第 27633 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-prefix-sample-mean；資料={"values":[0,1],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","observations":[1,0]}。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-prefix-sample-mean；資料={"values":[0,1],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","observations":[1,0]}。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-prefix-sample-mean]

derived = [[observations, common_mean, prefix_sample_mean, absolute_error, evidence_scope, almost_sure_limit_proof_claim, boundary]]

source roles = [infinite_iid]

derived = [ASSUME whole infinite sequence is IID with this common marginal]

SLLN ORIGINAL common marginal ORIGINAL mass sum0 := sum(0, 1/3)

SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 1/3

SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(1/3, 2/3)

SLLN ORIGINAL common marginal ORIGINAL mass sum1 = 1

SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(0, 1/3)

SLLN ORIGINAL common marginal ORIGINAL weighted value0 = 0

SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, 0)

SLLN ORIGINAL common marginal ORIGINAL mean sum0 = 0

SLLN ORIGINAL common marginal exact unbounded value square0 := mul(0, 0)

SLLN ORIGINAL common marginal exact unbounded value square0 = 0

SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(0, 1/3)

SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 0

SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 0)

SLLN ORIGINAL common marginal ORIGINAL second moment sum0 = 0

SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(1, 2/3)

SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 2/3

SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(0, 2/3)

SLLN ORIGINAL common marginal ORIGINAL mean sum1 = 2/3

SLLN ORIGINAL common marginal exact unbounded value square1 := mul(1, 1)

SLLN ORIGINAL common marginal exact unbounded value square1 = 1

SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(1, 2/3)

SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 2/3

SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(0, 2/3)

SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 2/3

SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(2/3, 2/3)

SLLN ORIGINAL common marginal ORIGINAL mean squared = 4/9

SLLN ORIGINAL common marginal ORIGINAL second minus mean squared := sub(2/3, 4/9)

SLLN ORIGINAL common marginal ORIGINAL second minus mean squared = 2/9

source roles = [SLLN ORIGINAL common marginal, [0, 1], [[1, 3], [2, 3]]]

derived = [[[0, 1/3], [1, 2/3]], 1, 2/3, 2/3, 2/9]

SLLN ORIGINAL prefix integer sum := sum(1, 0)

SLLN ORIGINAL prefix integer sum = 1

SLLN ORIGINAL prefix sample mean := div(1, 2)

SLLN ORIGINAL prefix sample mean = 1/2

SLLN ORIGINAL sample mean minus common mean := sub(1/2, 2/3)

SLLN ORIGINAL sample mean minus common mean = -1/6

source roles = [[1, 0], 2/3, 1/2, -1/6]

derived = [1/6, 否]

逐步反向核對

• S021 · 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。
• S020 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL sample mean minus common mean，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL prefix sample mean，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL prefix integer sum，不以小數近似替代。
• 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 — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second minus mean squared，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean squared，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second moment sum1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted square1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal exact unbounded value square1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean sum1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted value1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL second moment sum0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted square0，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal exact unbounded value square0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mean sum0，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL weighted value0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mass sum1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 SLLN ORIGINAL common marginal ORIGINAL mass sum0，不以小數近似替代。
• S002 · 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。
• 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

([1, 0], 2/3, 1/2, 1/6, fixed_prefix_only, 否, finite_prefix_does_not_prove_almost_sure_convergence)

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。