{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261004-advance-20261005-r06-expectation-0031", "pilot_number": "R0062-31631", "number": 31631, "family_id": "qf-expectation-affine", "title": { "zh-Hant": "第 31631 題：仿射變換的有限期望", "en": "Question 31631: Expectation of an Affine Transformation" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261004-advance-20261005-r06-expectation-0031", "batch_id": "batch-20261004-advance-20261005-r06-expectation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-expectation" ], "title_zh": "仿射變換的有限期望", "stem_zh": "非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, -4), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 7/12, 4/12, 0/12]，權重總和恰為一。令 Z=-2X+(0)，求精確 E[Z]。", "answer": { "kind": "expression", "value": "52/3", "explanation_zh": "期望=52/3。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "仿射變換的有限期望" ] }, "variant": { "family_id": "qf-expectation-affine", "seed": 202610056031, "parameters": { "S": [ 0, 1, 2, 3 ], "F": "powerset(S)", "values": [ -11, -11, -4, 1004 ], "weights": [ 1, 7, 4, 0 ], "denominator": 12, "a": -2, "b": 0, "operation": "affine_expectation", "semantic_template": "qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-2,\"b\":0,\"law\":[[\"-11\",\"2/3\"],[\"-4\",\"1/3\"]],\"operation\":\"affine_expectation\"}" } }, "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": "Codex independent producer (inherited model; exact runtime model not exposed)", "generated_at": "2026-10-04", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261004-advance-20261005-r06-expectation-0031", "batch_id": "batch-20261004-advance-20261005-r06-expectation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "e954eef8da4447147c2eb378807e98e26b5873365c6445d47911e47afc5755d7", "family_id": "qf-expectation-affine", "seed": 202610056031, "semantic_template": "qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-2,\"b\":0,\"law\":[[\"-11\",\"2/3\"],[\"-4\",\"1/3\"]],\"operation\":\"affine_expectation\"}" }, "fields": { "title": "Expectation of an Affine Transformation", "stem": "The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, -4), (3, 1004)}. In the displayed order of S, the nonnegative probability weights are [1/(12), 7/(12), 4/(12), 0/(12)], summing exactly to one. Let Z=(-2)X+(0). Find E[Z] exactly as a reduced fraction with positive denominator.", "answer": { "kind": "expression", "value": "52/3", "explanation": "Form Z at each outcome, then weight and sum to obtain 52/3, equivalently aE[X]+b. At a=0 only the constant b remains." }, "skills": [ "Expectation of an Affine Transformation" ], "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": "非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, -4), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 7/12, 4/12, 0/12]，權重總和恰為一。令 Z=-2X+(0)，求精確 E[Z]。", "en": "The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, -4), (3, 1004)}. In the displayed order of S, the nonnegative probability weights are [1/(12), 7/(12), 4/(12), 0/(12)], summing exactly to one. Let Z=(-2)X+(0). Find E[Z] exactly as a reduced fraction with positive denominator." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-expectation-affine, no-domain-field, affine_expectation, no-form-field]", "after": "derived = [[reduced_fraction_RETAIN1_STRING]]", "relation": "implies", "rule_id": "R62_EXACT_source_keys_domain_operation_form_AND_output_types", "conditions": [], "explanation": { "zh-Hant": "按原bank核欄位：BASE方程用form及論域，BASE事件／補集無operation，階乘無domain；ext18用local-hyphen，其他legacy用underscore。輸出型別不統一成tuple或fraction。", "en": "Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Expectation ORIGINAL weight total through slot0 := sum(0, 1)", "after": "Expectation ORIGINAL weight total through slot0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weight total through slot0，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weight total through slot0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1" ], "output": "1", "label": "Expectation ORIGINAL weight total through slot0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Expectation ORIGINAL weight total through slot1 := sum(1, 7)", "after": "Expectation ORIGINAL weight total through slot1 = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weight total through slot1，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weight total through slot1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "7" ], "output": "8", "label": "Expectation ORIGINAL weight total through slot1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Expectation ORIGINAL weight total through slot2 := sum(8, 4)", "after": "Expectation ORIGINAL weight total through slot2 = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weight total through slot2，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weight total through slot2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "8", "4" ], "output": "12", "label": "Expectation ORIGINAL weight total through slot2" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Expectation ORIGINAL weight total through slot3 := sum(12, 0)", "after": "Expectation ORIGINAL weight total through slot3 = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weight total through slot3，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weight total through slot3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "12", "0" ], "output": "12", "label": "Expectation ORIGINAL weight total through slot3" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [[0, 1, 2, 3], powerset(S), [1, 7, 4, 0], 12, [-11, -11, -4, 1004], not-supplied]", "after": "derived = [[0, 1, 2, 3], 12, 是]", "relation": "implies", "rule_id": "Expectation_FULL_model_RAW_slots_SAME_alignment_and_normalized_law", "conditions": [], "explanation": { "zh-Hant": "S非空、distinct且≤100，權重與X／Y依原S位置對齊。F必為powerset(S)，權重非負、den正、總和恰den；即使a0或零權重，也先驗全部raw欄位，不默改成uniform模型。", "en": "S is nonempty,distinct and≤100;weights,X/Y align to ORIGINAL S slots.F is exactly powerset(S),weights nonnegative,denominator positive,and total EXACTLYden.EVEN at a=0 or null weights validate ALL raw fields;never silently uniformize." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Expectation affine ORIGINAL slot0 a times X := mul(-2, -11)", "after": "Expectation affine ORIGINAL slot0 a times X = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation affine ORIGINAL slot0 a times X，不以小數近似替代。", "en": "Recompute Expectation affine ORIGINAL slot0 a times X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-11" ], "output": "22", "label": "Expectation affine ORIGINAL slot0 a times X" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Expectation affine ORIGINAL slot0 plus b := sum(22, 0)", "after": "Expectation affine ORIGINAL slot0 plus b = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation affine ORIGINAL slot0 plus b，不以小數近似替代。", "en": "Recompute Expectation affine ORIGINAL slot0 plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "22", "0" ], "output": "22", "label": "Expectation affine ORIGINAL slot0 plus b" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [0, 0, affine_expectation, -11, not-supplied, -2, 0, not-supplied]", "after": "derived = [22]", "relation": "implies", "rule_id": "Expectation_EACH_original_outcome_transformation_BEFORE_weight", "conditions": [], "explanation": { "zh-Hant": "每個原outcome先作本題變換，再用同位置權重；不能各自排序S／X／Y／weights。零權重的取值仍是原資料，不以刪除它們修題或推導range。", "en": "Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Expectation ORIGINAL slot0 weight times transformed value := mul(1, 22)", "after": "Expectation ORIGINAL slot0 weight times transformed value = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL slot0 weight times transformed value，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL slot0 weight times transformed value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "22" ], "output": "22", "label": "Expectation ORIGINAL slot0 weight times transformed value" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Expectation ORIGINAL weighted numerator through slot0 := sum(0, 22)", "after": "Expectation ORIGINAL weighted numerator through slot0 = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weighted numerator through slot0，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weighted numerator through slot0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "22" ], "output": "22", "label": "Expectation ORIGINAL weighted numerator through slot0" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Expectation affine ORIGINAL slot1 a times X := mul(-2, -11)", "after": "Expectation affine ORIGINAL slot1 a times X = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation affine ORIGINAL slot1 a times X，不以小數近似替代。", "en": "Recompute Expectation affine ORIGINAL slot1 a times X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-11" ], "output": "22", "label": "Expectation affine ORIGINAL slot1 a times X" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Expectation affine ORIGINAL slot1 plus b := sum(22, 0)", "after": "Expectation affine ORIGINAL slot1 plus b = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation affine ORIGINAL slot1 plus b，不以小數近似替代。", "en": "Recompute Expectation affine ORIGINAL slot1 plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "22", "0" ], "output": "22", "label": "Expectation affine ORIGINAL slot1 plus b" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "source roles = [1, 1, affine_expectation, -11, not-supplied, -2, 0, not-supplied]", "after": "derived = [22]", "relation": "implies", "rule_id": "Expectation_EACH_original_outcome_transformation_BEFORE_weight", "conditions": [], "explanation": { "zh-Hant": "每個原outcome先作本題變換，再用同位置權重；不能各自排序S／X／Y／weights。零權重的取值仍是原資料，不以刪除它們修題或推導range。", "en": "Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Expectation ORIGINAL slot1 weight times transformed value := mul(7, 22)", "after": "Expectation ORIGINAL slot1 weight times transformed value = 154", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL slot1 weight times transformed value，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL slot1 weight times transformed value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "22" ], "output": "154", "label": "Expectation ORIGINAL slot1 weight times transformed value" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Expectation ORIGINAL weighted numerator through slot1 := sum(22, 154)", "after": "Expectation ORIGINAL weighted numerator through slot1 = 176", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weighted numerator through slot1，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weighted numerator through slot1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "22", "154" ], "output": "176", "label": "Expectation ORIGINAL weighted numerator through slot1" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Expectation affine ORIGINAL slot2 a times X := mul(-2, -4)", "after": "Expectation affine ORIGINAL slot2 a times X = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation affine ORIGINAL slot2 a times X，不以小數近似替代。", "en": "Recompute Expectation affine ORIGINAL slot2 a times X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-4" ], "output": "8", "label": "Expectation affine ORIGINAL slot2 a times X" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Expectation affine ORIGINAL slot2 plus b := sum(8, 0)", "after": "Expectation affine ORIGINAL slot2 plus b = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation affine ORIGINAL slot2 plus b，不以小數近似替代。", "en": "Recompute Expectation affine ORIGINAL slot2 plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "8", "0" ], "output": "8", "label": "Expectation affine ORIGINAL slot2 plus b" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "source roles = [2, 2, affine_expectation, -4, not-supplied, -2, 0, not-supplied]", "after": "derived = [8]", "relation": "implies", "rule_id": "Expectation_EACH_original_outcome_transformation_BEFORE_weight", "conditions": [], "explanation": { "zh-Hant": "每個原outcome先作本題變換，再用同位置權重；不能各自排序S／X／Y／weights。零權重的取值仍是原資料，不以刪除它們修題或推導range。", "en": "Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Expectation ORIGINAL slot2 weight times transformed value := mul(4, 8)", "after": "Expectation ORIGINAL slot2 weight times transformed value = 32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL slot2 weight times transformed value，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL slot2 weight times transformed value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "8" ], "output": "32", "label": "Expectation ORIGINAL slot2 weight times transformed value" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Expectation ORIGINAL weighted numerator through slot2 := sum(176, 32)", "after": "Expectation ORIGINAL weighted numerator through slot2 = 208", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weighted numerator through slot2，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weighted numerator through slot2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "176", "32" ], "output": "208", "label": "Expectation ORIGINAL weighted numerator through slot2" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Expectation affine ORIGINAL slot3 a times X := mul(-2, 1004)", "after": "Expectation affine ORIGINAL slot3 a times X = -2008", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation affine ORIGINAL slot3 a times X，不以小數近似替代。", "en": "Recompute Expectation affine ORIGINAL slot3 a times X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1004" ], "output": "-2008", "label": "Expectation affine ORIGINAL slot3 a times X" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Expectation affine ORIGINAL slot3 plus b := sum(-2008, 0)", "after": "Expectation affine ORIGINAL slot3 plus b = -2008", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation affine ORIGINAL slot3 plus b，不以小數近似替代。", "en": "Recompute Expectation affine ORIGINAL slot3 plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2008", "0" ], "output": "-2008", "label": "Expectation affine ORIGINAL slot3 plus b" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "source roles = [3, 3, affine_expectation, 1004, not-supplied, -2, 0, not-supplied]", "after": "derived = [-2008]", "relation": "implies", "rule_id": "Expectation_EACH_original_outcome_transformation_BEFORE_weight", "conditions": [], "explanation": { "zh-Hant": "每個原outcome先作本題變換，再用同位置權重；不能各自排序S／X／Y／weights。零權重的取值仍是原資料，不以刪除它們修題或推導range。", "en": "Transform at EACH ORIGINAL outcome BEFORE using its aligned weight.Never sort S,X,Y,weights independently.Null-weight values remain original data;do not delete them to repair inputs or infer a range." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "Expectation ORIGINAL slot3 weight times transformed value := mul(0, -2008)", "after": "Expectation ORIGINAL slot3 weight times transformed value = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL slot3 weight times transformed value，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL slot3 weight times transformed value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-2008" ], "output": "0", "label": "Expectation ORIGINAL slot3 weight times transformed value" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "Expectation ORIGINAL weighted numerator through slot3 := sum(208, 0)", "after": "Expectation ORIGINAL weighted numerator through slot3 = 208", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weighted numerator through slot3，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weighted numerator through slot3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "208", "0" ], "output": "208", "label": "Expectation ORIGINAL weighted numerator through slot3" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "Expectation ORIGINAL weighted numerator over denominator := div(208, 12)", "after": "Expectation ORIGINAL weighted numerator over denominator = 52/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Expectation ORIGINAL weighted numerator over denominator，不以小數近似替代。", "en": "Recompute Expectation ORIGINAL weighted numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "208", "12" ], "output": "52/3", "label": "Expectation ORIGINAL weighted numerator over denominator" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "source roles = [[0, 1, 2, 3], [1, 7, 4, 0], 12, affine_expectation, [22, 22, 8, -2008]]", "after": "derived = [[22, 154, 32, 0], 208, 52/3]", "relation": "implies", "rule_id": "Expectation_LINEARITY_qualified_SAME_law_NOT_independence_or_sample_mean", "conditions": [], "explanation": { "zh-Hant": "依已合格同一概率law，E[Z]=Σw_i Z(s_i)/q。仿射為aE[X]+b、同模型相加為E[X]+E[Y]而不要求獨立，不能推成乘積期望公式。常數c的期望是c因權重總和1；負值或未被取到的期望合法，fraction保/1。", "en": "In the QUALIFIED SAME probability law,E[Z]=Σw_iZ(s_i)/q.Affine gives aE[X]+b and a same-model sum gives E[X]+E[Y]WITHOUT independence;this does not imply a product-expectation formula.A constant has mean c because total mass is1.Negative/unattained means are legal;retain/1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": "52/3" }, "conclusion": { "canonical_value": "52/3", "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-20261004-advance-20261005-r06-expectation-foundations", "batch_path": "public/data/questions/batches/batch-20261004-advance-20261005-r06-expectation-foundations/questions.jsonl", "batch_sha256": "dbe4ead22a73a9d7c0e38e8af216001d72f62c0733ec05f917cd2123c30e2b0a", "original_line": 31, "original_line_sha256": "c1618f475241d8e0e307e44d8f23d650f22929a2793a06a518cfaddbfc0d229d", "question_compact_sha256": "e954eef8da4447147c2eb378807e98e26b5873365c6445d47911e47afc5755d7", "english_row_compact_sha256": "99c4e3d1d5c3207b4f13d64ebbf9fc21de01ad0d5864ba9ad0e2e7446864ccce", "english_manifest_path": "public/data/i18n/editions/batch-20261004-advance-20261005-r06-expectation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261004-advance-20261005-r06-expectation-foundations.en.jsonl" }, "mko_bindings": { "mko-expectation": { "sha256": "71bc68faf69ca0df9cdd11d093055262aa277d07abd77982d0aa1afd849daf3c", "english_sha256": "636015fa1e3a6d9eae6091042ae3dd70c6cb996ecd9a409c890060a5b503333a", "original_path": "public/data/mko/expectation.json", "english_path": "public/data/i18n/editions/mko-expectation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "59d384ca3d958b27165f4295ff96efb155e7b12d3fb70bcdd0ed68148513c05a", "original_candidate_sha256": "c0429b6fa3905071450d533458a074799f1505d34c5961e20ae42c73c42077f4", "original_candidate_line_sha256": "c0429b6fa3905071450d533458a074799f1505d34c5961e20ae42c73c42077f4" }, "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-p63", "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": "963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd", "question_first_witness_sha256": "d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 28, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "dcbcf4a0d58ce9df765f4a818555e88bee06427e6d77cb9273749f68aa27bc3b", "arithmetic_steps": 21 }, "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." } }