第 18264 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-moment-at-n；資料={"masses":[[1,2],[1,2]],"coefficients":[14,-15],"shift":2,"p":2,"n":14}。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-moment-at-n；資料={"masses":[[1,2],[1,2]],"coefficients":[14,-15],"shift":2,"p":2,"n":14}。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-moment-at-n;original snapshot and mathematics retained

explicit ordered source output fields=[p, n, p_moment, 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, 14)

Lp coefficient atom0 absolute power product stage1 = 14

Lp coefficient atom0 absolute power product stage2 := mul(14, 14)

Lp coefficient atom0 absolute power product stage2 = 196

Lp constant weighted coefficient atom0 := mul(1/2, 196)

Lp constant weighted coefficient atom0 = 98

original atom=0;mass=1/2;SIGNED coefficient=14;fixed integer p=2

absolute coefficient power=196;weighted coefficient power=98

Lp coefficient atom1 absolute power product stage1 := mul(1, 15)

Lp coefficient atom1 absolute power product stage1 = 15

Lp coefficient atom1 absolute power product stage2 := mul(15, 15)

Lp coefficient atom1 absolute power product stage2 = 225

Lp constant weighted coefficient atom1 := mul(1/2, 225)

Lp constant weighted coefficient atom1 = 225/2

original atom=1;mass=1/2;SIGNED coefficient=-15;fixed integer p=2

absolute coefficient power=225;weighted coefficient power=225/2

Lp full weighted absolute coefficient power sum := sum(98, 225/2)

Lp full weighted absolute coefficient power sum = 421/2

SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[14, -15];shift=2;fixed p=2;weighted constant=421/2

for EVERY n>=0,p-moment E|error_n|^p=(421/2)/(n+2)^2;all moments finite and tend0;this is NOT the p-th-root norm value

selected Lp index common error denominator := sum(14, 2)

selected Lp index common error denominator = 16

selected Lp index atom0 absolute error amplitude := div(14, 16)

selected Lp index atom0 absolute error amplitude = 7/8

selected Lp index atom0 error power product stage1 := mul(1, 7/8)

selected Lp index atom0 error power product stage1 = 7/8

selected Lp index atom0 error power product stage2 := mul(7/8, 7/8)

selected Lp index atom0 error power product stage2 = 49/64

selected Lp index atom0 weighted moment term := mul(1/2, 49/64)

selected Lp index atom0 weighted moment term = 49/128

selected Lp index atom1 absolute error amplitude := div(15, 16)

selected Lp index atom1 absolute error amplitude = 15/16

selected Lp index atom1 error power product stage1 := mul(1, 15/16)

selected Lp index atom1 error power product stage1 = 15/16

selected Lp index atom1 error power product stage2 := mul(15/16, 15/16)

selected Lp index atom1 error power product stage2 = 225/256

selected Lp index atom1 weighted moment term := mul(1/2, 225/256)

selected Lp index atom1 weighted moment term = 225/512

selected Lp index full p-moment := sum(49/128, 225/512)

selected Lp index full p-moment = 421/512

selected Lp index:original n=14;fixed p=2;all weighted absolute-power terms=[49/128, 225/512]

exact p-moment=421/512;neither its p-th root nor unweighted atom mean

逐步反向核對

• 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 selected Lp index full p-moment，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 selected Lp index atom1 weighted moment term，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 selected Lp index atom1 error power product stage2，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 selected Lp index atom1 error power product stage1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 selected Lp index atom1 absolute error amplitude，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 selected Lp index atom0 weighted moment term，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 selected Lp index atom0 error power product stage2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 selected Lp index atom0 error power product stage1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 selected Lp index atom0 absolute error amplitude，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 selected Lp index common error denominator，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, 14, 421/512, exact_finite_atomic_moment)

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。