第 33784 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":9,"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":9,"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, 9)

Fair ORIGINAL positive minus negative count k=0 = -9

Fair ORIGINAL scale times signed count k=0 := mul(1, -9)

Fair ORIGINAL scale times signed count k=0 = -9

source roles = [9, 1, 0]

derived = [-9, 1]

Fair ORIGINAL binomial previous times remaining k=1 := mul(1, 9)

Fair ORIGINAL binomial previous times remaining k=1 = 9

Fair ORIGINAL binomial recurrence division k=1 := div(9, 1)

Fair ORIGINAL binomial recurrence division k=1 = 9

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, 9)

Fair ORIGINAL positive minus negative count k=1 = -7

Fair ORIGINAL scale times signed count k=1 := mul(1, -7)

Fair ORIGINAL scale times signed count k=1 = -7

source roles = [9, 1, 1]

derived = [-7, 9]

Fair ORIGINAL binomial previous times remaining k=2 := mul(9, 8)

Fair ORIGINAL binomial previous times remaining k=2 = 72

Fair ORIGINAL binomial recurrence division k=2 := div(72, 2)

Fair ORIGINAL binomial recurrence division k=2 = 36

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, 9)

Fair ORIGINAL positive minus negative count k=2 = -5

Fair ORIGINAL scale times signed count k=2 := mul(1, -5)

Fair ORIGINAL scale times signed count k=2 = -5

source roles = [9, 1, 2]

derived = [-5, 36]

Fair ORIGINAL binomial previous times remaining k=3 := mul(36, 7)

Fair ORIGINAL binomial previous times remaining k=3 = 252

Fair ORIGINAL binomial recurrence division k=3 := div(252, 3)

Fair ORIGINAL binomial recurrence division k=3 = 84

Fair ORIGINAL two positive signs k=3 := mul(2, 3)

Fair ORIGINAL two positive signs k=3 = 6

Fair ORIGINAL positive minus negative count k=3 := sub(6, 9)

Fair ORIGINAL positive minus negative count k=3 = -3

Fair ORIGINAL scale times signed count k=3 := mul(1, -3)

Fair ORIGINAL scale times signed count k=3 = -3

source roles = [9, 1, 3]

derived = [-3, 84]

Fair ORIGINAL binomial previous times remaining k=4 := mul(84, 6)

Fair ORIGINAL binomial previous times remaining k=4 = 504

Fair ORIGINAL binomial recurrence division k=4 := div(504, 4)

Fair ORIGINAL binomial recurrence division k=4 = 126

Fair ORIGINAL two positive signs k=4 := mul(2, 4)

Fair ORIGINAL two positive signs k=4 = 8

Fair ORIGINAL positive minus negative count k=4 := sub(8, 9)

Fair ORIGINAL positive minus negative count k=4 = -1

Fair ORIGINAL scale times signed count k=4 := mul(1, -1)

Fair ORIGINAL scale times signed count k=4 = -1

source roles = [9, 1, 4]

derived = [-1, 126]

Fair ORIGINAL binomial previous times remaining k=5 := mul(126, 5)

Fair ORIGINAL binomial previous times remaining k=5 = 630

Fair ORIGINAL binomial recurrence division k=5 := div(630, 5)

Fair ORIGINAL binomial recurrence division k=5 = 126

Fair ORIGINAL two positive signs k=5 := mul(2, 5)

Fair ORIGINAL two positive signs k=5 = 10

Fair ORIGINAL positive minus negative count k=5 := sub(10, 9)

Fair ORIGINAL positive minus negative count k=5 = 1

Fair ORIGINAL scale times signed count k=5 := mul(1, 1)

Fair ORIGINAL scale times signed count k=5 = 1

source roles = [9, 1, 5]

derived = [1, 126]

Fair ORIGINAL binomial previous times remaining k=6 := mul(126, 4)

Fair ORIGINAL binomial previous times remaining k=6 = 504

Fair ORIGINAL binomial recurrence division k=6 := div(504, 6)

Fair ORIGINAL binomial recurrence division k=6 = 84

Fair ORIGINAL two positive signs k=6 := mul(2, 6)

Fair ORIGINAL two positive signs k=6 = 12

Fair ORIGINAL positive minus negative count k=6 := sub(12, 9)

Fair ORIGINAL positive minus negative count k=6 = 3

Fair ORIGINAL scale times signed count k=6 := mul(1, 3)

Fair ORIGINAL scale times signed count k=6 = 3

source roles = [9, 1, 6]

derived = [3, 84]

Fair ORIGINAL binomial previous times remaining k=7 := mul(84, 3)

Fair ORIGINAL binomial previous times remaining k=7 = 252

Fair ORIGINAL binomial recurrence division k=7 := div(252, 7)

Fair ORIGINAL binomial recurrence division k=7 = 36

Fair ORIGINAL two positive signs k=7 := mul(2, 7)

Fair ORIGINAL two positive signs k=7 = 14

Fair ORIGINAL positive minus negative count k=7 := sub(14, 9)

Fair ORIGINAL positive minus negative count k=7 = 5

Fair ORIGINAL scale times signed count k=7 := mul(1, 5)

Fair ORIGINAL scale times signed count k=7 = 5

source roles = [9, 1, 7]

derived = [5, 36]

Fair ORIGINAL binomial previous times remaining k=8 := mul(36, 2)

Fair ORIGINAL binomial previous times remaining k=8 = 72

Fair ORIGINAL binomial recurrence division k=8 := div(72, 8)

Fair ORIGINAL binomial recurrence division k=8 = 9

Fair ORIGINAL two positive signs k=8 := mul(2, 8)

Fair ORIGINAL two positive signs k=8 = 16

Fair ORIGINAL positive minus negative count k=8 := sub(16, 9)

Fair ORIGINAL positive minus negative count k=8 = 7

Fair ORIGINAL scale times signed count k=8 := mul(1, 7)

Fair ORIGINAL scale times signed count k=8 = 7

source roles = [9, 1, 8]

derived = [7, 9]

Fair ORIGINAL binomial previous times remaining k=9 := mul(9, 1)

Fair ORIGINAL binomial previous times remaining k=9 = 9

Fair ORIGINAL binomial recurrence division k=9 := div(9, 9)

Fair ORIGINAL binomial recurrence division k=9 = 1

Fair ORIGINAL two positive signs k=9 := mul(2, 9)

Fair ORIGINAL two positive signs k=9 = 18

Fair ORIGINAL positive minus negative count k=9 := sub(18, 9)

Fair ORIGINAL positive minus negative count k=9 = 9

Fair ORIGINAL scale times signed count k=9 := mul(1, 9)

Fair ORIGINAL scale times signed count k=9 = 9

source roles = [9, 1, 9]

derived = [9, 1]

Fair ORIGINAL all binomial multiplicities := sum(1, 9, 36, 84, 126, 126, 84, 36, 9, 1)

Fair ORIGINAL all binomial multiplicities = 512

source roles = [9, 1, 512]

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], [time=2, prefixes=4, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=3, prefixes=8, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=4, prefixes=16, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=5, prefixes=32, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=6, prefixes=64, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=7, prefixes=128, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=8, prefixes=256, next_signs=+s,-s equiprobable, conditional_increment_mean=0]]

逐步反向核對

• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=9，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=9，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=9，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=9，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=9，不以小數近似替代。
• S053 · 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。
• S052 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=8，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=8，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=8，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=8，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=8，不以小數近似替代。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=7，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=7，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=7，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=7，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=7，不以小數近似替代。
• S041 · 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。
• S040 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=6，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=6，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=6，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=6，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=6，不以小數近似替代。
• S035 · 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。
• S034 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=5，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=5，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=5，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=5，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=5，不以小數近似替代。
• S029 · 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。
• S028 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=4，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=4，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=4，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=4，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=4，不以小數近似替代。
• S023 · 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。
• S022 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=3，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=3，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=3，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=3，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=3，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

(9, 1, [-9, -7, -5, -3, -1, 1, 3, 5, 7, 9], [1, 9, 36, 84, 126, 126, 84, 36, 9, 1], 2^9, 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。