第 15363 題：有限Martingale

原題與已知條件

同一有限probability space filtration與process。操作=martingale-expectation-invariant；資料={"masses":[[1,4],[1,4],[1,4],[1,4]],"partitions":[[0,0,0,0],[0,0,1,1]],"process_values":[[0,0,0,0],[0,0,0,0]]}。請依 [expectations,invariant,boundary] 欄序回答。

• G001 — 同一有限probability space filtration與process。操作=martingale-expectation-invariant；資料={"masses":[[1,4],[1,4],[1,4],[1,4]],"partitions":[[0,0,0,0],[0,0,1,1]],"process_values":[[0,0,0,0],[0,0,0,0]]}。請依 [expectations,invariant,boundary] 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-martingale

逐步推導

martingale full probability mass := sum(1/4, 1/4, 1/4, 1/4)

martingale full probability mass = 1

same finite probability carrier size=4;exact masses=[1/4, 1/4, 1/4, 1/4];partitions in time order=[[0, 0, 0, 0], [0, 0, 1, 1]]

total mass=1;all later-equal pairs with earlier-different labels=[]

time=0;block label=0;indices=[0, 1, 2, 3];current values in original order=[0, 0, 0, 0]

distinct current values first-encounter order=[0];constant on block=是

time=1;block label=0;indices=[0, 1];current values in original order=[0, 0]

distinct current values first-encounter order=[0];constant on block=是

time=1;block label=1;indices=[2, 3];current values in original order=[0, 0]

distinct current values first-encounter order=[0];constant on block=是

martingale time0 block0 mass := sum(1/4, 1/4, 1/4, 1/4)

martingale time0 block0 mass = 1

martingale time0 block0 next weighted outcome0 := mul(1/4, 0)

martingale time0 block0 next weighted outcome0 = 0

martingale time0 block0 next weighted outcome1 := mul(1/4, 0)

martingale time0 block0 next weighted outcome1 = 0

martingale time0 block0 next weighted outcome2 := mul(1/4, 0)

martingale time0 block0 next weighted outcome2 = 0

martingale time0 block0 next weighted outcome3 := mul(1/4, 0)

martingale time0 block0 next weighted outcome3 = 0

martingale time0 block0 next weighted sum := sum(0, 0, 0, 0)

martingale time0 block0 next weighted sum = 0

martingale time0 block0 next conditional mean := div(0, 1)

martingale time0 block0 next conditional mean = 0

time=0;block label=0;current distinct values=[0];next weighted sum=0;positive block mass=1

conditional mean=0;conditional step row=[0, 0, 0, [0], 是]

martingale expectation time0 weighted outcome0 := mul(1/4, 0)

martingale expectation time0 weighted outcome0 = 0

martingale expectation time0 weighted outcome1 := mul(1/4, 0)

martingale expectation time0 weighted outcome1 = 0

martingale expectation time0 weighted outcome2 := mul(1/4, 0)

martingale expectation time0 weighted outcome2 = 0

martingale expectation time0 weighted outcome3 := mul(1/4, 0)

martingale expectation time0 weighted outcome3 = 0

martingale expectation time0 full sum := sum(0, 0, 0, 0)

martingale expectation time0 full sum = 0

martingale expectation time1 weighted outcome0 := mul(1/4, 0)

martingale expectation time1 weighted outcome0 = 0

martingale expectation time1 weighted outcome1 := mul(1/4, 0)

martingale expectation time1 weighted outcome1 = 0

martingale expectation time1 weighted outcome2 := mul(1/4, 0)

martingale expectation time1 weighted outcome2 = 0

martingale expectation time1 weighted outcome3 := mul(1/4, 0)

martingale expectation time1 weighted outcome3 = 0

martingale expectation time1 full sum := sum(0, 0, 0, 0)

martingale expectation time1 full sum = 0

unconditional exact expectations in original time order=[0, 0];full conditional rows=[[0, 0, 0, [0], 是]]

expectation invariant=是;expectation_invariance_is_consequence_not_definition

逐步反向核對

• S024 · 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。
• S023 · premise_and_result_audit — 以精確有理數重算 martingale expectation time1 full sum，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 martingale expectation time1 weighted outcome3，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 martingale expectation time1 weighted outcome2，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 martingale expectation time1 weighted outcome1，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 martingale expectation time1 weighted outcome0，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 martingale expectation time0 full sum，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 martingale expectation time0 weighted outcome3，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 martingale expectation time0 weighted outcome2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 martingale expectation time0 weighted outcome1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 martingale expectation time0 weighted outcome0，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next conditional mean，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted sum，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted outcome3，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted outcome2，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted outcome1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted outcome0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 mass，不以小數近似替代。
• S005 · 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。
• S004 · 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。
• S003 · 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。
• S002 · 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。
• S001 · premise_and_result_audit — 以精確有理數重算 martingale full probability mass，不以小數近似替代。

結論

([0, 0], 是, expectation_invariance_is_consequence_not_definition)

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。