第 27642 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":[-1,0],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","observations":[-1,0,-1,0,-1]}。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":[-1,0],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","observations":[-1,0,-1,0,-1]}。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, 2/5)

SLLN ORIGINAL common marginal ORIGINAL mass sum0 = 2/5

SLLN ORIGINAL common marginal ORIGINAL mass sum1 := sum(2/5, 3/5)

SLLN ORIGINAL common marginal ORIGINAL mass sum1 = 1

SLLN ORIGINAL common marginal ORIGINAL weighted value0 := mul(-1, 2/5)

SLLN ORIGINAL common marginal ORIGINAL weighted value0 = -2/5

SLLN ORIGINAL common marginal ORIGINAL mean sum0 := sum(0, -2/5)

SLLN ORIGINAL common marginal ORIGINAL mean sum0 = -2/5

SLLN ORIGINAL common marginal exact unbounded value square0 := mul(-1, -1)

SLLN ORIGINAL common marginal exact unbounded value square0 = 1

SLLN ORIGINAL common marginal ORIGINAL weighted square0 := mul(1, 2/5)

SLLN ORIGINAL common marginal ORIGINAL weighted square0 = 2/5

SLLN ORIGINAL common marginal ORIGINAL second moment sum0 := sum(0, 2/5)

SLLN ORIGINAL common marginal ORIGINAL second moment sum0 = 2/5

SLLN ORIGINAL common marginal ORIGINAL weighted value1 := mul(0, 3/5)

SLLN ORIGINAL common marginal ORIGINAL weighted value1 = 0

SLLN ORIGINAL common marginal ORIGINAL mean sum1 := sum(-2/5, 0)

SLLN ORIGINAL common marginal ORIGINAL mean sum1 = -2/5

SLLN ORIGINAL common marginal exact unbounded value square1 := mul(0, 0)

SLLN ORIGINAL common marginal exact unbounded value square1 = 0

SLLN ORIGINAL common marginal ORIGINAL weighted square1 := mul(0, 3/5)

SLLN ORIGINAL common marginal ORIGINAL weighted square1 = 0

SLLN ORIGINAL common marginal ORIGINAL second moment sum1 := sum(2/5, 0)

SLLN ORIGINAL common marginal ORIGINAL second moment sum1 = 2/5

SLLN ORIGINAL common marginal ORIGINAL mean squared := mul(-2/5, -2/5)

SLLN ORIGINAL common marginal ORIGINAL mean squared = 4/25

SLLN ORIGINAL common marginal ORIGINAL second minus mean squared := sub(2/5, 4/25)

SLLN ORIGINAL common marginal ORIGINAL second minus mean squared = 6/25

source roles = [SLLN ORIGINAL common marginal, [-1, 0], [[2, 5], [3, 5]]]

derived = [[[-1, 2/5], [0, 3/5]], 1, -2/5, 2/5, 6/25]

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

SLLN ORIGINAL prefix integer sum = -3

SLLN ORIGINAL prefix sample mean := div(-3, 5)

SLLN ORIGINAL prefix sample mean = -3/5

SLLN ORIGINAL sample mean minus common mean := sub(-3/5, -2/5)

SLLN ORIGINAL sample mean minus common mean = -1/5

source roles = [[-1, 0, -1, 0, -1], -2/5, -3/5, -1/5]

derived = [1/5, 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, -1, 0, -1], -2/5, -3/5, 1/5, 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。