第 18212 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":[12,-13],"shift":2,"p":3,"epsilon":[1,21]}。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":[12,-13],"shift":2,"p":3,"epsilon":[1,21]}。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, 12)

Lp coefficient atom0 absolute power product stage1 = 12

Lp coefficient atom0 absolute power product stage2 := mul(12, 12)

Lp coefficient atom0 absolute power product stage2 = 144

Lp coefficient atom0 absolute power product stage3 := mul(144, 12)

Lp coefficient atom0 absolute power product stage3 = 1728

Lp constant weighted coefficient atom0 := mul(1/2, 1728)

Lp constant weighted coefficient atom0 = 864

original atom=0;mass=1/2;SIGNED coefficient=12;fixed integer p=3

absolute coefficient power=1728;weighted coefficient power=864

Lp coefficient atom1 absolute power product stage1 := mul(1, 13)

Lp coefficient atom1 absolute power product stage1 = 13

Lp coefficient atom1 absolute power product stage2 := mul(13, 13)

Lp coefficient atom1 absolute power product stage2 = 169

Lp coefficient atom1 absolute power product stage3 := mul(169, 13)

Lp coefficient atom1 absolute power product stage3 = 2197

Lp constant weighted coefficient atom1 := mul(1/2, 2197)

Lp constant weighted coefficient atom1 = 2197/2

original atom=1;mass=1/2;SIGNED coefficient=-13;fixed integer p=3

absolute coefficient power=2197;weighted coefficient power=2197/2

Lp full weighted absolute coefficient power sum := sum(864, 2197/2)

Lp full weighted absolute coefficient power sum = 3925/2

SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[12, -13];shift=2;fixed p=3;weighted constant=3925/2

for EVERY n>=0,p-moment E|error_n|^p=(3925/2)/(n+2)^3;all moments finite and tend0;this is NOT the p-th-root norm value

Lp strict moment comparison numerator := mul(3925, 21)

Lp strict moment comparison numerator = 82425

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

Lp strict moment comparison coefficient = 2

Lp least cutoff positive denominator := sum(33, 2)

Lp least cutoff positive denominator = 35

Lp cutoff denominator power product stage1 := mul(1, 35)

Lp cutoff denominator power product stage1 = 35

Lp cutoff denominator power product stage2 := mul(35, 35)

Lp cutoff denominator power product stage2 = 1225

Lp cutoff denominator power product stage3 := mul(1225, 35)

Lp cutoff denominator power product stage3 = 42875

Lp strict cutoff integer right side := mul(2, 42875)

Lp strict cutoff integer right side = 85750

Lp previous denominator := sub(35, 1)

Lp previous denominator = 34

Lp previous denominator power product stage1 := mul(1, 34)

Lp previous denominator power product stage1 = 34

Lp previous denominator power product stage2 := mul(34, 34)

Lp previous denominator power product stage2 = 1156

Lp previous denominator power product stage3 := mul(1156, 34)

Lp previous denominator power product stage3 = 39304

Lp previous integer right side := mul(2, 39304)

Lp previous integer right side = 78608

cutoff N=33;denominator=35;integer left=82425;current right=85750;previous right=78608

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

Lp cutoff evaluation common error denominator := sum(33, 2)

Lp cutoff evaluation common error denominator = 35

Lp cutoff evaluation atom0 absolute error amplitude := div(12, 35)

Lp cutoff evaluation atom0 absolute error amplitude = 12/35

Lp cutoff evaluation atom0 error power product stage1 := mul(1, 12/35)

Lp cutoff evaluation atom0 error power product stage1 = 12/35

Lp cutoff evaluation atom0 error power product stage2 := mul(12/35, 12/35)

Lp cutoff evaluation atom0 error power product stage2 = 144/1225

Lp cutoff evaluation atom0 error power product stage3 := mul(144/1225, 12/35)

Lp cutoff evaluation atom0 error power product stage3 = 1728/42875

Lp cutoff evaluation atom0 weighted moment term := mul(1/2, 1728/42875)

Lp cutoff evaluation atom0 weighted moment term = 864/42875

Lp cutoff evaluation atom1 absolute error amplitude := div(13, 35)

Lp cutoff evaluation atom1 absolute error amplitude = 13/35

Lp cutoff evaluation atom1 error power product stage1 := mul(1, 13/35)

Lp cutoff evaluation atom1 error power product stage1 = 13/35

Lp cutoff evaluation atom1 error power product stage2 := mul(13/35, 13/35)

Lp cutoff evaluation atom1 error power product stage2 = 169/1225

Lp cutoff evaluation atom1 error power product stage3 := mul(169/1225, 13/35)

Lp cutoff evaluation atom1 error power product stage3 = 2197/42875

Lp cutoff evaluation atom1 weighted moment term := mul(1/2, 2197/42875)

Lp cutoff evaluation atom1 weighted moment term = 2197/85750

Lp cutoff evaluation full p-moment := sum(864/42875, 2197/85750)

Lp cutoff evaluation full p-moment = 157/3430

Lp cutoff evaluation:original n=33;fixed p=3;all weighted absolute-power terms=[864/42875, 2197/85750]

exact p-moment=157/3430;neither its p-th root nor unweighted atom mean

moment_tolerance epsilon=1/21;fixed p=3;least N=33;EXACT moment at N=157/3430

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, 1/21, 33, 157/3430, 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。