{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261110-r04-law-of-total-variance-0023", "pilot_number": "R0044-22648", "number": 22648, "family_id": "qf-mod41-totalvar-partition-qualification", "title": { "zh-Hant": "第 22648 題：全變異數公式", "en": "Question 22648: Nonempty Cells May Have Zero Mass without Becoming Empty Events" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261110-r04-law-of-total-variance-0023", "batch_id": "batch-20261014-advance-20261110-r04-law-of-total-variance-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-law-of-total-variance" ], "title_zh": "全變異數公式", "stem_zh": "同一有限law與完整disjoint partition生成的G。操作=totalvar-partition-qualification；資料={\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"values\":[22,23,25,121],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([[0, 1], [2, 3]], [[0, [0, 1], 3/32], [1, [2, 3], 29/32]], 是, 793/32, 495/1024)", "explanation_zh": "Wave6第四層核對=([[0, 1], [2, 3]], [[0, [0, 1], 3/32], [1, [2, 3], 29/32]], 是, 793/32, 495/1024)；依完整明示資料獨立重算；within與between terms使用同一law及同一cells且兩項皆不可省略；zero-mass cell禁止conditional division且外層加權貢獻為0；不要求partition events independent。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "全變異數公式" ] }, "variant": { "family_id": "qf-mod41-totalvar-partition-qualification", "seed": 202611104023, "parameters": { "masses": [ [ 1, 32 ], [ 2, 32 ], [ 29, 32 ], [ 0, 1 ] ], "values": [ 22, 23, 25, 121 ], "partition": [ [ 0, 1 ], [ 2, 3 ] ], "domain": "Finite_partition_total_variance", "operation": "totalvar-partition-qualification", "semantic_template": "qf-mod41-totalvar-partition-qualification/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[22,23,25,121]}" } }, "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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、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-20261110-r04-law-of-total-variance-0023", "batch_id": "batch-20261014-advance-20261110-r04-law-of-total-variance-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5040807800d89ad338491e7f0e9b7cb28729450412aec9880e69c6f0c86e8303", "family_id": "qf-mod41-totalvar-partition-qualification", "seed": 202611104023, "semantic_template": "qf-mod41-totalvar-partition-qualification/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[22,23,25,121]}" }, "fields": { "title": "Nonempty Cells May Have Zero Mass without Becoming Empty Events", "stem": "Use the same complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1], [2, 3]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).", "answer": { "kind": "expression", "value": "([[0, 1], [2, 3]], [[0, [0, 1], 3/32], [1, [2, 3], 29/32]], Yes, 793/32, 495/1024)", "explanation": "Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/32], [1, [2, 3], 29/32]], Yes, 793/32, 495/1024)." }, "skills": [ "Nonempty Cells May Have Zero Mass without Becoming Empty Events" ], "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law與完整disjoint partition生成的G。操作=totalvar-partition-qualification；資料={\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"values\":[22,23,25,121],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the same complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1], [2, 3]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=totalvar-partition-qualification;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[partition, cell_mass_rows, complete_disjoint_certificate, mean, variance]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "保留原題歷史介面標記，直接列來源欄序，不需JavaScript。tag不是額外證據，固定No依原聲稱範圍解讀；排序輔助只處理協定，不是數學oracle。", "en": "Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness." }, "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 variance complete probability mass := sum(1/32, 1/16, 29/32, 0)", "after": "Total variance complete probability mass = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance complete probability mass，不以小數近似替代。", "en": "Recompute Total variance complete probability mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/32", "1/16", "29/32", "0" ], "output": "1", "label": "Total variance complete probability mass" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "ORIGINAL masses=[1/32, 1/16, 29/32, 0];SIGNED original X=[22, 23, 25, 121];ORIGINAL cell order=[[0, 1], [2, 3]]", "after": "ALL cells NONEMPTY and every atom occurs EXACTLY ONCE,including null atoms;normalized mass=1;no independence premise", "relation": "implies", "rule_id": "totalvar_NONEMPTY_cells_complete_disjoint_same_law", "conditions": [], "explanation": { "zh-Hant": "本題與全期望的空cell慣例不同，這裡每格必非空；零質量格仍可含真正原子。先核同一law及完整不交partition，X可負，不需要independence。", "en": "Unlike the earlier total-expectation convention,every cell here is nonempty.A null cell may still contain actual atoms.Use the same law and a complete disjoint partition;X may be signed and 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": "S004", "uses": [ "G001", "S003" ], "before": "Total variance original atom0 weighted X := mul(1/32, 22)", "after": "Total variance original atom0 weighted X = 11/16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom0 weighted X，不以小數近似替代。", "en": "Recompute Total variance original atom0 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/32", "22" ], "output": "11/16", "label": "Total variance original atom0 weighted X" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Total variance original atom1 weighted X := mul(1/16, 23)", "after": "Total variance original atom1 weighted X = 23/16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom1 weighted X，不以小數近似替代。", "en": "Recompute Total variance original atom1 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/16", "23" ], "output": "23/16", "label": "Total variance original atom1 weighted X" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Total variance original atom2 weighted X := mul(29/32, 25)", "after": "Total variance original atom2 weighted X = 725/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom2 weighted X，不以小數近似替代。", "en": "Recompute Total variance original atom2 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "29/32", "25" ], "output": "725/32", "label": "Total variance original atom2 weighted X" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Total variance original atom3 weighted X := mul(0, 121)", "after": "Total variance original atom3 weighted X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom3 weighted X，不以小數近似替代。", "en": "Recompute Total variance original atom3 weighted X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "121" ], "output": "0", "label": "Total variance original atom3 weighted X" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Total variance ORIGINAL global mean := sum(11/16, 23/16, 725/32, 0)", "after": "Total variance ORIGINAL global mean = 793/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance ORIGINAL global mean，不以小數近似替代。", "en": "Recompute Total variance ORIGINAL global mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "11/16", "23/16", "725/32", "0" ], "output": "793/32", "label": "Total variance ORIGINAL global mean" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Total variance original atom0 centered X := sub(22, 793/32)", "after": "Total variance original atom0 centered X = -89/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom0 centered X，不以小數近似替代。", "en": "Recompute Total variance original atom0 centered X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "22", "793/32" ], "output": "-89/32", "label": "Total variance original atom0 centered X" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Total variance original atom0 squared deviation := mul(-89/32, -89/32)", "after": "Total variance original atom0 squared deviation = 7921/1024", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom0 squared deviation，不以小數近似替代。", "en": "Recompute Total variance original atom0 squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-89/32", "-89/32" ], "output": "7921/1024", "label": "Total variance original atom0 squared deviation" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Total variance original atom0 weighted squared deviation := mul(1/32, 7921/1024)", "after": "Total variance original atom0 weighted squared deviation = 7921/32768", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom0 weighted squared deviation，不以小數近似替代。", "en": "Recompute Total variance original atom0 weighted squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/32", "7921/1024" ], "output": "7921/32768", "label": "Total variance original atom0 weighted squared deviation" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Total variance original atom1 centered X := sub(23, 793/32)", "after": "Total variance original atom1 centered X = -57/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom1 centered X，不以小數近似替代。", "en": "Recompute Total variance original atom1 centered X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "23", "793/32" ], "output": "-57/32", "label": "Total variance original atom1 centered X" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Total variance original atom1 squared deviation := mul(-57/32, -57/32)", "after": "Total variance original atom1 squared deviation = 3249/1024", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom1 squared deviation，不以小數近似替代。", "en": "Recompute Total variance original atom1 squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-57/32", "-57/32" ], "output": "3249/1024", "label": "Total variance original atom1 squared deviation" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Total variance original atom1 weighted squared deviation := mul(1/16, 3249/1024)", "after": "Total variance original atom1 weighted squared deviation = 3249/16384", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom1 weighted squared deviation，不以小數近似替代。", "en": "Recompute Total variance original atom1 weighted squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/16", "3249/1024" ], "output": "3249/16384", "label": "Total variance original atom1 weighted squared deviation" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Total variance original atom2 centered X := sub(25, 793/32)", "after": "Total variance original atom2 centered X = 7/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom2 centered X，不以小數近似替代。", "en": "Recompute Total variance original atom2 centered X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "25", "793/32" ], "output": "7/32", "label": "Total variance original atom2 centered X" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Total variance original atom2 squared deviation := mul(7/32, 7/32)", "after": "Total variance original atom2 squared deviation = 49/1024", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom2 squared deviation，不以小數近似替代。", "en": "Recompute Total variance original atom2 squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7/32", "7/32" ], "output": "49/1024", "label": "Total variance original atom2 squared deviation" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Total variance original atom2 weighted squared deviation := mul(29/32, 49/1024)", "after": "Total variance original atom2 weighted squared deviation = 1421/32768", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom2 weighted squared deviation，不以小數近似替代。", "en": "Recompute Total variance original atom2 weighted squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "29/32", "49/1024" ], "output": "1421/32768", "label": "Total variance original atom2 weighted squared deviation" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Total variance original atom3 centered X := sub(121, 793/32)", "after": "Total variance original atom3 centered X = 3079/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom3 centered X，不以小數近似替代。", "en": "Recompute Total variance original atom3 centered X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "121", "793/32" ], "output": "3079/32", "label": "Total variance original atom3 centered X" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Total variance original atom3 squared deviation := mul(3079/32, 3079/32)", "after": "Total variance original atom3 squared deviation = 9480241/1024", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom3 squared deviation，不以小數近似替代。", "en": "Recompute Total variance original atom3 squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3079/32", "3079/32" ], "output": "9480241/1024", "label": "Total variance original atom3 squared deviation" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Total variance original atom3 weighted squared deviation := mul(0, 9480241/1024)", "after": "Total variance original atom3 weighted squared deviation = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance original atom3 weighted squared deviation，不以小數近似替代。", "en": "Recompute Total variance original atom3 weighted squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "9480241/1024" ], "output": "0", "label": "Total variance original atom3 weighted squared deviation" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Total variance ORIGINAL global variance := sum(7921/32768, 3249/16384, 1421/32768, 0)", "after": "Total variance ORIGINAL global variance = 495/1024", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance ORIGINAL global variance，不以小數近似替代。", "en": "Recompute Total variance ORIGINAL global variance with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "7921/32768", "3249/16384", "1421/32768", "0" ], "output": "495/1024", "label": "Total variance ORIGINAL global variance" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Total variance cell0 mass := sum(1/32, 1/16)", "after": "Total variance cell0 mass = 3/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 mass，不以小數近似替代。", "en": "Recompute Total variance cell0 mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/32", "1/16" ], "output": "3/32", "label": "Total variance cell0 mass" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Total variance cell0 weighted numerator := sum(11/16, 23/16)", "after": "Total variance cell0 weighted numerator = 17/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 weighted numerator，不以小數近似替代。", "en": "Recompute Total variance cell0 weighted numerator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "11/16", "23/16" ], "output": "17/8", "label": "Total variance cell0 weighted numerator" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "Total variance positive cell0 mean := div(17/8, 3/32)", "after": "Total variance positive cell0 mean = 68/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance positive cell0 mean，不以小數近似替代。", "en": "Recompute Total variance positive cell0 mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "17/8", "3/32" ], "output": "68/3", "label": "Total variance positive cell0 mean" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "Total variance cell0 atom0 conditional centering := sub(22, 68/3)", "after": "Total variance cell0 atom0 conditional centering = -2/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom0 conditional centering，不以小數近似替代。", "en": "Recompute Total variance cell0 atom0 conditional centering with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "22", "68/3" ], "output": "-2/3", "label": "Total variance cell0 atom0 conditional centering" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "Total variance cell0 atom0 conditional square := mul(-2/3, -2/3)", "after": "Total variance cell0 atom0 conditional square = 4/9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom0 conditional square，不以小數近似替代。", "en": "Recompute Total variance cell0 atom0 conditional square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2/3", "-2/3" ], "output": "4/9", "label": "Total variance cell0 atom0 conditional square" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "Total variance cell0 atom0 normalized conditional mass := div(1/32, 3/32)", "after": "Total variance cell0 atom0 normalized conditional mass = 1/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom0 normalized conditional mass，不以小數近似替代。", "en": "Recompute Total variance cell0 atom0 normalized conditional mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/32", "3/32" ], "output": "1/3", "label": "Total variance cell0 atom0 normalized conditional mass" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "Total variance cell0 atom0 conditional variance term := mul(1/3, 4/9)", "after": "Total variance cell0 atom0 conditional variance term = 4/27", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom0 conditional variance term，不以小數近似替代。", "en": "Recompute Total variance cell0 atom0 conditional variance term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/3", "4/9" ], "output": "4/27", "label": "Total variance cell0 atom0 conditional variance term" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "Total variance cell0 atom0 weighted residual := mul(1/32, -2/3)", "after": "Total variance cell0 atom0 weighted residual = -1/48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom0 weighted residual，不以小數近似替代。", "en": "Recompute Total variance cell0 atom0 weighted residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/32", "-2/3" ], "output": "-1/48", "label": "Total variance cell0 atom0 weighted residual" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "Total variance cell0 atom1 conditional centering := sub(23, 68/3)", "after": "Total variance cell0 atom1 conditional centering = 1/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom1 conditional centering，不以小數近似替代。", "en": "Recompute Total variance cell0 atom1 conditional centering with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "23", "68/3" ], "output": "1/3", "label": "Total variance cell0 atom1 conditional centering" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "Total variance cell0 atom1 conditional square := mul(1/3, 1/3)", "after": "Total variance cell0 atom1 conditional square = 1/9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom1 conditional square，不以小數近似替代。", "en": "Recompute Total variance cell0 atom1 conditional square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/3", "1/3" ], "output": "1/9", "label": "Total variance cell0 atom1 conditional square" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "Total variance cell0 atom1 normalized conditional mass := div(1/16, 3/32)", "after": "Total variance cell0 atom1 normalized conditional mass = 2/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom1 normalized conditional mass，不以小數近似替代。", "en": "Recompute Total variance cell0 atom1 normalized conditional mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1/16", "3/32" ], "output": "2/3", "label": "Total variance cell0 atom1 normalized conditional mass" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "Total variance cell0 atom1 conditional variance term := mul(2/3, 1/9)", "after": "Total variance cell0 atom1 conditional variance term = 2/27", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom1 conditional variance term，不以小數近似替代。", "en": "Recompute Total variance cell0 atom1 conditional variance term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2/3", "1/9" ], "output": "2/27", "label": "Total variance cell0 atom1 conditional variance term" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "Total variance cell0 atom1 weighted residual := mul(1/16, 1/3)", "after": "Total variance cell0 atom1 weighted residual = 1/48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 atom1 weighted residual，不以小數近似替代。", "en": "Recompute Total variance cell0 atom1 weighted residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/16", "1/3" ], "output": "1/48", "label": "Total variance cell0 atom1 weighted residual" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "Total variance positive cell0 conditional variance := sum(4/27, 2/27)", "after": "Total variance positive cell0 conditional variance = 2/9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance positive cell0 conditional variance，不以小數近似替代。", "en": "Recompute Total variance positive cell0 conditional variance with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4/27", "2/27" ], "output": "2/9", "label": "Total variance positive cell0 conditional variance" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "Total variance positive cell0 residual mean zero := sum(-1/48, 1/48)", "after": "Total variance positive cell0 residual mean zero = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance positive cell0 residual mean zero，不以小數近似替代。", "en": "Recompute Total variance positive cell0 residual mean zero with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1/48", "1/48" ], "output": "0", "label": "Total variance positive cell0 residual mean zero" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "ORIGINAL cell index=0;actual atoms=[0, 1];mass=3/32;numerator=17/8", "after": "conditional mean=68/3;conditional variance=2/9;positive-cell weighted residual sum0,so cross term vanishes", "relation": "implies", "rule_id": "totalvar_conditional_mean_variance_null_and_cross_term", "conditions": [], "explanation": { "zh-Hant": "正質量cell用p_i/cellmass算條件變異數，且Σp_i(X_i−cellmean)=0。零質量cell不除0，均值與變異數都保null，兩個外層貢獻為0；不是把未定義比值算成0。", "en": "A positive cell uses p_i/cellmass for its conditional variance,and its weighted residual sum is0.A nonempty null cell is never divided:both mean and variance remain null,while both outer contributions are0.Undefined quotients are not computed zeros." }, "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": "S038", "uses": [ "G001", "S037" ], "before": "Total variance cell1 mass := sum(29/32, 0)", "after": "Total variance cell1 mass = 29/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 mass，不以小數近似替代。", "en": "Recompute Total variance cell1 mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "29/32", "0" ], "output": "29/32", "label": "Total variance cell1 mass" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "Total variance cell1 weighted numerator := sum(725/32, 0)", "after": "Total variance cell1 weighted numerator = 725/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 weighted numerator，不以小數近似替代。", "en": "Recompute Total variance cell1 weighted numerator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "725/32", "0" ], "output": "725/32", "label": "Total variance cell1 weighted numerator" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "Total variance positive cell1 mean := div(725/32, 29/32)", "after": "Total variance positive cell1 mean = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance positive cell1 mean，不以小數近似替代。", "en": "Recompute Total variance positive cell1 mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "725/32", "29/32" ], "output": "25", "label": "Total variance positive cell1 mean" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "Total variance cell1 atom2 conditional centering := sub(25, 25)", "after": "Total variance cell1 atom2 conditional centering = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom2 conditional centering，不以小數近似替代。", "en": "Recompute Total variance cell1 atom2 conditional centering with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "25", "25" ], "output": "0", "label": "Total variance cell1 atom2 conditional centering" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "Total variance cell1 atom2 conditional square := mul(0, 0)", "after": "Total variance cell1 atom2 conditional square = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom2 conditional square，不以小數近似替代。", "en": "Recompute Total variance cell1 atom2 conditional square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Total variance cell1 atom2 conditional square" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "Total variance cell1 atom2 normalized conditional mass := div(29/32, 29/32)", "after": "Total variance cell1 atom2 normalized conditional mass = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom2 normalized conditional mass，不以小數近似替代。", "en": "Recompute Total variance cell1 atom2 normalized conditional mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "29/32", "29/32" ], "output": "1", "label": "Total variance cell1 atom2 normalized conditional mass" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "Total variance cell1 atom2 conditional variance term := mul(1, 0)", "after": "Total variance cell1 atom2 conditional variance term = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom2 conditional variance term，不以小數近似替代。", "en": "Recompute Total variance cell1 atom2 conditional variance term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "0" ], "output": "0", "label": "Total variance cell1 atom2 conditional variance term" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "Total variance cell1 atom2 weighted residual := mul(29/32, 0)", "after": "Total variance cell1 atom2 weighted residual = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom2 weighted residual，不以小數近似替代。", "en": "Recompute Total variance cell1 atom2 weighted residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "29/32", "0" ], "output": "0", "label": "Total variance cell1 atom2 weighted residual" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "Total variance cell1 atom3 conditional centering := sub(121, 25)", "after": "Total variance cell1 atom3 conditional centering = 96", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom3 conditional centering，不以小數近似替代。", "en": "Recompute Total variance cell1 atom3 conditional centering with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "121", "25" ], "output": "96", "label": "Total variance cell1 atom3 conditional centering" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "Total variance cell1 atom3 conditional square := mul(96, 96)", "after": "Total variance cell1 atom3 conditional square = 9216", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom3 conditional square，不以小數近似替代。", "en": "Recompute Total variance cell1 atom3 conditional square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "96", "96" ], "output": "9216", "label": "Total variance cell1 atom3 conditional square" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "Total variance cell1 atom3 normalized conditional mass := div(0, 29/32)", "after": "Total variance cell1 atom3 normalized conditional mass = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom3 normalized conditional mass，不以小數近似替代。", "en": "Recompute Total variance cell1 atom3 normalized conditional mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "29/32" ], "output": "0", "label": "Total variance cell1 atom3 normalized conditional mass" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "Total variance cell1 atom3 conditional variance term := mul(0, 9216)", "after": "Total variance cell1 atom3 conditional variance term = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom3 conditional variance term，不以小數近似替代。", "en": "Recompute Total variance cell1 atom3 conditional variance term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "9216" ], "output": "0", "label": "Total variance cell1 atom3 conditional variance term" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "Total variance cell1 atom3 weighted residual := mul(0, 96)", "after": "Total variance cell1 atom3 weighted residual = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 atom3 weighted residual，不以小數近似替代。", "en": "Recompute Total variance cell1 atom3 weighted residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "96" ], "output": "0", "label": "Total variance cell1 atom3 weighted residual" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "Total variance positive cell1 conditional variance := sum(0, 0)", "after": "Total variance positive cell1 conditional variance = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance positive cell1 conditional variance，不以小數近似替代。", "en": "Recompute Total variance positive cell1 conditional variance with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "Total variance positive cell1 conditional variance" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "Total variance positive cell1 residual mean zero := sum(0, 0)", "after": "Total variance positive cell1 residual mean zero = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance positive cell1 residual mean zero，不以小數近似替代。", "en": "Recompute Total variance positive cell1 residual mean zero with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "Total variance positive cell1 residual mean zero" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "ORIGINAL cell index=1;actual atoms=[2, 3];mass=29/32;numerator=725/32", "after": "conditional mean=25;conditional variance=0;positive-cell weighted residual sum0,so cross term vanishes", "relation": "implies", "rule_id": "totalvar_conditional_mean_variance_null_and_cross_term", "conditions": [], "explanation": { "zh-Hant": "正質量cell用p_i/cellmass算條件變異數，且Σp_i(X_i−cellmean)=0。零質量cell不除0，均值與變異數都保null，兩個外層貢獻為0；不是把未定義比值算成0。", "en": "A positive cell uses p_i/cellmass for its conditional variance,and its weighted residual sum is0.A nonempty null cell is never divided:both mean and variance remain null,while both outer contributions are0.Undefined quotients are not computed zeros." }, "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": "S054", "uses": [ "G001", "S053" ], "before": "Total variance cell0 weighted WITHIN term := mul(3/32, 2/9)", "after": "Total variance cell0 weighted WITHIN term = 1/48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 weighted WITHIN term，不以小數近似替代。", "en": "Recompute Total variance cell0 weighted WITHIN term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3/32", "2/9" ], "output": "1/48", "label": "Total variance cell0 weighted WITHIN term" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "Total variance cell0 mean minus global mean := sub(68/3, 793/32)", "after": "Total variance cell0 mean minus global mean = -203/96", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 mean minus global mean，不以小數近似替代。", "en": "Recompute Total variance cell0 mean minus global mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "68/3", "793/32" ], "output": "-203/96", "label": "Total variance cell0 mean minus global mean" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "Total variance cell0 BETWEEN squared deviation := mul(-203/96, -203/96)", "after": "Total variance cell0 BETWEEN squared deviation = 41209/9216", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 BETWEEN squared deviation，不以小數近似替代。", "en": "Recompute Total variance cell0 BETWEEN squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-203/96", "-203/96" ], "output": "41209/9216", "label": "Total variance cell0 BETWEEN squared deviation" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "Total variance cell0 weighted BETWEEN term := mul(3/32, 41209/9216)", "after": "Total variance cell0 weighted BETWEEN term = 41209/98304", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell0 weighted BETWEEN term，不以小數近似替代。", "en": "Recompute Total variance cell0 weighted BETWEEN term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3/32", "41209/9216" ], "output": "41209/98304", "label": "Total variance cell0 weighted BETWEEN term" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "Total variance cell1 weighted WITHIN term := mul(29/32, 0)", "after": "Total variance cell1 weighted WITHIN term = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 weighted WITHIN term，不以小數近似替代。", "en": "Recompute Total variance cell1 weighted WITHIN term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "29/32", "0" ], "output": "0", "label": "Total variance cell1 weighted WITHIN term" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "Total variance cell1 mean minus global mean := sub(25, 793/32)", "after": "Total variance cell1 mean minus global mean = 7/32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 mean minus global mean，不以小數近似替代。", "en": "Recompute Total variance cell1 mean minus global mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "25", "793/32" ], "output": "7/32", "label": "Total variance cell1 mean minus global mean" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "Total variance cell1 BETWEEN squared deviation := mul(7/32, 7/32)", "after": "Total variance cell1 BETWEEN squared deviation = 49/1024", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 BETWEEN squared deviation，不以小數近似替代。", "en": "Recompute Total variance cell1 BETWEEN squared deviation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7/32", "7/32" ], "output": "49/1024", "label": "Total variance cell1 BETWEEN squared deviation" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "Total variance cell1 weighted BETWEEN term := mul(29/32, 49/1024)", "after": "Total variance cell1 weighted BETWEEN term = 1421/32768", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance cell1 weighted BETWEEN term，不以小數近似替代。", "en": "Recompute Total variance cell1 weighted BETWEEN term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "29/32", "49/1024" ], "output": "1421/32768", "label": "Total variance cell1 weighted BETWEEN term" } }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "Total variance sum WITHIN terms := sum(1/48, 0)", "after": "Total variance sum WITHIN terms = 1/48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance sum WITHIN terms，不以小數近似替代。", "en": "Recompute Total variance sum WITHIN terms with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/48", "0" ], "output": "1/48", "label": "Total variance sum WITHIN terms" } }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "Total variance sum BETWEEN terms := sum(41209/98304, 1421/32768)", "after": "Total variance sum BETWEEN terms = 1421/3072", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance sum BETWEEN terms，不以小數近似替代。", "en": "Recompute Total variance sum BETWEEN terms with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "41209/98304", "1421/32768" ], "output": "1421/3072", "label": "Total variance sum BETWEEN terms" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "Total variance WITHIN plus BETWEEN := sum(1/48, 1421/3072)", "after": "Total variance WITHIN plus BETWEEN = 495/1024", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Total variance WITHIN plus BETWEEN，不以小數近似替代。", "en": "Recompute Total variance WITHIN plus BETWEEN with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/48", "1421/3072" ], "output": "495/1024", "label": "Total variance WITHIN plus BETWEEN" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "ORIGINAL SAME-law mean=793/32;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1], 3/32, 68/3, 2/9], [1, [2, 3], 29/32, 25, 0]]", "after": "Var(X)=495/1024;within=1/48;between=1421/3072;sum=495/1024;X-mu=(X-c_j)+(c_j-mu),and each positive-cell cross term2(c_j-mu)*sum_i p_i(X_i-c_j)=0", "relation": "implies", "rule_id": "totalvar_general_centered_square_decomposition_same_partition", "conditions": [], "explanation": { "zh-Hant": "逐cell展開平方，第一項合為cellmass乘conditional variance，第二項為cellmean對globalmean的加權平方，交叉項由已核residual0消失。null cell全原子質量0，不能用獨立性或省掉某一項代替此恆等。", "en": "Expand the square cellwise.The first term gives mass times conditional variance,the second the weighted squared cell-mean deviation,and the cross term vanishes by the checked residual identity.Null atoms contribute0.No independence assumption or omitted term replaces this decomposition." }, "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 ], [ 2, 3 ] ], [ [ 0, [ 0, 1 ], "3/32" ], [ 1, [ 2, 3 ], "29/32" ] ], true, "793/32", "495/1024" ] }, "conclusion": { "canonical_value": "([[0, 1], [2, 3]], [[0, [0, 1], 3/32], [1, [2, 3], 29/32]], 是, 793/32, 495/1024)", "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-20261110-r04-law-of-total-variance-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261110-r04-law-of-total-variance-foundations/questions.jsonl", "batch_sha256": "7930f5fcd163cd3fdb88aff705a30592d289987e3ba0ec0f2229f844966ac6ac", "original_line": 23, "original_line_sha256": "2d2d32aeb556311b3c07730c3bd76414690a341d8ad8f94fae3c93388fa48d19", "question_compact_sha256": "5040807800d89ad338491e7f0e9b7cb28729450412aec9880e69c6f0c86e8303", "english_row_compact_sha256": "12708cad6a86bdd7f647286388d4fd7aa7dbf1a2b2b77e80a39ac1e0cd5a2af6", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261110-r04-law-of-total-variance-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261110-r04-law-of-total-variance-foundations.en.jsonl" }, "mko_bindings": { "mko-law-of-total-variance": { "sha256": "603199e10801a37db817e6bb93a0c0e3caac715f1667385f7837bbca5fd374f3", "english_sha256": "ae67ca99bc2e4fe0a83cf4359affb5afe7aa7722c78181f9c5216e49d4a5fed7", "original_path": "public/data/mko/law-of-total-variance.json", "english_path": "public/data/i18n/editions/mko-law-of-total-variance.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "4db6f09f48d29a15f89787c4aac3ae0abe7981d9f906e3fbecc88b60dd686db5", "original_candidate_sha256": "ace721c641e61a308a8c6a22400cb0c21a8e53505b8004c1c4151535c47b0ce4", "original_candidate_line_sha256": "ace721c641e61a308a8c6a22400cb0c21a8e53505b8004c1c4151535c47b0ce4" }, "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-p45", "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": "89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81", "question_first_witness_sha256": "b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 65, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "8b2a8a60febb6c91250846c13a096ac25d619755b726be6d25b0cf34cb78f1a7", "arithmetic_steps": 60 }, "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." } }