第 27699 題：有限值IID強大數律

原題與已知條件

無限IID process certificate、finite marginal law與almost-sure limit責任。操作=slln-scope-boundary；資料={"mode":"finite_prefix","model":{"values":[1,2],"probabilities":[[1,5],[4,5]],"observations":[1,2,2]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 無限IID process certificate、finite marginal law與almost-sure limit責任。操作=slln-scope-boundary；資料={"mode":"finite_prefix","model":{"values":[1,2],"probabilities":[[1,5],[4,5]],"observations":[1,2,2]}}。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-strong-law-large-numbers-finite-valued-iid

逐步推導

source roles = [slln-scope-boundary]

derived = [[mode, observations, common_mean, infinite_process_or_limit_proof_claim, boundary]]

SLLN boundary prefix law ORIGINAL mass sum0 := sum(0, 1/5)

SLLN boundary prefix law ORIGINAL mass sum0 = 1/5

SLLN boundary prefix law ORIGINAL mass sum1 := sum(1/5, 4/5)

SLLN boundary prefix law ORIGINAL mass sum1 = 1

SLLN boundary prefix law ORIGINAL weighted value0 := mul(1, 1/5)

SLLN boundary prefix law ORIGINAL weighted value0 = 1/5

SLLN boundary prefix law ORIGINAL mean sum0 := sum(0, 1/5)

SLLN boundary prefix law ORIGINAL mean sum0 = 1/5

SLLN boundary prefix law exact unbounded value square0 := mul(1, 1)

SLLN boundary prefix law exact unbounded value square0 = 1

SLLN boundary prefix law ORIGINAL weighted square0 := mul(1, 1/5)

SLLN boundary prefix law ORIGINAL weighted square0 = 1/5

SLLN boundary prefix law ORIGINAL second moment sum0 := sum(0, 1/5)

SLLN boundary prefix law ORIGINAL second moment sum0 = 1/5

SLLN boundary prefix law ORIGINAL weighted value1 := mul(2, 4/5)

SLLN boundary prefix law ORIGINAL weighted value1 = 8/5

SLLN boundary prefix law ORIGINAL mean sum1 := sum(1/5, 8/5)

SLLN boundary prefix law ORIGINAL mean sum1 = 9/5

SLLN boundary prefix law exact unbounded value square1 := mul(2, 2)

SLLN boundary prefix law exact unbounded value square1 = 4

SLLN boundary prefix law ORIGINAL weighted square1 := mul(4, 4/5)

SLLN boundary prefix law ORIGINAL weighted square1 = 16/5

SLLN boundary prefix law ORIGINAL second moment sum1 := sum(1/5, 16/5)

SLLN boundary prefix law ORIGINAL second moment sum1 = 17/5

SLLN boundary prefix law ORIGINAL mean squared := mul(9/5, 9/5)

SLLN boundary prefix law ORIGINAL mean squared = 81/25

SLLN boundary prefix law ORIGINAL second minus mean squared := sub(17/5, 81/25)

SLLN boundary prefix law ORIGINAL second minus mean squared = 4/25

source roles = [SLLN boundary prefix law, [1, 2], [[1, 5], [4, 5]]]

derived = [[[1, 1/5], [2, 4/5]], 1, 9/5, 17/5, 4/25]

source roles = [[1, 2, 2], [[1, 1/5], [2, 4/5]]]

derived = [9/5, 否]

逐步反向核對

• S017 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S016 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S015 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL second minus mean squared，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL mean squared，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL second moment sum1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL weighted square1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law exact unbounded value square1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL mean sum1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL weighted value1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL second moment sum0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL weighted square0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law exact unbounded value square0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL mean sum0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL weighted value0，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL mass sum1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 SLLN boundary prefix law ORIGINAL mass sum0，不以小數近似替代。
• S001 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。

結論

(finite_prefix, [1, 2, 2], 9/5, 否, finite_prefix_cannot_establish_infinite_process_or_a.s._limit)

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。