第 18220 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":[20,-21],"shift":2,"p":2,"epsilon":[1,29]}。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":[20,-21],"shift":2,"p":2,"epsilon":[1,29]}。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, 20)

Lp coefficient atom0 absolute power product stage1 = 20

Lp coefficient atom0 absolute power product stage2 := mul(20, 20)

Lp coefficient atom0 absolute power product stage2 = 400

Lp constant weighted coefficient atom0 := mul(1/2, 400)

Lp constant weighted coefficient atom0 = 200

original atom=0;mass=1/2;SIGNED coefficient=20;fixed integer p=2

absolute coefficient power=400;weighted coefficient power=200

Lp coefficient atom1 absolute power product stage1 := mul(1, 21)

Lp coefficient atom1 absolute power product stage1 = 21

Lp coefficient atom1 absolute power product stage2 := mul(21, 21)

Lp coefficient atom1 absolute power product stage2 = 441

Lp constant weighted coefficient atom1 := mul(1/2, 441)

Lp constant weighted coefficient atom1 = 441/2

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

absolute coefficient power=441;weighted coefficient power=441/2

Lp full weighted absolute coefficient power sum := sum(200, 441/2)

Lp full weighted absolute coefficient power sum = 841/2

SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[20, -21];shift=2;fixed p=2;weighted constant=841/2

for EVERY n>=0,p-moment E|error_n|^p=(841/2)/(n+2)^2;all moments finite and tend0;this is NOT the p-th-root norm value

Lp strict moment comparison numerator := mul(841, 29)

Lp strict moment comparison numerator = 24389

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

Lp strict moment comparison coefficient = 2

Lp least cutoff positive denominator := sum(109, 2)

Lp least cutoff positive denominator = 111

Lp cutoff denominator power product stage1 := mul(1, 111)

Lp cutoff denominator power product stage1 = 111

Lp cutoff denominator power product stage2 := mul(111, 111)

Lp cutoff denominator power product stage2 = 12321

Lp strict cutoff integer right side := mul(2, 12321)

Lp strict cutoff integer right side = 24642

Lp previous denominator := sub(111, 1)

Lp previous denominator = 110

Lp previous denominator power product stage1 := mul(1, 110)

Lp previous denominator power product stage1 = 110

Lp previous denominator power product stage2 := mul(110, 110)

Lp previous denominator power product stage2 = 12100

Lp previous integer right side := mul(2, 12100)

Lp previous integer right side = 24200

cutoff N=109;denominator=111;integer left=24389;current right=24642;previous right=24200

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

Lp cutoff evaluation common error denominator := sum(109, 2)

Lp cutoff evaluation common error denominator = 111

Lp cutoff evaluation atom0 absolute error amplitude := div(20, 111)

Lp cutoff evaluation atom0 absolute error amplitude = 20/111

Lp cutoff evaluation atom0 error power product stage1 := mul(1, 20/111)

Lp cutoff evaluation atom0 error power product stage1 = 20/111

Lp cutoff evaluation atom0 error power product stage2 := mul(20/111, 20/111)

Lp cutoff evaluation atom0 error power product stage2 = 400/12321

Lp cutoff evaluation atom0 weighted moment term := mul(1/2, 400/12321)

Lp cutoff evaluation atom0 weighted moment term = 200/12321

Lp cutoff evaluation atom1 absolute error amplitude := div(21, 111)

Lp cutoff evaluation atom1 absolute error amplitude = 7/37

Lp cutoff evaluation atom1 error power product stage1 := mul(1, 7/37)

Lp cutoff evaluation atom1 error power product stage1 = 7/37

Lp cutoff evaluation atom1 error power product stage2 := mul(7/37, 7/37)

Lp cutoff evaluation atom1 error power product stage2 = 49/1369

Lp cutoff evaluation atom1 weighted moment term := mul(1/2, 49/1369)

Lp cutoff evaluation atom1 weighted moment term = 49/2738

Lp cutoff evaluation full p-moment := sum(200/12321, 49/2738)

Lp cutoff evaluation full p-moment = 841/24642

Lp cutoff evaluation:original n=109;fixed p=2;all weighted absolute-power terms=[200/12321, 49/2738]

exact p-moment=841/24642;neither its p-th root nor unweighted atom mean

moment_tolerance epsilon=1/29;fixed p=2;least N=109;EXACT moment at N=841/24642

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, 1/29, 109, 841/24642, 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。