{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0084", "pilot_number": "R0057-29184", "number": 29184, "family_id": "qf-str22-distribution-pushforward", "title": { "zh-Hant": "第 29184 題：有限離散PMF", "en": "Question 29184: Merge All Masses with a Common Image" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0084", "batch_id": "batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-discrete-probability-distribution" ], "title_zh": "有限離散PMF", "stem_zh": "有限離散精確PMF。操作=distribution-pushforward；資料={\"outcomes\":[8,9,11],\"masses\":[[1,4],[1,4],[1,2]],\"images\":[2,0,2]}。請依 [outcomes,masses,images,pushforward_rows] 欄序回答。", "answer": { "kind": "expression", "value": "([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])", "explanation_zh": "結構基礎核對=([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])；依完整明示資料重算；完整明示的精確切片不是一般定理或經驗聲明；質量非負且精確和為1；數學分布不是資料擬合證據。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "有限離散PMF" ] }, "variant": { "family_id": "qf-str22-distribution-pushforward", "seed": 202610227084, "parameters": { "outcomes": [ 8, 9, 11 ], "masses": [ [ 1, 4 ], [ 1, 4 ], [ 1, 2 ] ], "images": [ 2, 0, 2 ], "domain": "Exact_finite_discrete_pmf", "operation": "distribution-pushforward", "semantic_template": "qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[8,9,11]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、投資建議、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-20261022-r07-finite-discrete-probability-distribution-0084", "batch_id": "batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1a26215be7c5054ddade175f1cb09af9759047bc14d1a3bf4209141998db8564", "family_id": "qf-str22-distribution-pushforward", "seed": 202610227084, "semantic_template": "qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[8,9,11]}" }, "fields": { "title": "Merge All Masses with a Common Image", "stem": "Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.", "answer": { "kind": "expression", "value": "([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])", "explanation": "Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])." }, "skills": [ "Merge All Masses with a Common Image" ], "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": "有限離散精確PMF。操作=distribution-pushforward；資料={\"outcomes\":[8,9,11],\"masses\":[[1,4],[1,4],[1,2]],\"images\":[2,0,2]}。請依 [outcomes,masses,images,pushforward_rows] 欄序回答。", "en": "Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-str22-distribution-pushforward, distribution-pushforward]", "after": "derived = [[outcomes, masses, images, pushforward_rows]]", "relation": "implies", "rule_id": "STR22_exact_interface_and_source_ORDERED_tuple_roles", "conditions": [], "explanation": { "zh-Hant": "此銀行依原tuple欄序與hyphen operation。fraction約分、正分母且整數省略/1，與前輪舊契約不同；不改原中英快照或題意。", "en": "Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [PMF raw mass slot0, [1, 4]]", "after": "derived = [1/4, 1, 4]", "relation": "implies", "rule_id": "STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization", "conditions": [], "explanation": { "zh-Hant": "原rational必須二元pair且raw分子／分母各在±1e6、分母非0。先正分母與gcd約分；不能接受scalar或先約分再繞過raw輸入界。", "en": "A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [PMF raw mass slot1, [1, 4]]", "after": "derived = [1/4, 1, 4]", "relation": "implies", "rule_id": "STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization", "conditions": [], "explanation": { "zh-Hant": "原rational必須二元pair且raw分子／分母各在±1e6、分母非0。先正分母與gcd約分；不能接受scalar或先約分再繞過raw輸入界。", "en": "A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [PMF raw mass slot2, [1, 2]]", "after": "derived = [1/2, 1, 2]", "relation": "implies", "rule_id": "STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization", "conditions": [], "explanation": { "zh-Hant": "原rational必須二元pair且raw分子／分母各在±1e6、分母非0。先正分母與gcd約分；不能接受scalar或先約分再繞過raw輸入界。", "en": "A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "PMF ORIGINAL normalization running sum0 := sum(0, 1/4)", "after": "PMF ORIGINAL normalization running sum0 = 1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 PMF ORIGINAL normalization running sum0，不以小數近似替代。", "en": "Recompute PMF ORIGINAL normalization running sum0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1/4" ], "output": "1/4", "label": "PMF ORIGINAL normalization running sum0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [PMF ORIGINAL normalization running sum0, sum, [0, 1/4]]", "after": "derived = [[1], [1, 4], 1/4]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "PMF ORIGINAL normalization running sum1 := sum(1/4, 1/4)", "after": "PMF ORIGINAL normalization running sum1 = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 PMF ORIGINAL normalization running sum1，不以小數近似替代。", "en": "Recompute PMF ORIGINAL normalization running sum1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/4", "1/4" ], "output": "1/2", "label": "PMF ORIGINAL normalization running sum1" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [PMF ORIGINAL normalization running sum1, sum, [1/4, 1/4]]", "after": "derived = [[4], [2, 4], 1/2]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "PMF ORIGINAL normalization running sum2 := sum(1/2, 1/2)", "after": "PMF ORIGINAL normalization running sum2 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 PMF ORIGINAL normalization running sum2，不以小數近似替代。", "en": "Recompute PMF ORIGINAL normalization running sum2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/2" ], "output": "1", "label": "PMF ORIGINAL normalization running sum2" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [PMF ORIGINAL normalization running sum2, sum, [1/2, 1/2]]", "after": "derived = [[2], [2, 2], 1]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [[8, 9, 11], [1/4, 1/4, 1/2]]", "after": "derived = [[[8, 1/4], [9, 1/4], [11, 1/2]], 是, 1, 是, 是]", "relation": "implies", "rule_id": "PMF_ALL_original_outcome_mass_alignment_NONNEG_AND_EXACTsum1", "conditions": [], "explanation": { "zh-Hant": "保原outcomes與mass逐slot配對，逐原順序求完整和。非負與總和1是分開責任；qualification可合法No，不重新normalize或把負值clip0。", "en": "Preserve ORIGINAL outcome-mass slot alignment and sum every mass in original order.Nonnegativity and total1 are separate obligations.Qualification may legally be No;do not renormalize or clip negative masses." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "PMF ORIGINAL pushforward slot0 image2 := sum(0, 1/4)", "after": "PMF ORIGINAL pushforward slot0 image2 = 1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 PMF ORIGINAL pushforward slot0 image2，不以小數近似替代。", "en": "Recompute PMF ORIGINAL pushforward slot0 image2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1/4" ], "output": "1/4", "label": "PMF ORIGINAL pushforward slot0 image2" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "source roles = [PMF ORIGINAL pushforward slot0 image2, sum, [0, 1/4]]", "after": "derived = [[1], [1, 4], 1/4]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "PMF ORIGINAL pushforward slot1 image0 := sum(0, 1/4)", "after": "PMF ORIGINAL pushforward slot1 image0 = 1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 PMF ORIGINAL pushforward slot1 image0，不以小數近似替代。", "en": "Recompute PMF ORIGINAL pushforward slot1 image0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1/4" ], "output": "1/4", "label": "PMF ORIGINAL pushforward slot1 image0" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "source roles = [PMF ORIGINAL pushforward slot1 image0, sum, [0, 1/4]]", "after": "derived = [[1], [1, 4], 1/4]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "PMF ORIGINAL pushforward slot2 image2 := sum(1/4, 1/2)", "after": "PMF ORIGINAL pushforward slot2 image2 = 3/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "精確代原操作數，不用浮點；只有明定的來源rational operations套自己的profile，額外教學整數算式不另縮原題域。", "en": "Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 PMF ORIGINAL pushforward slot2 image2，不以小數近似替代。", "en": "Recompute PMF ORIGINAL pushforward slot2 image2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/4", "1/2" ], "output": "3/4", "label": "PMF ORIGINAL pushforward slot2 image2" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [PMF ORIGINAL pushforward slot2 image2, sum, [1/4, 1/2]]", "after": "derived = [[2], [3, 4], 3/4]", "relation": "implies", "rule_id": "STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate", "conditions": [], "explanation": { "zh-Hant": "來源加法先取分母gcd，乘法先cross-cancel，再驗本步raw numerator與denominator各<=1e6，最後才約分；除法另先驗非零及reciprocal。不能誤用未消約乘積界，也不能先最終約分才驗界。", "en": "SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "source roles = [[8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2]]", "after": "derived = [[[0, 1/4], [2, 3/4]]]", "relation": "implies", "rule_id": "PMF_pushforward_ALL_fibres_ORIGINAL_alignment_ZERO_rows_retained", "conditions": [], "explanation": { "zh-Hant": "由P(f(Y)=z)=sum_{i:f(y_i)=z}p_i，把每個原slot的mass恰加一次到完整fiber，之後才依image數值排序。即使某image只有零mass也保row；不是改配對或只列正support。", "en": "P(f(Y)=z)=sum_{i:f(y_i)=z}p_i.Add each ORIGINAL slot mass exactly once into its COMPLETE fibre,then sort image rows numerically.Retain a row even if its combined mass is0;do not alter pairing or keep only positive support." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原完整資料反向核對每一座標／表項／配對與嚴格前提，不能只看最後答案。保全有限候選、一般理由、形式證明及發布的不同責任。", "en": "Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct." } }, "arithmetic": null } ], "math_model": { "result": [ [ 8, 9, 11 ], [ "1/4", "1/4", "1/2" ], [ 2, 0, 2 ], [ [ 0, "1/4" ], [ 2, "3/4" ] ] ] }, "conclusion": { "canonical_value": "([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])", "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-20261022-r07-finite-discrete-probability-distribution-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations/questions.jsonl", "batch_sha256": "3f2a66842adfe33f8df8991b000989787d14e3c6f434dd23efde40240975c6b4", "original_line": 84, "original_line_sha256": "9a2984d2cd930ec52c9c28098e6df0c8627a353794a91bb058d64be9e324b1c1", "question_compact_sha256": "1a26215be7c5054ddade175f1cb09af9759047bc14d1a3bf4209141998db8564", "english_row_compact_sha256": "f2db0c5b9e8295ecf1455811be951d864e6e0a883b28a385cb9b1eebb8db4f4f", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-discrete-probability-distribution": { "sha256": "a77aa248d456c6fff168a9df6f8b4ad85be410912a036130257ff1d3e0aa1b2a", "english_sha256": "2d1802954848cf7c228069df4e5277244c06e16e2e2bd85ef8da7a4334958a87", "original_path": "public/data/mko/finite-discrete-probability-distribution.json", "english_path": "public/data/i18n/editions/mko-finite-discrete-probability-distribution.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "cd714b09e116fbc0088007895c4d956bc0f4fad5a61507e4a16f09649d05a3b0", "original_candidate_sha256": "bab46a40bc68b477bdbfcbc03567961dce5613891fba3cc7fd1d621f01df155c", "original_candidate_line_sha256": "bab46a40bc68b477bdbfcbc03567961dce5613891fba3cc7fd1d621f01df155c" }, "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-p58", "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": "db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6", "question_first_witness_sha256": "58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 18, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "42c2acb4aaf999be2c3f58eb7f90c023bc6878cec3b4d77dcae2e6173819d49d", "arithmetic_steps": 6 }, "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." } }