第 27689 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":"non_iid","model":{"values_a":[1,2],"probabilities_a":[[1,4],[3,4]],"values_b":[1,2],"probabilities_b":[[1,7],[6,7]]}}。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":"non_iid","model":{"values_a":[1,2],"probabilities_a":[[1,4],[3,4]],"values_b":[1,2],"probabilities_b":[[1,7],[6,7]]}}。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, identical_marginals, this_finite_valued_IID_theorem_applicable, scope]]

SLLN boundary first marginal ORIGINAL mass sum0 := sum(0, 1/4)

SLLN boundary first marginal ORIGINAL mass sum0 = 1/4

SLLN boundary first marginal ORIGINAL mass sum1 := sum(1/4, 3/4)

SLLN boundary first marginal ORIGINAL mass sum1 = 1

SLLN boundary first marginal ORIGINAL weighted value0 := mul(1, 1/4)

SLLN boundary first marginal ORIGINAL weighted value0 = 1/4

SLLN boundary first marginal ORIGINAL mean sum0 := sum(0, 1/4)

SLLN boundary first marginal ORIGINAL mean sum0 = 1/4

SLLN boundary first marginal exact unbounded value square0 := mul(1, 1)

SLLN boundary first marginal exact unbounded value square0 = 1

SLLN boundary first marginal ORIGINAL weighted square0 := mul(1, 1/4)

SLLN boundary first marginal ORIGINAL weighted square0 = 1/4

SLLN boundary first marginal ORIGINAL second moment sum0 := sum(0, 1/4)

SLLN boundary first marginal ORIGINAL second moment sum0 = 1/4

SLLN boundary first marginal ORIGINAL weighted value1 := mul(2, 3/4)

SLLN boundary first marginal ORIGINAL weighted value1 = 3/2

SLLN boundary first marginal ORIGINAL mean sum1 := sum(1/4, 3/2)

SLLN boundary first marginal ORIGINAL mean sum1 = 7/4

SLLN boundary first marginal exact unbounded value square1 := mul(2, 2)

SLLN boundary first marginal exact unbounded value square1 = 4

SLLN boundary first marginal ORIGINAL weighted square1 := mul(4, 3/4)

SLLN boundary first marginal ORIGINAL weighted square1 = 3

SLLN boundary first marginal ORIGINAL second moment sum1 := sum(1/4, 3)

SLLN boundary first marginal ORIGINAL second moment sum1 = 13/4

SLLN boundary first marginal ORIGINAL mean squared := mul(7/4, 7/4)

SLLN boundary first marginal ORIGINAL mean squared = 49/16

SLLN boundary first marginal ORIGINAL second minus mean squared := sub(13/4, 49/16)

SLLN boundary first marginal ORIGINAL second minus mean squared = 3/16

source roles = [SLLN boundary first marginal, [1, 2], [[1, 4], [3, 4]]]

derived = [[[1, 1/4], [2, 3/4]], 1, 7/4, 13/4, 3/16]

SLLN boundary second marginal ORIGINAL mass sum0 := sum(0, 1/7)

SLLN boundary second marginal ORIGINAL mass sum0 = 1/7

SLLN boundary second marginal ORIGINAL mass sum1 := sum(1/7, 6/7)

SLLN boundary second marginal ORIGINAL mass sum1 = 1

SLLN boundary second marginal ORIGINAL weighted value0 := mul(1, 1/7)

SLLN boundary second marginal ORIGINAL weighted value0 = 1/7

SLLN boundary second marginal ORIGINAL mean sum0 := sum(0, 1/7)

SLLN boundary second marginal ORIGINAL mean sum0 = 1/7

SLLN boundary second marginal exact unbounded value square0 := mul(1, 1)

SLLN boundary second marginal exact unbounded value square0 = 1

SLLN boundary second marginal ORIGINAL weighted square0 := mul(1, 1/7)

SLLN boundary second marginal ORIGINAL weighted square0 = 1/7

SLLN boundary second marginal ORIGINAL second moment sum0 := sum(0, 1/7)

SLLN boundary second marginal ORIGINAL second moment sum0 = 1/7

SLLN boundary second marginal ORIGINAL weighted value1 := mul(2, 6/7)

SLLN boundary second marginal ORIGINAL weighted value1 = 12/7

SLLN boundary second marginal ORIGINAL mean sum1 := sum(1/7, 12/7)

SLLN boundary second marginal ORIGINAL mean sum1 = 13/7

SLLN boundary second marginal exact unbounded value square1 := mul(2, 2)

SLLN boundary second marginal exact unbounded value square1 = 4

SLLN boundary second marginal ORIGINAL weighted square1 := mul(4, 6/7)

SLLN boundary second marginal ORIGINAL weighted square1 = 24/7

SLLN boundary second marginal ORIGINAL second moment sum1 := sum(1/7, 24/7)

SLLN boundary second marginal ORIGINAL second moment sum1 = 25/7

SLLN boundary second marginal ORIGINAL mean squared := mul(13/7, 13/7)

SLLN boundary second marginal ORIGINAL mean squared = 169/49

SLLN boundary second marginal ORIGINAL second minus mean squared := sub(25/7, 169/49)

SLLN boundary second marginal ORIGINAL second minus mean squared = 6/49

source roles = [SLLN boundary second marginal, [1, 2], [[1, 7], [6, 7]]]

derived = [[[1, 1/7], [2, 6/7]], 1, 13/7, 25/7, 6/49]

source roles = [[[1, 1/4], [2, 3/4]], [[1, 1/7], [2, 6/7]]]

derived = [否, 否]

逐步反向核對

• S032 · 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。
• S031 · 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。
• S030 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL second minus mean squared，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL mean squared，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL second moment sum1，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL weighted square1，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal exact unbounded value square1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL mean sum1，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL weighted value1，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL second moment sum0，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL weighted square0，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal exact unbounded value square0，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL mean sum0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL weighted value0，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL mass sum1，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 SLLN boundary second marginal ORIGINAL mass sum0，不以小數近似替代。
• 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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 first marginal 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

(non_iid, 否, 否, finite_valued_IID_SLLN_not_applicable_not_general_failure)

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。