 0, 0, 1)", "after": "original E_XY = 5/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original E_XY，不以小數近似替代。", "en": "Recompute original E_XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/4", "0", "0", "1" ], "output": "5/4", "label": "original E_XY" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "original product of means := mul(0, 1)", "after": "original product of means = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original product of means，不以小數近似替代。", "en": "Recompute original product of means with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "original product of means" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "original covariance := sub(5/4, 0)", "after": "original covariance = 5/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 original covariance，不以小數近似替代。", "en": "Recompute original covariance with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "5/4", "0" ], "output": "5/4", "label": "original covariance" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "original:E_X=0;E_Y=1;E_XY=5/4", "after": "Cov=E_XY-E_X*E_Y=5/4", "relation": "implies", "rule_id": "covariance_same_joint_three_moments", "conditions": [], "explanation": { "zh-Hant": "三個期望全部使用相同原子與質量，不能混合不同邊際模型，亦不假設獨立。", "en": "All three expectations use the same atoms and masses.Do not mix marginal models or assume independence." }, "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": "S025", "uses": [ "G001", "S024" ], "before": "affine atom0 scaled X := mul(0, -2)", "after": "affine atom0 scaled X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom0 scaled X，不以小數近似替代。", "en": "Recompute affine atom0 scaled X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-2" ], "output": "0", "label": "affine atom0 scaled X" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "affine atom0 U := sum(0, -2)", "after": "affine atom0 U = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom0 U，不以小數近似替代。", "en": "Recompute affine atom0 U with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-2" ], "output": "-2", "label": "affine atom0 U" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "affine atom0 scaled Y := mul(-2, -1)", "after": "affine atom0 scaled Y = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom0 scaled Y，不以小數近似替代。", "en": "Recompute affine atom0 scaled Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-1" ], "output": "2", "label": "affine atom0 scaled Y" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "affine atom0 V := sum(2, 2)", "after": "affine atom0 V = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom0 V，不以小數近似替代。", "en": "Recompute affine atom0 V with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "2" ], "output": "4", "label": "affine atom0 V" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "affine atom1 scaled X := mul(0, -1)", "after": "affine atom1 scaled X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom1 scaled X，不以小數近似替代。", "en": "Recompute affine atom1 scaled X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-1" ], "output": "0", "label": "affine atom1 scaled X" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "affine atom1 U := sum(0, -2)", "after": "affine atom1 U = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom1 U，不以小數近似替代。", "en": "Recompute affine atom1 U with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-2" ], "output": "-2", "label": "affine atom1 U" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "affine atom1 scaled Y := mul(-2, 0)", "after": "affine atom1 scaled Y = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom1 scaled Y，不以小數近似替代。", "en": "Recompute affine atom1 scaled Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "0" ], "output": "0", "label": "affine atom1 scaled Y" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "affine atom1 V := sum(0, 2)", "after": "affine atom1 V = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom1 V，不以小數近似替代。", "en": "Recompute affine atom1 V with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "2" ], "output": "2", "label": "affine atom1 V" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "affine atom2 scaled X := mul(0, 0)", "after": "affine atom2 scaled X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom2 scaled X，不以小數近似替代。", "en": "Recompute affine atom2 scaled X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "affine atom2 scaled X" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "affine atom2 U := sum(0, -2)", "after": "affine atom2 U = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom2 U，不以小數近似替代。", "en": "Recompute affine atom2 U with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-2" ], "output": "-2", "label": "affine atom2 U" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "affine atom2 scaled Y := mul(-2, 1)", "after": "affine atom2 scaled Y = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom2 scaled Y，不以小數近似替代。", "en": "Recompute affine atom2 scaled Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "affine atom2 scaled Y" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "affine atom2 V := sum(-2, 2)", "after": "affine atom2 V = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom2 V，不以小數近似替代。", "en": "Recompute affine atom2 V with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "2" ], "output": "0", "label": "affine atom2 V" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "affine atom3 scaled X := mul(0, 1)", "after": "affine atom3 scaled X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom3 scaled X，不以小數近似替代。", "en": "Recompute affine atom3 scaled X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "affine atom3 scaled X" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "affine atom3 U := sum(0, -2)", "after": "affine atom3 U = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom3 U，不以小數近似替代。", "en": "Recompute affine atom3 U with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-2" ], "output": "-2", "label": "affine atom3 U" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "affine atom3 scaled Y := mul(-2, 2)", "after": "affine atom3 scaled Y = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom3 scaled Y，不以小數近似替代。", "en": "Recompute affine atom3 scaled Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "2" ], "output": "-4", "label": "affine atom3 scaled Y" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "affine atom3 V := sum(-4, 2)", "after": "affine atom3 V = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine atom3 V，不以小數近似替代。", "en": "Recompute affine atom3 V with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "2" ], "output": "-2", "label": "affine atom3 V" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "affine transformed atom0 weighted X := mul(1/8, -2)", "after": "affine transformed atom0 weighted X = -1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom0 weighted X，不以小數近似替代。", "en": "Recompute affine transformed atom0 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "-2" ], "output": "-1/4", "label": "affine transformed atom0 weighted X" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "affine transformed atom0 weighted Y := mul(1/8, 4)", "after": "affine transformed atom0 weighted Y = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom0 weighted Y，不以小數近似替代。", "en": "Recompute affine transformed atom0 weighted Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "4" ], "output": "1/2", "label": "affine transformed atom0 weighted Y" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "affine transformed atom0 XY := mul(-2, 4)", "after": "affine transformed atom0 XY = -8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom0 XY，不以小數近似替代。", "en": "Recompute affine transformed atom0 XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "4" ], "output": "-8", "label": "affine transformed atom0 XY" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "affine transformed atom0 weighted XY := mul(1/8, -8)", "after": "affine transformed atom0 weighted XY = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom0 weighted XY，不以小數近似替代。", "en": "Recompute affine transformed atom0 weighted XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "-8" ], "output": "-1", "label": "affine transformed atom0 weighted XY" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "affine transformed atom1 weighted X := mul(1/4, -2)", "after": "affine transformed atom1 weighted X = -1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom1 weighted X，不以小數近似替代。", "en": "Recompute affine transformed atom1 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "-2" ], "output": "-1/2", "label": "affine transformed atom1 weighted X" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "affine transformed atom1 weighted Y := mul(1/4, 2)", "after": "affine transformed atom1 weighted Y = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom1 weighted Y，不以小數近似替代。", "en": "Recompute affine transformed atom1 weighted Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "2" ], "output": "1/2", "label": "affine transformed atom1 weighted Y" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "affine transformed atom1 XY := mul(-2, 2)", "after": "affine transformed atom1 XY = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom1 XY，不以小數近似替代。", "en": "Recompute affine transformed atom1 XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "2" ], "output": "-4", "label": "affine transformed atom1 XY" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "affine transformed atom1 weighted XY := mul(1/4, -4)", "after": "affine transformed atom1 weighted XY = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom1 weighted XY，不以小數近似替代。", "en": "Recompute affine transformed atom1 weighted XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "-4" ], "output": "-1", "label": "affine transformed atom1 weighted XY" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "affine transformed atom2 weighted X := mul(1/8, -2)", "after": "affine transformed atom2 weighted X = -1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom2 weighted X，不以小數近似替代。", "en": "Recompute affine transformed atom2 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "-2" ], "output": "-1/4", "label": "affine transformed atom2 weighted X" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "affine transformed atom2 weighted Y := mul(1/8, 0)", "after": "affine transformed atom2 weighted Y = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom2 weighted Y，不以小數近似替代。", "en": "Recompute affine transformed atom2 weighted Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "0" ], "output": "0", "label": "affine transformed atom2 weighted Y" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "affine transformed atom2 XY := mul(-2, 0)", "after": "affine transformed atom2 XY = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom2 XY，不以小數近似替代。", "en": "Recompute affine transformed atom2 XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "0" ], "output": "0", "label": "affine transformed atom2 XY" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "affine transformed atom2 weighted XY := mul(1/8, 0)", "after": "affine transformed atom2 weighted XY = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom2 weighted XY，不以小數近似替代。", "en": "Recompute affine transformed atom2 weighted XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/8", "0" ], "output": "0", "label": "affine transformed atom2 weighted XY" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "affine transformed atom3 weighted X := mul(1/2, -2)", "after": "affine transformed atom3 weighted X = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom3 weighted X，不以小數近似替代。", "en": "Recompute affine transformed atom3 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "-2" ], "output": "-1", "label": "affine transformed atom3 weighted X" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "affine transformed atom3 weighted Y := mul(1/2, -2)", "after": "affine transformed atom3 weighted Y = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom3 weighted Y，不以小數近似替代。", "en": "Recompute affine transformed atom3 weighted Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "-2" ], "output": "-1", "label": "affine transformed atom3 weighted Y" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "affine transformed atom3 XY := mul(-2, -2)", "after": "affine transformed atom3 XY = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom3 XY，不以小數近似替代。", "en": "Recompute affine transformed atom3 XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "affine transformed atom3 XY" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "affine transformed atom3 weighted XY := mul(1/2, 4)", "after": "affine transformed atom3 weighted XY = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed atom3 weighted XY，不以小數近似替代。", "en": "Recompute affine transformed atom3 weighted XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "4" ], "output": "2", "label": "affine transformed atom3 weighted XY" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "affine transformed E_X := sum(-1/4, -1/2, -1/4, -1)", "after": "affine transformed E_X = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed E_X，不以小數近似替代。", "en": "Recompute affine transformed E_X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1/4", "-1/2", "-1/4", "-1" ], "output": "-2", "label": "affine transformed E_X" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "affine transformed E_Y := sum(1/2, 1/2, 0, -1)", "after": "affine transformed E_Y = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed E_Y，不以小數近似替代。", "en": "Recompute affine transformed E_Y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2", "0", "-1" ], "output": "0", "label": "affine transformed E_Y" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "affine transformed E_XY := sum(-1, -1, 0, 2)", "after": "affine transformed E_XY = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed E_XY，不以小數近似替代。", "en": "Recompute affine transformed E_XY with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1", "-1", "0", "2" ], "output": "0", "label": "affine transformed E_XY" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "affine transformed product of means := mul(-2, 0)", "after": "affine transformed product of means = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed product of means，不以小數近似替代。", "en": "Recompute affine transformed product of means with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "0" ], "output": "0", "label": "affine transformed product of means" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "affine transformed covariance := sub(0, 0)", "after": "affine transformed covariance = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine transformed covariance，不以小數近似替代。", "en": "Recompute affine transformed covariance with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "affine transformed covariance" } }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "affine transformed:E_X=-2;E_Y=0;E_XY=0", "after": "Cov=E_XY-E_X*E_Y=0", "relation": "implies", "rule_id": "covariance_same_joint_three_moments", "conditions": [], "explanation": { "zh-Hant": "三個期望全部使用相同原子與質量，不能混合不同邊際模型，亦不假設獨立。", "en": "All three expectations use the same atoms and masses.Do not mix marginal models or assume independence." }, "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": "S063", "uses": [ "G001", "S062" ], "before": "affine two scale product := mul(0, -2)", "after": "affine two scale product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine two scale product，不以小數近似替代。", "en": "Recompute affine two scale product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-2" ], "output": "0", "label": "affine two scale product" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "affine scaled original covariance := mul(0, 5/4)", "after": "affine scaled original covariance = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。", "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine scaled original covariance，不以小數近似替代。", "en": "Recompute affine scaled original covariance with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "5/4" ], "output": "0", "label": "affine scaled original covariance" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "U=(0)*X+(-2);V=(-2)*Y+(2);same indexed atoms and weights", "after": "E_UV-E_U*E_V=(0)*(-2)*(E_XY-E_X*E_Y)=0", "relation": "implies", "rule_id": "affine_covariance_shift_cancellation_not_new_joint_support", "conditions": [], "explanation": { "zh-Hant": "展開E[UV]=acE[XY]+adE[X]+bcE[Y]+bd，再減(aE[X]+b)(cE[Y]+d)，平移項全消。即使縮放0使像點重複，也保留原機率空間原子，不把像誤當新的不重複support。", "en": "Expand E[UV]=acE[XY]+adE[X]+bcE[Y]+bd and subtract(aE[X]+b)(cE[Y]+d);all shift terms cancel.Even zero scaling may duplicate image points:retain original probability atoms rather than treating images as a new distinct support." }, "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": [ 0, -2, -2, 2, "5/4", "0", "Cov(aX+b,cY+d)=ac Cov(X,Y)" ] }, "conclusion": { "canonical_value": "(0, -2, -2, 2, 5/4, 0, Cov(aX+b,cY+d)=ac Cov(X,Y))", "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-20261026-r05-covariance-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261026-r05-covariance-foundations/questions.jsonl", "batch_sha256": "bbd1b50d44aff10183fe3178034a74eede1636496e7e219652bc8417573a0746", "original_line": 61, "original_line_sha256": "b898913f74a97cdeafbd45c8fdc254c9dcbc491073f1787d89f70d86d3c1af2d", "question_compact_sha256": "19bfae2c0461d8346dc0bb96cb8ba82e3be4e57307030ee7125c1d69e420baeb", "english_row_compact_sha256": "2ab35afb33014ae826331fbb27177ea65193e6664987cdc0ffefb0c813d20fab", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261026-r05-covariance-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261026-r05-covariance-foundations.en.jsonl" }, "mko_bindings": { "mko-covariance": { "sha256": "3b0480b894efd088fcbcb2c9c856fb6e46bc8b62a223285723b060bf0070143d", "english_sha256": "df82b21fa123d4f51a7320f03eab21717099356a4a1b8959e5cec7b86c26e42a", "original_path": "public/data/mko/covariance.json", "english_path": "public/data/i18n/editions/mko-covariance.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "c6ae588be9bc8b3dd8461be32c073abfa25a0f827b4602cf28bf291ce532d507", "original_candidate_sha256": "8192698389e93649262d1591d70485e279f4d82c5de1a42a91c2f68b6f825989", "original_candidate_line_sha256": "8192698389e93649262d1591d70485e279f4d82c5de1a42a91c2f68b6f825989" }, "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-20261026-006-p24", "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": "f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff", "question_first_witness_sha256": "49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 65, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "d2ef9f45cb39e19e5375e4eaedae076df0bcabfea381e292e9c6aceec75c230d", "arithmetic_steps": 61 }, "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." } }