第 18211 題：Lp收斂

原題與已知條件

同一有限原子probability space、固定整數p大於等於1與symbolic moment tail。操作=lp-epsilon-witness；資料={"masses":[[1,2],[1,2]],"coefficients":[11,-12],"shift":2,"p":2,"epsilon":[1,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 同一有限原子probability space、固定整數p大於等於1與symbolic moment tail。操作=lp-epsilon-witness；資料={"masses":[[1,2],[1,2]],"coefficients":[11,-12],"shift":2,"p":2,"epsilon":[1,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

• mko-lp-convergence

逐步推導

source operation=lp-epsilon-witness;original snapshot and mathematics retained

explicit ordered source output fields=[p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate]

Lp complete atomic probability mass := sum(1/2, 1/2)

Lp complete atomic probability mass = 1

Lp coefficient atom0 absolute power product stage1 := mul(1, 11)

Lp coefficient atom0 absolute power product stage1 = 11

Lp coefficient atom0 absolute power product stage2 := mul(11, 11)

Lp coefficient atom0 absolute power product stage2 = 121

Lp constant weighted coefficient atom0 := mul(1/2, 121)

Lp constant weighted coefficient atom0 = 121/2

original atom=0;mass=1/2;SIGNED coefficient=11;fixed integer p=2

absolute coefficient power=121;weighted coefficient power=121/2

Lp coefficient atom1 absolute power product stage1 := mul(1, 12)

Lp coefficient atom1 absolute power product stage1 = 12

Lp coefficient atom1 absolute power product stage2 := mul(12, 12)

Lp coefficient atom1 absolute power product stage2 = 144

Lp constant weighted coefficient atom1 := mul(1/2, 144)

Lp constant weighted coefficient atom1 = 72

original atom=1;mass=1/2;SIGNED coefficient=-12;fixed integer p=2

absolute coefficient power=144;weighted coefficient power=72

Lp full weighted absolute coefficient power sum := sum(121/2, 72)

Lp full weighted absolute coefficient power sum = 265/2

SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[11, -12];shift=2;fixed p=2;weighted constant=265/2

for EVERY n>=0,p-moment E|error_n|^p=(265/2)/(n+2)^2;all moments finite and tend0;this is NOT the p-th-root norm value

Lp strict moment comparison numerator := mul(265, 20)

Lp strict moment comparison numerator = 5300

Lp strict moment comparison coefficient := mul(2, 1)

Lp strict moment comparison coefficient = 2

Lp least cutoff positive denominator := sum(50, 2)

Lp least cutoff positive denominator = 52

Lp cutoff denominator power product stage1 := mul(1, 52)

Lp cutoff denominator power product stage1 = 52

Lp cutoff denominator power product stage2 := mul(52, 52)

Lp cutoff denominator power product stage2 = 2704

Lp strict cutoff integer right side := mul(2, 2704)

Lp strict cutoff integer right side = 5408

Lp previous denominator := sub(52, 1)

Lp previous denominator = 51

Lp previous denominator power product stage1 := mul(1, 51)

Lp previous denominator power product stage1 = 51

Lp previous denominator power product stage2 := mul(51, 51)

Lp previous denominator power product stage2 = 2601

Lp previous integer right side := mul(2, 2601)

Lp previous integer right side = 5202

cutoff N=50;denominator=52;integer left=5300;current right=5408;previous right=5202

strict comparison holds at N and fails at N-1;all smaller allowed indices fail

Lp cutoff evaluation common error denominator := sum(50, 2)

Lp cutoff evaluation common error denominator = 52

Lp cutoff evaluation atom0 absolute error amplitude := div(11, 52)

Lp cutoff evaluation atom0 absolute error amplitude = 11/52

Lp cutoff evaluation atom0 error power product stage1 := mul(1, 11/52)

Lp cutoff evaluation atom0 error power product stage1 = 11/52

Lp cutoff evaluation atom0 error power product stage2 := mul(11/52, 11/52)

Lp cutoff evaluation atom0 error power product stage2 = 121/2704

Lp cutoff evaluation atom0 weighted moment term := mul(1/2, 121/2704)

Lp cutoff evaluation atom0 weighted moment term = 121/5408

Lp cutoff evaluation atom1 absolute error amplitude := div(12, 52)

Lp cutoff evaluation atom1 absolute error amplitude = 3/13

Lp cutoff evaluation atom1 error power product stage1 := mul(1, 3/13)

Lp cutoff evaluation atom1 error power product stage1 = 3/13

Lp cutoff evaluation atom1 error power product stage2 := mul(3/13, 3/13)

Lp cutoff evaluation atom1 error power product stage2 = 9/169

Lp cutoff evaluation atom1 weighted moment term := mul(1/2, 9/169)

Lp cutoff evaluation atom1 weighted moment term = 9/338

Lp cutoff evaluation full p-moment := sum(121/5408, 9/338)

Lp cutoff evaluation full p-moment = 265/5408

Lp cutoff evaluation:original n=50;fixed p=2;all weighted absolute-power terms=[121/5408, 9/338]

exact p-moment=265/5408;neither its p-th root nor unweighted atom mean

moment_tolerance epsilon=1/20;fixed p=2;least N=50;EXACT moment at N=265/5408

for EVERY later n,moment=K/(n+shift)^p<=moment_at_N<epsilon;norm bound with SAME epsilon would instead require moment<epsilon^2

逐步反向核對

• S035 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S034 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S033 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation full p-moment，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom1 weighted moment term，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom1 error power product stage2，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom1 error power product stage1，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom1 absolute error amplitude，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom0 weighted moment term，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom0 error power product stage2，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom0 error power product stage1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom0 absolute error amplitude，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation common error denominator，不以小數近似替代。
• S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · premise_and_result_audit — 以精確有理數重算 Lp previous integer right side，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Lp previous denominator power product stage2，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Lp previous denominator power product stage1，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Lp previous denominator，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Lp strict cutoff integer right side，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Lp cutoff denominator power product stage2，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Lp cutoff denominator power product stage1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Lp least cutoff positive denominator，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Lp strict moment comparison coefficient，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Lp strict moment comparison numerator，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 Lp full weighted absolute coefficient power sum，不以小數近似替代。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 以精確有理數重算 Lp constant weighted coefficient atom1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Lp coefficient atom1 absolute power product stage2，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Lp coefficient atom1 absolute power product stage1，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 Lp constant weighted coefficient atom0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Lp coefficient atom0 absolute power product stage2，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Lp coefficient atom0 absolute power product stage1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Lp complete atomic probability mass，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(2, 1/20, 50, 265/5408, 是, forall_n_ge_N_symbolic_monotone_tail)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。