第 18221 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":[21,-22],"shift":2,"p":3,"epsilon":[1,30]}。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":[21,-22],"shift":2,"p":3,"epsilon":[1,30]}。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, 21)

Lp coefficient atom0 absolute power product stage1 = 21

Lp coefficient atom0 absolute power product stage2 := mul(21, 21)

Lp coefficient atom0 absolute power product stage2 = 441

Lp coefficient atom0 absolute power product stage3 := mul(441, 21)

Lp coefficient atom0 absolute power product stage3 = 9261

Lp constant weighted coefficient atom0 := mul(1/2, 9261)

Lp constant weighted coefficient atom0 = 9261/2

original atom=0;mass=1/2;SIGNED coefficient=21;fixed integer p=3

absolute coefficient power=9261;weighted coefficient power=9261/2

Lp coefficient atom1 absolute power product stage1 := mul(1, 22)

Lp coefficient atom1 absolute power product stage1 = 22

Lp coefficient atom1 absolute power product stage2 := mul(22, 22)

Lp coefficient atom1 absolute power product stage2 = 484

Lp coefficient atom1 absolute power product stage3 := mul(484, 22)

Lp coefficient atom1 absolute power product stage3 = 10648

Lp constant weighted coefficient atom1 := mul(1/2, 10648)

Lp constant weighted coefficient atom1 = 5324

original atom=1;mass=1/2;SIGNED coefficient=-22;fixed integer p=3

absolute coefficient power=10648;weighted coefficient power=5324

Lp full weighted absolute coefficient power sum := sum(9261/2, 5324)

Lp full weighted absolute coefficient power sum = 19909/2

SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[21, -22];shift=2;fixed p=3;weighted constant=19909/2

for EVERY n>=0,p-moment E|error_n|^p=(19909/2)/(n+2)^3;all moments finite and tend0;this is NOT the p-th-root norm value

Lp strict moment comparison numerator := mul(19909, 30)

Lp strict moment comparison numerator = 597270

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

Lp strict moment comparison coefficient = 2

Lp least cutoff positive denominator := sum(65, 2)

Lp least cutoff positive denominator = 67

Lp cutoff denominator power product stage1 := mul(1, 67)

Lp cutoff denominator power product stage1 = 67

Lp cutoff denominator power product stage2 := mul(67, 67)

Lp cutoff denominator power product stage2 = 4489

Lp cutoff denominator power product stage3 := mul(4489, 67)

Lp cutoff denominator power product stage3 = 300763

Lp strict cutoff integer right side := mul(2, 300763)

Lp strict cutoff integer right side = 601526

Lp previous denominator := sub(67, 1)

Lp previous denominator = 66

Lp previous denominator power product stage1 := mul(1, 66)

Lp previous denominator power product stage1 = 66

Lp previous denominator power product stage2 := mul(66, 66)

Lp previous denominator power product stage2 = 4356

Lp previous denominator power product stage3 := mul(4356, 66)

Lp previous denominator power product stage3 = 287496

Lp previous integer right side := mul(2, 287496)

Lp previous integer right side = 574992

cutoff N=65;denominator=67;integer left=597270;current right=601526;previous right=574992

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

Lp cutoff evaluation common error denominator := sum(65, 2)

Lp cutoff evaluation common error denominator = 67

Lp cutoff evaluation atom0 absolute error amplitude := div(21, 67)

Lp cutoff evaluation atom0 absolute error amplitude = 21/67

Lp cutoff evaluation atom0 error power product stage1 := mul(1, 21/67)

Lp cutoff evaluation atom0 error power product stage1 = 21/67

Lp cutoff evaluation atom0 error power product stage2 := mul(21/67, 21/67)

Lp cutoff evaluation atom0 error power product stage2 = 441/4489

Lp cutoff evaluation atom0 error power product stage3 := mul(441/4489, 21/67)

Lp cutoff evaluation atom0 error power product stage3 = 9261/300763

Lp cutoff evaluation atom0 weighted moment term := mul(1/2, 9261/300763)

Lp cutoff evaluation atom0 weighted moment term = 9261/601526

Lp cutoff evaluation atom1 absolute error amplitude := div(22, 67)

Lp cutoff evaluation atom1 absolute error amplitude = 22/67

Lp cutoff evaluation atom1 error power product stage1 := mul(1, 22/67)

Lp cutoff evaluation atom1 error power product stage1 = 22/67

Lp cutoff evaluation atom1 error power product stage2 := mul(22/67, 22/67)

Lp cutoff evaluation atom1 error power product stage2 = 484/4489

Lp cutoff evaluation atom1 error power product stage3 := mul(484/4489, 22/67)

Lp cutoff evaluation atom1 error power product stage3 = 10648/300763

Lp cutoff evaluation atom1 weighted moment term := mul(1/2, 10648/300763)

Lp cutoff evaluation atom1 weighted moment term = 5324/300763

Lp cutoff evaluation full p-moment := sum(9261/601526, 5324/300763)

Lp cutoff evaluation full p-moment = 19909/601526

Lp cutoff evaluation:original n=65;fixed p=3;all weighted absolute-power terms=[9261/601526, 5324/300763]

exact p-moment=19909/601526;neither its p-th root nor unweighted atom mean

moment_tolerance epsilon=1/30;fixed p=3;least N=65;EXACT moment at N=19909/601526

for EVERY later n,moment=K/(n+shift)^p<=moment_at_N<epsilon;norm bound with SAME epsilon would instead require moment<epsilon^3

逐步反向核對

• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• S037 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom1 error power product stage3，不以小數近似替代。
• S036 · 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。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · premise_and_result_audit — 以精確有理數重算 Lp cutoff evaluation atom0 error power product stage3，不以小數近似替代。
• S031 · 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。
• S030 · 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。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · premise_and_result_audit — 以精確有理數重算 Lp previous denominator power product stage3，不以小數近似替代。
• S024 · 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。
• S023 · 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。
• S022 · premise_and_result_audit — 以精確有理數重算 Lp previous denominator，不以小數近似替代。
• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 Lp cutoff denominator power product stage3，不以小數近似替代。
• S019 · 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。
• S018 · 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。
• S017 · 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。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• S013 · 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。
• 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 constant weighted coefficient atom1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Lp coefficient atom1 absolute power product stage3，不以小數近似替代。
• S009 · 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。
• S008 · 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。
• S007 · 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。
• S006 · 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。
• S005 · premise_and_result_audit — 以精確有理數重算 Lp coefficient atom0 absolute power product stage3，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

(3, 1/30, 65, 19909/601526, 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。