第 33777 題：Martingale深度

原題與已知條件

同一有限機率空間與filtration的精確模型。操作=martdeep-fair-partial-sum；資料={"steps":2,"scale":1}。請依 [steps,scale,support,multiplicities,total_paths,martingale,certificate] 欄序回答。

• G001 — 同一有限機率空間與filtration的精確模型。操作=martdeep-fair-partial-sum；資料={"steps":2,"scale":1}。請依 [steps,scale,support,multiplicities,total_paths,martingale,certificate] 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

source roles = [qf-mod31-martdeep-fair-partial-sum, [domain, operation, scale, steps]]

derived = [martdeep-fair-partial-sum, Exact_finite_martingale_construction]

Fair ORIGINAL two positive signs k=0 := mul(2, 0)

Fair ORIGINAL two positive signs k=0 = 0

Fair ORIGINAL positive minus negative count k=0 := sub(0, 2)

Fair ORIGINAL positive minus negative count k=0 = -2

Fair ORIGINAL scale times signed count k=0 := mul(1, -2)

Fair ORIGINAL scale times signed count k=0 = -2

source roles = [2, 1, 0]

derived = [-2, 1]

Fair ORIGINAL binomial previous times remaining k=1 := mul(1, 2)

Fair ORIGINAL binomial previous times remaining k=1 = 2

Fair ORIGINAL binomial recurrence division k=1 := div(2, 1)

Fair ORIGINAL binomial recurrence division k=1 = 2

Fair ORIGINAL two positive signs k=1 := mul(2, 1)

Fair ORIGINAL two positive signs k=1 = 2

Fair ORIGINAL positive minus negative count k=1 := sub(2, 2)

Fair ORIGINAL positive minus negative count k=1 = 0

Fair ORIGINAL scale times signed count k=1 := mul(1, 0)

Fair ORIGINAL scale times signed count k=1 = 0

source roles = [2, 1, 1]

derived = [0, 2]

Fair ORIGINAL binomial previous times remaining k=2 := mul(2, 1)

Fair ORIGINAL binomial previous times remaining k=2 = 2

Fair ORIGINAL binomial recurrence division k=2 := div(2, 2)

Fair ORIGINAL binomial recurrence division k=2 = 1

Fair ORIGINAL two positive signs k=2 := mul(2, 2)

Fair ORIGINAL two positive signs k=2 = 4

Fair ORIGINAL positive minus negative count k=2 := sub(4, 2)

Fair ORIGINAL positive minus negative count k=2 = 2

Fair ORIGINAL scale times signed count k=2 := mul(1, 2)

Fair ORIGINAL scale times signed count k=2 = 2

source roles = [2, 1, 2]

derived = [2, 1]

Fair ORIGINAL all binomial multiplicities := sum(1, 2, 1)

Fair ORIGINAL all binomial multiplicities = 4

source roles = [2, 1, 4]

derived = [[time=0, prefixes=1, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=1, prefixes=2, next_signs=+s,-s equiprobable, conditional_increment_mean=0]]

逐步反向核對

• S019 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S018 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL all binomial multiplicities，不以小數近似替代。
• S017 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S016 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=2，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=2，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=2，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=2，不以小數近似替代。
• S011 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S010 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=1，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=1，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=1，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=1，不以小數近似替代。
• S005 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S004 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=0，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=0，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=0，不以小數近似替代。
• S001 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。

結論

(2, 1, [-2, 0, 2], [1, 2, 1], 2^2, 是, independent_fair_signed_increments_have_conditional_mean_zero)

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。