{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261108-r04-chebyshev-inequality-0075", "pilot_number": "R0040-20725", "number": 20725, "family_id": "qf-mod39-cheb-bound-comparison", "title": { "zh-Hant": "第 20725 題：Chebyshev不等式", "en": "Question 20725: Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261108-r04-chebyshev-inequality-0075", "batch_id": "batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-chebyshev-inequality" ], "title_zh": "Chebyshev不等式", "stem_zh": "完整有限law、finite mean與variance及positive epsilon。操作=cheb-bound-comparison；資料={\"masses\":[[1,30],[2,30],[27,30]],\"values\":[0,25,28],\"epsilon\":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(1, 5726/225, 5726/225, 是, upper_bound_not_tail_equality_and_no_independence_needed)", "explanation_zh": "Wave6第二層核對=(1, 5726/225, 5726/225, 是, upper_bound_not_tail_equality_and_no_independence_needed)；依完整明示資料獨立重算；P(abs(X-mu)>=epsilon)小於等於Var(X)/epsilon squared且不需要independence；對nonnegative squared deviation套Markov；bound不等於tail並保持zero-mass與zero-variance a.s.語義。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "Chebyshev不等式" ] }, "variant": { "family_id": "qf-mod39-cheb-bound-comparison", "seed": 202611084075, "parameters": { "masses": [ [ 1, 30 ], [ 2, 30 ], [ 27, 30 ] ], "values": [ 0, 25, 28 ], "epsilon": [ 1, 1 ], "domain": "Finite_rational_Chebyshev_inequality", "operation": "cheb-bound-comparison", "semantic_template": "qf-mod39-cheb-bound-comparison/{\"epsilon\":[1,1],\"masses\":[[1,30],[2,30],[27,30]],\"values\":[0,25,28]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、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-20261108-r04-chebyshev-inequality-0075", "batch_id": "batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "d53bd7f9547371e05dbd392befab0d155ea3b705f404693ef0ee0bdc1130bb3a", "family_id": "qf-mod39-cheb-bound-comparison", "seed": 202611084075, "semantic_template": "qf-mod39-cheb-bound-comparison/{\"epsilon\":[1,1],\"masses\":[[1,30],[2,30],[27,30]],\"values\":[0,25,28]}" }, "fields": { "title": "Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence", "stem": "Use complete normalized finite law masses=[1/30, 2/30, 27/30] and aligned values X=[0, 25, 28]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).", "answer": { "kind": "expression", "value": "(1, 5726/225, 5726/225, Yes, upper_bound_not_tail_equality_and_no_independence_needed)", "explanation": "Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 5726/225, 5726/225, Yes, upper_bound_not_tail_equality_and_no_independence_needed)." }, "skills": [ "Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence" ], "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": "完整有限law、finite mean與variance及positive epsilon。操作=cheb-bound-comparison；資料={\"masses\":[[1,30],[2,30],[27,30]],\"values\":[0,25,28],\"epsilon\":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete normalized finite law masses=[1/30, 2/30, 27/30] and aligned values X=[0, 25, 28]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=cheb-bound-comparison;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[tail_probability, variance, raw_upper_bound, bound_holds, certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。", "en": "Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Chebyshev complete probability law mass := sum(1/30, 1/15, 9/10)", "after": "Chebyshev complete probability law mass = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev complete probability law mass，不以小數近似替代。", "en": "Recompute Chebyshev complete probability law mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/30", "1/15", "9/10" ], "output": "1", "label": "Chebyshev complete probability law mass" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Chebyshev original atom0 weighted SIGNED X := mul(1/30, 0)", "after": "Chebyshev original atom0 weighted SIGNED X = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom0 weighted SIGNED X，不以小數近似替代。", "en": "Recompute Chebyshev original atom0 weighted SIGNED X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/30", "0" ], "output": "0", "label": "Chebyshev original atom0 weighted SIGNED X" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Chebyshev original atom1 weighted SIGNED X := mul(1/15, 25)", "after": "Chebyshev original atom1 weighted SIGNED X = 5/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom1 weighted SIGNED X，不以小數近似替代。", "en": "Recompute Chebyshev original atom1 weighted SIGNED X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/15", "25" ], "output": "5/3", "label": "Chebyshev original atom1 weighted SIGNED X" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Chebyshev original atom2 weighted SIGNED X := mul(9/10, 28)", "after": "Chebyshev original atom2 weighted SIGNED X = 126/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom2 weighted SIGNED X，不以小數近似替代。", "en": "Recompute Chebyshev original atom2 weighted SIGNED X with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9/10", "28" ], "output": "126/5", "label": "Chebyshev original atom2 weighted SIGNED X" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Chebyshev complete SAME-law mean := sum(0, 5/3, 126/5)", "after": "Chebyshev complete SAME-law mean = 403/15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev complete SAME-law mean，不以小數近似替代。", "en": "Recompute Chebyshev complete SAME-law mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "5/3", "126/5" ], "output": "403/15", "label": "Chebyshev complete SAME-law mean" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Chebyshev original atom0 X minus SAME mean := sub(0, 403/15)", "after": "Chebyshev original atom0 X minus SAME mean = -403/15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom0 X minus SAME mean，不以小數近似替代。", "en": "Recompute Chebyshev original atom0 X minus SAME mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "403/15" ], "output": "-403/15", "label": "Chebyshev original atom0 X minus SAME mean" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Chebyshev original atom0 centered SQUARE := mul(-403/15, -403/15)", "after": "Chebyshev original atom0 centered SQUARE = 162409/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom0 centered SQUARE，不以小數近似替代。", "en": "Recompute Chebyshev original atom0 centered SQUARE with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-403/15", "-403/15" ], "output": "162409/225", "label": "Chebyshev original atom0 centered SQUARE" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Chebyshev original atom0 weighted centered square := mul(1/30, 162409/225)", "after": "Chebyshev original atom0 weighted centered square = 162409/6750", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom0 weighted centered square，不以小數近似替代。", "en": "Recompute Chebyshev original atom0 weighted centered square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/30", "162409/225" ], "output": "162409/6750", "label": "Chebyshev original atom0 weighted centered square" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Chebyshev original atom1 X minus SAME mean := sub(25, 403/15)", "after": "Chebyshev original atom1 X minus SAME mean = -28/15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom1 X minus SAME mean，不以小數近似替代。", "en": "Recompute Chebyshev original atom1 X minus SAME mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "25", "403/15" ], "output": "-28/15", "label": "Chebyshev original atom1 X minus SAME mean" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Chebyshev original atom1 centered SQUARE := mul(-28/15, -28/15)", "after": "Chebyshev original atom1 centered SQUARE = 784/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom1 centered SQUARE，不以小數近似替代。", "en": "Recompute Chebyshev original atom1 centered SQUARE with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-28/15", "-28/15" ], "output": "784/225", "label": "Chebyshev original atom1 centered SQUARE" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Chebyshev original atom1 weighted centered square := mul(1/15, 784/225)", "after": "Chebyshev original atom1 weighted centered square = 784/3375", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom1 weighted centered square，不以小數近似替代。", "en": "Recompute Chebyshev original atom1 weighted centered square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/15", "784/225" ], "output": "784/3375", "label": "Chebyshev original atom1 weighted centered square" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Chebyshev original atom2 X minus SAME mean := sub(28, 403/15)", "after": "Chebyshev original atom2 X minus SAME mean = 17/15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom2 X minus SAME mean，不以小數近似替代。", "en": "Recompute Chebyshev original atom2 X minus SAME mean with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "28", "403/15" ], "output": "17/15", "label": "Chebyshev original atom2 X minus SAME mean" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Chebyshev original atom2 centered SQUARE := mul(17/15, 17/15)", "after": "Chebyshev original atom2 centered SQUARE = 289/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom2 centered SQUARE，不以小數近似替代。", "en": "Recompute Chebyshev original atom2 centered SQUARE with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "17/15", "17/15" ], "output": "289/225", "label": "Chebyshev original atom2 centered SQUARE" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Chebyshev original atom2 weighted centered square := mul(9/10, 289/225)", "after": "Chebyshev original atom2 weighted centered square = 289/250", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom2 weighted centered square，不以小數近似替代。", "en": "Recompute Chebyshev original atom2 weighted centered square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9/10", "289/225" ], "output": "289/250", "label": "Chebyshev original atom2 weighted centered square" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Chebyshev full variance E centered square := sum(162409/6750, 784/3375, 289/250)", "after": "Chebyshev full variance E centered square = 5726/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev full variance E centered square，不以小數近似替代。", "en": "Recompute Chebyshev full variance E centered square with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "162409/6750", "784/3375", "289/250" ], "output": "5726/225", "label": "Chebyshev full variance E centered square" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "ORIGINAL complete normalized masses=[1/30, 1/15, 9/10];ALL aligned SIGNED values X=[0, 25, 28];SAME-law mean=403/15;all deviations=[-403/15, -28/15, 17/15]", "after": "ALL nonnegative squared deviations Y=[162409/225, 784/225, 289/225];variance=E[Y]=5726/225;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms", "relation": "implies", "rule_id": "cheb_same_law_finite_signed_variable_no_independence_premise", "conditions": [], "explanation": { "zh-Hant": "此題是完整law的一個隨機變數，X可有正質量負值，不要偷加X≥0或independence。平方Y在每個原子都非負；零變異只迫正質量處偏差0，null-outlier仍保原資料。", "en": "This is one random variable under a complete law.X may be negative at positive mass;do not add X≥0 or independence.Each squared Y is nonnegative,and zero variance forces zero deviations only at positive-mass atoms;retain null outliers." }, "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": "S018", "uses": [ "G001", "S017" ], "before": "Chebyshev original atom0 absolute-event probability mass := mul(1/30, 1)", "after": "Chebyshev original atom0 absolute-event probability mass = 1/30", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom0 absolute-event probability mass，不以小數近似替代。", "en": "Recompute Chebyshev original atom0 absolute-event probability mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/30", "1" ], "output": "1/30", "label": "Chebyshev original atom0 absolute-event probability mass" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "ORIGINAL atom=0;SIGNED centered deviation=-403/15;absolute deviation=403/15;positive epsilon=1", "after": "event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/30", "relation": "implies", "rule_id": "cheb_every_absolute_event_atom_including_equality", "conditions": [], "explanation": { "zh-Hant": "先對同一law的mean中心化再取絕對值，事件用≥而非>；正epsilon本身的嚴格資格不會把尾事件改成嚴格。", "en": "Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict." }, "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": "S020", "uses": [ "G001", "S019" ], "before": "Chebyshev original atom1 absolute-event probability mass := mul(1/15, 1)", "after": "Chebyshev original atom1 absolute-event probability mass = 1/15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom1 absolute-event probability mass，不以小數近似替代。", "en": "Recompute Chebyshev original atom1 absolute-event probability mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/15", "1" ], "output": "1/15", "label": "Chebyshev original atom1 absolute-event probability mass" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "ORIGINAL atom=1;SIGNED centered deviation=-28/15;absolute deviation=28/15;positive epsilon=1", "after": "event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=1/15", "relation": "implies", "rule_id": "cheb_every_absolute_event_atom_including_equality", "conditions": [], "explanation": { "zh-Hant": "先對同一law的mean中心化再取絕對值，事件用≥而非>；正epsilon本身的嚴格資格不會把尾事件改成嚴格。", "en": "Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict." }, "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": "S022", "uses": [ "G001", "S021" ], "before": "Chebyshev original atom2 absolute-event probability mass := mul(9/10, 1)", "after": "Chebyshev original atom2 absolute-event probability mass = 9/10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev original atom2 absolute-event probability mass，不以小數近似替代。", "en": "Recompute Chebyshev original atom2 absolute-event probability mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9/10", "1" ], "output": "9/10", "label": "Chebyshev original atom2 absolute-event probability mass" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "ORIGINAL atom=2;SIGNED centered deviation=17/15;absolute deviation=17/15;positive epsilon=1", "after": "event abs(deviation)>=epsilon=是;EQUALITY INCLUDED;probability contribution=9/10", "relation": "implies", "rule_id": "cheb_every_absolute_event_atom_including_equality", "conditions": [], "explanation": { "zh-Hant": "先對同一law的mean中心化再取絕對值，事件用≥而非>；正epsilon本身的嚴格資格不會把尾事件改成嚴格。", "en": "Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict." }, "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": "S024", "uses": [ "G001", "S023" ], "before": "Chebyshev COMPLETE absolute-deviation tail probability := sum(1/30, 1/15, 9/10)", "after": "Chebyshev COMPLETE absolute-deviation tail probability = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev COMPLETE absolute-deviation tail probability，不以小數近似替代。", "en": "Recompute Chebyshev COMPLETE absolute-deviation tail probability with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/30", "1/15", "9/10" ], "output": "1", "label": "Chebyshev COMPLETE absolute-deviation tail probability" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "Chebyshev POSITIVE epsilon SQUARED threshold := mul(1, 1)", "after": "Chebyshev POSITIVE epsilon SQUARED threshold = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev POSITIVE epsilon SQUARED threshold，不以小數近似替代。", "en": "Recompute Chebyshev POSITIVE epsilon SQUARED threshold with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Chebyshev POSITIVE epsilon SQUARED threshold" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "positive original epsilon=1;consider ANY real centered deviation d;let r=abs(d)>=0", "after": "r+epsilon>0,so r>=epsilon iff(r-epsilon)*(r+epsilon)>=0 iff d^2>=epsilon^2;POINTWISE event identity,not just accidental probability equality", "relation": "implies", "rule_id": "cheb_square_absolute_event_iff_positive_epsilon", "conditions": [], "explanation": { "zh-Hant": "差平方因式分解配合r+epsilon正才保同一不等號，故每個實d的兩事件等價。這是對signed X轉成非負Y的真正橋樑，不能只因兩個數值機率相等就說事件相同。", "en": "Factoring the difference of squares with r+epsilon positive preserves the inequality,proving the events equivalent for every real d.This is the bridge from signed X to nonnegative Y,not an inference of event identity from equal numerical probabilities." }, "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": "S027", "uses": [ "G001", "S026" ], "before": "ORIGINAL atom=0;deviation=-403/15;its SQUARE=162409/225;EPSILON SQUARED=1", "after": "absolute event=是;squared event=是;same pointwise membership=是", "relation": "implies", "rule_id": "cheb_each_original_squared_event_same_membership", "conditions": [], "explanation": { "zh-Hant": "逐每個原原子保同一對應，不只比加權後的總數；null質量處也保持事件membership的數學正確。", "en": "Check membership at every original atom,not just equality of weighted totals.Keep the mathematical event correspondence correct even on null atoms." }, "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": "S028", "uses": [ "G001", "S027" ], "before": "ORIGINAL atom=1;deviation=-28/15;its SQUARE=784/225;EPSILON SQUARED=1", "after": "absolute event=是;squared event=是;same pointwise membership=是", "relation": "implies", "rule_id": "cheb_each_original_squared_event_same_membership", "conditions": [], "explanation": { "zh-Hant": "逐每個原原子保同一對應，不只比加權後的總數；null質量處也保持事件membership的數學正確。", "en": "Check membership at every original atom,not just equality of weighted totals.Keep the mathematical event correspondence correct even on null atoms." }, "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": "S029", "uses": [ "G001", "S028" ], "before": "ORIGINAL atom=2;deviation=17/15;its SQUARE=289/225;EPSILON SQUARED=1", "after": "absolute event=是;squared event=是;same pointwise membership=是", "relation": "implies", "rule_id": "cheb_each_original_squared_event_same_membership", "conditions": [], "explanation": { "zh-Hant": "逐每個原原子保同一對應，不只比加權後的總數；null質量處也保持事件membership的數學正確。", "en": "Check membership at every original atom,not just equality of weighted totals.Keep the mathematical event correspondence correct even on null atoms." }, "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": "S030", "uses": [ "G001", "S029" ], "before": "Chebyshev RAW variance over epsilon SQUARED := div(5726/225, 1)", "after": "Chebyshev RAW variance over epsilon SQUARED = 5726/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev RAW variance over epsilon SQUARED，不以小數近似替代。", "en": "Recompute Chebyshev RAW variance over epsilon SQUARED with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "5726/225", "1" ], "output": "5726/225", "label": "Chebyshev RAW variance over epsilon SQUARED" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "Chebyshev Markov atom0 squared-threshold indicator := mul(1, 1)", "after": "Chebyshev Markov atom0 squared-threshold indicator = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom0 squared-threshold indicator，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom0 squared-threshold indicator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Chebyshev Markov atom0 squared-threshold indicator" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "Chebyshev Markov atom0 Y minus threshold indicator := sub(162409/225, 1)", "after": "Chebyshev Markov atom0 Y minus threshold indicator = 162184/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom0 Y minus threshold indicator，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom0 Y minus threshold indicator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "162409/225", "1" ], "output": "162184/225", "label": "Chebyshev Markov atom0 Y minus threshold indicator" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "Chebyshev Markov atom0 weighted nonnegative slack := mul(1/30, 162184/225)", "after": "Chebyshev Markov atom0 weighted nonnegative slack = 81092/3375", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom0 weighted nonnegative slack，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom0 weighted nonnegative slack with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/30", "162184/225" ], "output": "81092/3375", "label": "Chebyshev Markov atom0 weighted nonnegative slack" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "Chebyshev Markov atom1 squared-threshold indicator := mul(1, 1)", "after": "Chebyshev Markov atom1 squared-threshold indicator = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom1 squared-threshold indicator，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom1 squared-threshold indicator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Chebyshev Markov atom1 squared-threshold indicator" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "Chebyshev Markov atom1 Y minus threshold indicator := sub(784/225, 1)", "after": "Chebyshev Markov atom1 Y minus threshold indicator = 559/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom1 Y minus threshold indicator，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom1 Y minus threshold indicator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "784/225", "1" ], "output": "559/225", "label": "Chebyshev Markov atom1 Y minus threshold indicator" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "Chebyshev Markov atom1 weighted nonnegative slack := mul(1/15, 559/225)", "after": "Chebyshev Markov atom1 weighted nonnegative slack = 559/3375", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom1 weighted nonnegative slack，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom1 weighted nonnegative slack with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/15", "559/225" ], "output": "559/3375", "label": "Chebyshev Markov atom1 weighted nonnegative slack" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "Chebyshev Markov atom2 squared-threshold indicator := mul(1, 1)", "after": "Chebyshev Markov atom2 squared-threshold indicator = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom2 squared-threshold indicator，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom2 squared-threshold indicator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Chebyshev Markov atom2 squared-threshold indicator" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "Chebyshev Markov atom2 Y minus threshold indicator := sub(289/225, 1)", "after": "Chebyshev Markov atom2 Y minus threshold indicator = 64/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom2 Y minus threshold indicator，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom2 Y minus threshold indicator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "289/225", "1" ], "output": "64/225", "label": "Chebyshev Markov atom2 Y minus threshold indicator" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "Chebyshev Markov atom2 weighted nonnegative slack := mul(9/10, 64/225)", "after": "Chebyshev Markov atom2 weighted nonnegative slack = 32/125", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov atom2 weighted nonnegative slack，不以小數近似替代。", "en": "Recompute Chebyshev Markov atom2 weighted nonnegative slack with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9/10", "64/225" ], "output": "32/125", "label": "Chebyshev Markov atom2 weighted nonnegative slack" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "Chebyshev Markov ALL weighted slacks sum := sum(81092/3375, 559/3375, 32/125)", "after": "Chebyshev Markov ALL weighted slacks sum = 5501/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev Markov ALL weighted slacks sum，不以小數近似替代。", "en": "Recompute Chebyshev Markov ALL weighted slacks sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "81092/3375", "559/3375", "32/125" ], "output": "5501/225", "label": "Chebyshev Markov ALL weighted slacks sum" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "Chebyshev epsilon squared times SAME event probability := mul(1, 1)", "after": "Chebyshev epsilon squared times SAME event probability = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev epsilon squared times SAME event probability，不以小數近似替代。", "en": "Recompute Chebyshev epsilon squared times SAME event probability with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Chebyshev epsilon squared times SAME event probability" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "Chebyshev variance minus threshold-event mass := sub(5726/225, 1)", "after": "Chebyshev variance minus threshold-event mass = 5501/225", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Chebyshev variance minus threshold-event mass，不以小數近似替代。", "en": "Recompute Chebyshev variance minus threshold-event mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "5726/225", "1" ], "output": "5501/225", "label": "Chebyshev variance minus threshold-event mass" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "Y=ORIGINAL squared deviations=[162409/225, 784/225, 289/225] nonnegative at EVERY atom;E[Y]=variance=5726/225;positive threshold epsilon^2=1;complete weighted slacks=[81092/3375, 559/3375, 32/125]", "after": "variance-epsilon^2*tail=5501/225>=0;divide POSITIVE epsilon^2 to obtain actual tail=1<=RAW bound=5726/225;no independence and no equality/clipping claim", "relation": "implies", "rule_id": "cheb_full_Markov_on_squared_variable_not_signed_X_or_IID", "conditions": [], "explanation": { "zh-Hant": "事件內Y≥epsilon²，事件外Y≥0，乘原law質量全和再除正epsilon²即可。套用的是Y，不是可能signed的X；不需要independence。raw bound可>1，不等於尾機率，σ²=0情形也直接涵蓋。", "en": "On the event Y≥epsilon squared;off it Y≥0.Weight by the complete original law,sum and divide by positive epsilon squared.Apply Markov to Y,not potentially signed X.No independence is needed.The raw bound may exceed1 and need not equal the tail;variance0 is included." }, "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": [ "1", "5726/225", "5726/225", true, "upper_bound_not_tail_equality_and_no_independence_needed" ] }, "conclusion": { "canonical_value": "(1, 5726/225, 5726/225, 是, upper_bound_not_tail_equality_and_no_independence_needed)", "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-20261108-r04-chebyshev-inequality-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations/questions.jsonl", "batch_sha256": "c90bc449291772f89479d9f3a367a2ffcec971aa6328ff0535e2cd63991a9827", "original_line": 75, "original_line_sha256": "9013295f6f5ef906548fabc2460f5629dcebedf255fb95e04a86692926e895d1", "question_compact_sha256": "d53bd7f9547371e05dbd392befab0d155ea3b705f404693ef0ee0bdc1130bb3a", "english_row_compact_sha256": "54b11f79ec8f2de723435d2815d750399e9936425486b1c9bb31a4d6256266b0", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations.en.jsonl" }, "mko_bindings": { "mko-chebyshev-inequality": { "sha256": "53f883dd0d24e24dd2c178a77c932cdf6caf9b65e2b443bee96a237be01b73e3", "english_sha256": "af0da60b30b2be710d729655f44307db4c48210cbe727eded9d899eefcd264a5", "original_path": "public/data/mko/chebyshev-inequality.json", "english_path": "public/data/i18n/editions/mko-chebyshev-inequality.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "f1ed3239c9185524aa4f0839ab4cfbe8f7d74200b283ea09e850a792f0178a3b", "original_candidate_sha256": "b5487c50aa704a70da1c0d8c13c6931195445dde1021e80cc762bd7dd99b05a4", "original_candidate_line_sha256": "b5487c50aa704a70da1c0d8c13c6931195445dde1021e80cc762bd7dd99b05a4" }, "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-20261026-008-p41", "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": "c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d", "question_first_witness_sha256": "e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 43, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "e405aca4d94d8e0f9da3d81e70f549c76fd0544943673cccc0307e0537ec4155", "arithmetic_steps": 33 }, "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." } }