{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261109-r04-law-of-total-expectation-0087", "pilot_number": "R0042-21987", "number": 21987, "family_id": "qf-mod40-totalexp-zero-mass-boundary", "title": { "zh-Hant": "第 21987 題：全期望公式", "en": "Question 21987: Null Cells Retain Undefined Quotients and Zero Weighted Contributions" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261109-r04-law-of-total-expectation-0087", "batch_id": "batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-law-of-total-expectation" ], "title_zh": "全期望公式", "stem_zh": "同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"values\":[11,12,14,110],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)", "explanation_zh": "Wave6第三層核對=([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "全期望公式" ] }, "variant": { "family_id": "qf-mod40-totalexp-zero-mass-boundary", "seed": 202611094087, "parameters": { "masses": [ [ 1, 21 ], [ 2, 21 ], [ 18, 21 ], [ 0, 1 ] ], "values": [ 11, 12, 14, 110 ], "partition": [ [ 0, 1, 2 ], [ 3 ] ], "domain": "Finite_partition_total_expectation", "operation": "totalexp-zero-mass-boundary", "semantic_template": "qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[11,12,14,110]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261109-r04-law-of-total-expectation-0087", "batch_id": "batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5565d3a672318a08c949592c7b6d16de842a968301672f2b245b7b9e4fb81aa2", "family_id": "qf-mod40-totalexp-zero-mass-boundary", "seed": 202611094087, "semantic_template": "qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[11,12,14,110]}" }, "fields": { "title": "Null Cells Retain Undefined Quotients and Zero Weighted Contributions", "stem": "Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).", "answer": { "kind": "expression", "value": "([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)", "explanation": "Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)." }, "skills": [ "Null Cells Retain Undefined Quotients and Zero Weighted Contributions" ], "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": "同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"values\":[11,12,14,110],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=totalexp-zero-mass-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題歷史介面標記保留，本頁直接列來源有序欄位，不需JavaScript。tag不是額外證據，固定No保留其原聲稱範圍。", "en": "Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope." }, "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": "S002", "uses": [ "G001", "S001" ], "before": "Total expectation complete original mass := sum(1/21, 2/21, 6/7, 0)", "after": "Total expectation complete original mass = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation complete original mass，不以小數近似替代。", "en": "Recompute Total expectation complete original mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/21", "2/21", "6/7", "0" ], "output": "1", "label": "Total expectation complete original mass" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Total expectation original atom0 weighted X := mul(1/21, 11)", "after": "Total expectation original atom0 weighted X = 11/21", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation original atom0 weighted X，不以小數近似替代。", "en": "Recompute Total expectation original atom0 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/21", "11" ], "output": "11/21", "label": "Total expectation original atom0 weighted X" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Total expectation original atom1 weighted X := mul(2/21, 12)", "after": "Total expectation original atom1 weighted X = 8/7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation original atom1 weighted X，不以小數近似替代。", "en": "Recompute Total expectation original atom1 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2/21", "12" ], "output": "8/7", "label": "Total expectation original atom1 weighted X" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Total expectation original atom2 weighted X := mul(6/7, 14)", "after": "Total expectation original atom2 weighted X = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation original atom2 weighted X，不以小數近似替代。", "en": "Recompute Total expectation original atom2 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6/7", "14" ], "output": "12", "label": "Total expectation original atom2 weighted X" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Total expectation original atom3 weighted X := mul(0, 110)", "after": "Total expectation original atom3 weighted X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation original atom3 weighted X，不以小數近似替代。", "en": "Recompute Total expectation original atom3 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "110" ], "output": "0", "label": "Total expectation original atom3 weighted X" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Total expectation full original mean := sum(11/21, 8/7, 12, 0)", "after": "Total expectation full original mean = 41/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation full original mean，不以小數近似替代。", "en": "Recompute Total expectation full original mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "11/21", "8/7", "12", "0" ], "output": "41/3", "label": "Total expectation full original mean" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "ORIGINAL masses=[1/21, 2/21, 6/7, 0],SIGNED values X=[11, 12, 14, 110];ORIGINAL cell order=[[0, 1, 2], [3]]", "after": "flattened original atom labels=[0, 1, 2, 3] cover EVERY atom exactly ONCE including null atoms;total mass=1;integrable finite mean=41/3;NO independence premise", "relation": "implies", "rule_id": "totalexp_same_law_complete_disjoint_partition_all_atoms", "conditions": [], "explanation": { "zh-Hant": "分割必涵蓋全部原子，包括零質量點；cell順序保原輸入，cell內按來源排序。X可負，有限值可積；獨立性不是全期望公式的前提。", "en": "The partition covers all original atoms,including null atoms,exactly once.Keep original cell order and source-normalized internal labels.X may be signed and is integrable in this finite model;independence is not required." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "Total expectation cell0 mass := sum(1/21, 2/21, 6/7)", "after": "Total expectation cell0 mass = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation cell0 mass，不以小數近似替代。", "en": "Recompute Total expectation cell0 mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/21", "2/21", "6/7" ], "output": "1", "label": "Total expectation cell0 mass" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Total expectation cell0 numerator := sum(11/21, 8/7, 12)", "after": "Total expectation cell0 numerator = 41/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation cell0 numerator，不以小數近似替代。", "en": "Recompute Total expectation cell0 numerator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "11/21", "8/7", "12" ], "output": "41/3", "label": "Total expectation cell0 numerator" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Total expectation cell0 positive-mass conditional mean := div(41/3, 1)", "after": "Total expectation cell0 positive-mass conditional mean = 41/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation cell0 positive-mass conditional mean，不以小數近似替代。", "en": "Recompute Total expectation cell0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "41/3", "1" ], "output": "41/3", "label": "Total expectation cell0 positive-mass conditional mean" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "ORIGINAL cell index=0,atom labels=[0, 1, 2];mass=1;weighted numerator=41/3", "after": "conditional quotient=41/3;positive denominator qualifies division", "relation": "implies", "rule_id": "totalexp_cell_quotient_positive_mass_only_null_preserved", "conditions": [], "explanation": { "zh-Hant": "正cell質量才可取條件均值比值。零cell中所有非負原子質量皆0，故分子0但0/0未定義；null不能當已算出的0。空cell亦屬此情況。", "en": "Divide only for positive cell mass.In a null cell,all nonnegative atom masses are0,so the numerator is0 but0/0 remains undefined.Keep null rather than an asserted quotient0,including empty cells." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "Total expectation cell1 mass := sum(0)", "after": "Total expectation cell1 mass = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation cell1 mass，不以小數近似替代。", "en": "Recompute Total expectation cell1 mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0" ], "output": "0", "label": "Total expectation cell1 mass" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Total expectation cell1 numerator := sum(0)", "after": "Total expectation cell1 numerator = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation cell1 numerator，不以小數近似替代。", "en": "Recompute Total expectation cell1 numerator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0" ], "output": "0", "label": "Total expectation cell1 numerator" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "ORIGINAL cell index=1,atom labels=[3];mass=0;weighted numerator=0", "after": "conditional quotient=null;null cell:every nonnegative atom mass is0,numerator0;NO division performed", "relation": "implies", "rule_id": "totalexp_cell_quotient_positive_mass_only_null_preserved", "conditions": [], "explanation": { "zh-Hant": "正cell質量才可取條件均值比值。零cell中所有非負原子質量皆0，故分子0但0/0未定義；null不能當已算出的0。空cell亦屬此情況。", "en": "Divide only for positive cell mass.In a null cell,all nonnegative atom masses are0,so the numerator is0 but0/0 remains undefined.Keep null rather than an asserted quotient0,including empty cells." }, "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": "S016", "uses": [ "G001", "S015" ], "before": "Total expectation positive cell0 weighted mean := mul(1, 41/3)", "after": "Total expectation positive cell0 weighted mean = 41/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation positive cell0 weighted mean，不以小數近似替代。", "en": "Recompute Total expectation positive cell0 weighted mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "41/3" ], "output": "41/3", "label": "Total expectation positive cell0 weighted mean" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Total expectation sum positive-cell weighted means := sum(41/3)", "after": "Total expectation sum positive-cell weighted means = 41/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原題角色、順序與條件，逐項精確代值，不作浮點近似。", "en": "Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total expectation sum positive-cell weighted means，不以小數近似替代。", "en": "Recompute Total expectation sum positive-cell weighted means with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "41/3" ], "output": "41/3", "label": "Total expectation sum positive-cell weighted means" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1, 2], 1, 41/3, 41/3], [1, [3], 0, null, 0]]", "after": "positive-cell weighted sum=41/3;ORIGINAL global mean=41/3;equal=是;disjoint complete cells regroup EVERY original p_i*X_i exactly ONCE,null contributions0", "relation": "implies", "rule_id": "totalexp_regroup_complete_partition_general_identity", "conditions": [], "explanation": { "zh-Hant": "每個正cell的mass乘條件均值恰還原分子，零cell貢獻0；全部不交且完整分割才可重組為原全體加權和。這是一般恆等理由，不是無權平均cell均值或一次數值湊巧相等。", "en": "Each positive mass times its conditional mean recovers its numerator;null cells contribute0.Disjoint complete cells then regroup every original weighted term exactly once.This proves the finite identity,not an unweighted average or an accidental numerical equality." }, "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": "S019", "uses": [ "G001", "S018" ], "before": "ACTUAL null cells=[1];positive weighted sum=41/3;mean=41/3", "after": "null_cell_division_claim fixed No;undefined cell quotients remain null,while the weighted total-expectation identity still holds", "relation": "implies", "rule_id": "totalexp_null_division_nonclaim_not_tower_failure", "conditions": [], "explanation": { "zh-Hant": "最後No拒絕null-cell除法，不是否定恆等式。null cell上的條件期望版本可另選常數，但不能說由0/0唯一算出。", "en": "The final No rejects null-cell division,not the identity.A conditional-expectation version may choose a value on a null cell,but not as a uniquely computed0/0 quotient." }, "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, "1", "41/3" ], [ 1, "0", null ] ], "41/3", "41/3", true, false, "zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero" ] }, "conclusion": { "canonical_value": "([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)", "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-20261014-advance-20261109-r04-law-of-total-expectation-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations/questions.jsonl", "batch_sha256": "3daaadbaf91eceddfd25e8cd1196ed1ec667feb7cb26e57494d4857b7ba168c8", "original_line": 87, "original_line_sha256": "6c5a5eeb59fb18f8200a7e2c578c83a4278b845018f03ae05a37c55971c79d9f", "question_compact_sha256": "5565d3a672318a08c949592c7b6d16de842a968301672f2b245b7b9e4fb81aa2", "english_row_compact_sha256": "c3e90ee27051576bd075d67b49a99dfcdd4a68383195ac9913af7c639e54f894", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations.en.jsonl" }, "mko_bindings": { "mko-law-of-total-expectation": { "sha256": "135b6dedfa00c0fd52813c51ffe3c6af6b5eca3527d0e536110c62f0e27392ac", "english_sha256": "90d94b2ee88d08ce7a31ddf914834dec4c8f41f84a8ef5839bc5508ac80c7e0a", "original_path": "public/data/mko/law-of-total-expectation.json", "english_path": "public/data/i18n/editions/mko-law-of-total-expectation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a84fbe07a4e46e8e47a8ad331f35fd3f81b821aef6cb27e7852de854aaa96b9a", "original_candidate_sha256": "4d2cd8ab23c1312078cddc1c5434e473a025d8fa3e2b4c492862578bf73b5b7e", "original_candidate_line_sha256": "4d2cd8ab23c1312078cddc1c5434e473a025d8fa3e2b4c492862578bf73b5b7e" }, "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-20261026-009-p43", "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": "dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a", "question_first_witness_sha256": "9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 19, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "0bd6014ebf135454134554af5ecfe9c796f84f606041a496b1671ba15f598838", "arithmetic_steps": 13 }, "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." } }