{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-lp-convergence-01", "mko_id": "mko-lp-convergence", "title": { "zh-Hant": "有限law的L平方精確尾端", "en": "An exact L-squared tail on a finite law" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-lp-convergence-01", "source": { "mko_id": "mko-lp-convergence", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "aec69290c870938d15b40e7ffb6849ffa5fa53265e0fa29d59021aa1f01ce47f", "english_sha256": "c406a88686066c71f8dd62d25eb6931b8411e40497c57a3e0826bbddfc8655cb" }, "mode": "worked_example", "audience_prerequisites": [ "work" ], "goal": { "zh-Hant": "計算二次誤差並提供任意容忍值的尾端。", "en": "Compute the second error moment and give a tail for any tolerance." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "Ω={a,b}質量1/4、3/4；X(a)=1、X(b)=-1；所有n≥1，X_n(a)=1+2/(n+1)，X_n(b)=-1-1/(n+1)。", "en": "Omega={a,b} has masses 1/4,3/4; X(a)=1, X(b)=-1; for every n≥1, X_n(a)=1+2/(n+1), X_n(b)=-1-1/(n+1)." } } ], "steps": [ { "id": "S01", "uses": [ "G1", "work" ], "before_tex": "p=2,\\quad1/4+3/4=1", "after_tex": "X_n,X\\in L^2(\\Omega,P)\\quad(n\\ge1)", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "同空間有限有理值保證每一二次矩存在。", "en": "Finite rational values on one space ensure each second moment exists." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "work" ], "explanation": { "zh-Hant": "p固定為正整數2，不隨n改。", "en": "The positive integer p=2 is fixed across n." } }, "leaf_basis": "work" }, { "id": "S02", "uses": [ "G1", "S01", "work" ], "before_tex": "X_n-X", "after_tex": "(2/(n+1),-1/(n+1))", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "逐原子先算有號誤差。", "en": "Compute signed errors atom by atom first." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "work" ], "explanation": { "zh-Hant": "b的負誤差在平方前不能誤加到a。", "en": "Do not cancel b's negative error against a before squaring." } }, "leaf_basis": "work" }, { "id": "S03", "uses": [ "G1", "S02", "work" ], "before_tex": "|X_n-X|^2", "after_tex": "(4/(n+1)^2,1/(n+1)^2)", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "取絕對值平方，兩項非負。", "en": "Absolute squares make both terms nonnegative." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02", "work" ], "explanation": { "zh-Hant": "分母也必須平方。", "en": "The denominator must be squared too." } }, "leaf_basis": "work" }, { "id": "S04", "uses": [ "G1", "S03", "work" ], "before_tex": "E|X_n-X|^2", "after_tex": "\\frac14\\frac4{(n+1)^2}+\\frac34\\frac1{(n+1)^2}=\\frac7{4(n+1)^2}", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "以原law權重求精確二次誤差。", "en": "Use the original masses for the exact second error moment." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03", "work" ], "explanation": { "zh-Hant": "常數為7/4，不是4+1。", "en": "The weighted constant is 7/4, not 4+1." } }, "leaf_basis": "work" }, { "id": "S05", "uses": [ "G1", "S04", "work" ], "before_tex": "\\|X_n-X\\|_2^2=7/[4(n+1)^2]", "after_tex": "\\|X_n-X\\|_2=\\sqrt7/[2(n+1)]", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "L2 norm是二次矩的平方根，不是二次矩本身。", "en": "The L2 norm is the square root of the second moment, not that moment itself." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S04", "work" ], "explanation": { "zh-Hant": "平方根分母因n+1正而化成2(n+1)。", "en": "The denominator simplifies because n+1 is positive." } }, "leaf_basis": "work" }, { "id": "S06", "uses": [ "G1", "S05", "work" ], "before_tex": "\\eta>0,\\quad N=\\lfloor7/(4\\eta)\\rfloor+1", "after_tex": "n\\ge N\\implies\\frac7{4(n+1)^2}\\le\\frac7{4(n+1)}<\\eta", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "用較鬆有理界仍給明示全稱尾端。", "en": "A looser rational bound still yields an explicit universal tail." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S05", "work" ], "explanation": { "zh-Hant": "n+1≥1才可用平方分母較大。", "en": "The comparison of denominators uses n+1≥1." } }, "leaf_basis": "work" }, { "id": "S07", "uses": [ "G1", "S06", "work" ], "before_tex": "\\eta=\\varepsilon^2\\quad(\\varepsilon>0)", "after_tex": "n\\ge N(\\varepsilon^2)\\implies\\|X_n-X\\|_2<\\varepsilon", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "以epsilon平方控制moment即可控制norm。", "en": "Controlling the moment by epsilon squared controls the norm." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S06", "work" ], "explanation": { "zh-Hant": "避免混同norm誤差與moment誤差容忍值。", "en": "Keep norm tolerance distinct from moment tolerance." } }, "leaf_basis": "work" }, { "id": "S08", "uses": [ "G1", "S07", "work" ], "before_tex": "\\forall\\varepsilon>0\\ \\exists N\\ \\forall n\\ge N:\\|X_n-X\\|_2<\\varepsilon", "after_tex": "X_n\\to X\\text{ in }L^2", "relation": "implies", "rule": "work", "conditions": [], "explanation": { "zh-Hant": "完整量詞證L2收斂，而不是列前幾個moment。", "en": "The full quantifiers prove L2 convergence, not a finite moment list." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S07", "work" ], "explanation": { "zh-Hant": "未以此有限計算概括一般跨mode定理。", "en": "No general cross-mode theorem is inferred from this calculation." } }, "leaf_basis": "work" } ], "result": { "zh-Hant": "E|X_n-X|²=7/[4(n+1)²]→0，因此L2收斂。", "en": "E|X_n-X|²=7/[4(n+1)²] tends to 0, proving L2 convergence." }, "negative_controls": [ { "claim": { "zh-Hant": "先平均有號誤差再平方就是L2 moment。", "en": "Square the average signed error for the L2 moment." }, "expected": "reject", "reason": { "zh-Hant": "須先逐原子平方再平均。", "en": "Square atomwise before averaging." } }, { "claim": { "zh-Hant": "norm等於7/[4(n+1)²]。", "en": "The norm is 7/[4(n+1)²]." }, "expected": "reject", "reason": { "zh-Hant": "那是norm平方。", "en": "That is the squared norm." } }, { "claim": { "zh-Hant": "有限十個moment下降即證收斂。", "en": "Ten decreasing moments prove convergence." }, "expected": "reject", "reason": { "zh-Hant": "須有所有n≥N的尾端證書。", "en": "A certificate for every n≥N is required." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "本輪限定正整數p及完整有限原子law，使用symbolic error amplitudes c_i/(n+s)精確計算p-moment與epsilon tail。finite prefix不足；不無條件推a.s.、其他Lq或反向機率模式。", "en": "This slice uses positive integer p, a complete finite atomic law and symbolic error amplitudes c_i/(n+s). Compute exact p-moments and epsilon tail bounds. A finite prefix is insufficient; unrestricted almost-sure, other-Lq or reverse probability-mode implications are not inferred without their additional theorem assumptions." }, "source": { "mko_id": "mko-lp-convergence", "sha256": "aec69290c870938d15b40e7ffb6849ffa5fa53265e0fa29d59021aa1f01ce47f", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } }, { "id": "work", "kind": "prerequisite", "statement": { "zh-Hant": "有限law的p-moment為加權絕對p次方和；L2 norm是二次矩平方根；L2收斂定義為誤差L2 norm趨零；floor(u)+1>u。", "en": "Finite-law p-moments sum weighted absolute pth powers; the L2 norm is the square root of the second moment; L2 convergence means the L2 error norm tends to zero; floor(u)+1>u." } } ], "revision_log": [], "numeric_checks": [ { "id": "N01", "lhs": "(1/4)*4+(3/4)*1", "rhs": "7/4" }, { "id": "N02", "lhs": "7/(4*(9+1)**2)", "rhs": "7/400" } ] }, "presentation": { "md_scope": { "zh-Hant": "固定p=2；不由有限prefix推其他模式。", "en": "Fix p=2; no other mode is inferred from a finite prefix." }, "md_symbols": [ [ "p", "固定指數2", "fixed exponent2" ], [ "η", "moment容忍值", "moment tolerance" ], [ "ε", "norm容忍值", "norm tolerance" ] ], "md_error_controls": [ { "title": "權重 / Weights", "zh": "常數1/4乘4再加3/4乘1得7/4。", "en": "The weighted constant is(1/4)4+(3/4)1=7/4." }, { "title": "兩種容忍值 / Two tolerances", "zh": "將moment容忍值設ε²後才推norm<ε。", "en": "Set the moment tolerance to epsilon squared before concluding norm