第 18224 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":[24,-25],"shift":2,"p":3,"epsilon":[1,33]}。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":[24,-25],"shift":2,"p":3,"epsilon":[1,33]}。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, 24)

Lp coefficient atom0 absolute power product stage1 = 24

Lp coefficient atom0 absolute power product stage2 := mul(24, 24)

Lp coefficient atom0 absolute power product stage2 = 576

Lp coefficient atom0 absolute power product stage3 := mul(576, 24)

Lp coefficient atom0 absolute power product stage3 = 13824

Lp constant weighted coefficient atom0 := mul(1/2, 13824)

Lp constant weighted coefficient atom0 = 6912

original atom=0;mass=1/2;SIGNED coefficient=24;fixed integer p=3

absolute coefficient power=13824;weighted coefficient power=6912

Lp coefficient atom1 absolute power product stage1 := mul(1, 25)

Lp coefficient atom1 absolute power product stage1 = 25

Lp coefficient atom1 absolute power product stage2 := mul(25, 25)

Lp coefficient atom1 absolute power product stage2 = 625

Lp coefficient atom1 absolute power product stage3 := mul(625, 25)

Lp coefficient atom1 absolute power product stage3 = 15625

Lp constant weighted coefficient atom1 := mul(1/2, 15625)

Lp constant weighted coefficient atom1 = 15625/2

original atom=1;mass=1/2;SIGNED coefficient=-25;fixed integer p=3

absolute coefficient power=15625;weighted coefficient power=15625/2

Lp full weighted absolute coefficient power sum := sum(6912, 15625/2)

Lp full weighted absolute coefficient power sum = 29449/2

SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[24, -25];shift=2;fixed p=3;weighted constant=29449/2

for EVERY n>=0,p-moment E|error_n|^p=(29449/2)/(n+2)^3;all moments finite and tend0;this is NOT the p-th-root norm value

Lp strict moment comparison numerator := mul(29449, 33)

Lp strict moment comparison numerator = 971817

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

Lp strict moment comparison coefficient = 2

Lp least cutoff positive denominator := sum(77, 2)

Lp least cutoff positive denominator = 79

Lp cutoff denominator power product stage1 := mul(1, 79)

Lp cutoff denominator power product stage1 = 79

Lp cutoff denominator power product stage2 := mul(79, 79)

Lp cutoff denominator power product stage2 = 6241

Lp cutoff denominator power product stage3 := mul(6241, 79)

Lp cutoff denominator power product stage3 = 493039

Lp strict cutoff integer right side := mul(2, 493039)

Lp strict cutoff integer right side = 986078

Lp previous denominator := sub(79, 1)

Lp previous denominator = 78

Lp previous denominator power product stage1 := mul(1, 78)

Lp previous denominator power product stage1 = 78

Lp previous denominator power product stage2 := mul(78, 78)

Lp previous denominator power product stage2 = 6084

Lp previous denominator power product stage3 := mul(6084, 78)

Lp previous denominator power product stage3 = 474552

Lp previous integer right side := mul(2, 474552)

Lp previous integer right side = 949104

cutoff N=77;denominator=79;integer left=971817;current right=986078;previous right=949104

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

Lp cutoff evaluation common error denominator := sum(77, 2)

Lp cutoff evaluation common error denominator = 79

Lp cutoff evaluation atom0 absolute error amplitude := div(24, 79)

Lp cutoff evaluation atom0 absolute error amplitude = 24/79

Lp cutoff evaluation atom0 error power product stage1 := mul(1, 24/79)

Lp cutoff evaluation atom0 error power product stage1 = 24/79

Lp cutoff evaluation atom0 error power product stage2 := mul(24/79, 24/79)

Lp cutoff evaluation atom0 error power product stage2 = 576/6241

Lp cutoff evaluation atom0 error power product stage3 := mul(576/6241, 24/79)

Lp cutoff evaluation atom0 error power product stage3 = 13824/493039

Lp cutoff evaluation atom0 weighted moment term := mul(1/2, 13824/493039)

Lp cutoff evaluation atom0 weighted moment term = 6912/493039

Lp cutoff evaluation atom1 absolute error amplitude := div(25, 79)

Lp cutoff evaluation atom1 absolute error amplitude = 25/79

Lp cutoff evaluation atom1 error power product stage1 := mul(1, 25/79)

Lp cutoff evaluation atom1 error power product stage1 = 25/79

Lp cutoff evaluation atom1 error power product stage2 := mul(25/79, 25/79)

Lp cutoff evaluation atom1 error power product stage2 = 625/6241

Lp cutoff evaluation atom1 error power product stage3 := mul(625/6241, 25/79)

Lp cutoff evaluation atom1 error power product stage3 = 15625/493039

Lp cutoff evaluation atom1 weighted moment term := mul(1/2, 15625/493039)

Lp cutoff evaluation atom1 weighted moment term = 15625/986078

Lp cutoff evaluation full p-moment := sum(6912/493039, 15625/986078)

Lp cutoff evaluation full p-moment = 29449/986078

Lp cutoff evaluation:original n=77;fixed p=3;all weighted absolute-power terms=[6912/493039, 15625/986078]

exact p-moment=29449/986078;neither its p-th root nor unweighted atom mean

moment_tolerance epsilon=1/33;fixed p=3;least N=77;EXACT moment at N=29449/986078

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, 1/33, 77, 29449/986078, 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。