第 33798 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":11,"scale":2}。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":11,"scale":2}。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, 11)

Fair ORIGINAL positive minus negative count k=0 = -11

Fair ORIGINAL scale times signed count k=0 := mul(2, -11)

Fair ORIGINAL scale times signed count k=0 = -22

source roles = [11, 2, 0]

derived = [-22, 1]

Fair ORIGINAL binomial previous times remaining k=1 := mul(1, 11)

Fair ORIGINAL binomial previous times remaining k=1 = 11

Fair ORIGINAL binomial recurrence division k=1 := div(11, 1)

Fair ORIGINAL binomial recurrence division k=1 = 11

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, 11)

Fair ORIGINAL positive minus negative count k=1 = -9

Fair ORIGINAL scale times signed count k=1 := mul(2, -9)

Fair ORIGINAL scale times signed count k=1 = -18

source roles = [11, 2, 1]

derived = [-18, 11]

Fair ORIGINAL binomial previous times remaining k=2 := mul(11, 10)

Fair ORIGINAL binomial previous times remaining k=2 = 110

Fair ORIGINAL binomial recurrence division k=2 := div(110, 2)

Fair ORIGINAL binomial recurrence division k=2 = 55

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, 11)

Fair ORIGINAL positive minus negative count k=2 = -7

Fair ORIGINAL scale times signed count k=2 := mul(2, -7)

Fair ORIGINAL scale times signed count k=2 = -14

source roles = [11, 2, 2]

derived = [-14, 55]

Fair ORIGINAL binomial previous times remaining k=3 := mul(55, 9)

Fair ORIGINAL binomial previous times remaining k=3 = 495

Fair ORIGINAL binomial recurrence division k=3 := div(495, 3)

Fair ORIGINAL binomial recurrence division k=3 = 165

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, 11)

Fair ORIGINAL positive minus negative count k=3 = -5

Fair ORIGINAL scale times signed count k=3 := mul(2, -5)

Fair ORIGINAL scale times signed count k=3 = -10

source roles = [11, 2, 3]

derived = [-10, 165]

Fair ORIGINAL binomial previous times remaining k=4 := mul(165, 8)

Fair ORIGINAL binomial previous times remaining k=4 = 1320

Fair ORIGINAL binomial recurrence division k=4 := div(1320, 4)

Fair ORIGINAL binomial recurrence division k=4 = 330

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, 11)

Fair ORIGINAL positive minus negative count k=4 = -3

Fair ORIGINAL scale times signed count k=4 := mul(2, -3)

Fair ORIGINAL scale times signed count k=4 = -6

source roles = [11, 2, 4]

derived = [-6, 330]

Fair ORIGINAL binomial previous times remaining k=5 := mul(330, 7)

Fair ORIGINAL binomial previous times remaining k=5 = 2310

Fair ORIGINAL binomial recurrence division k=5 := div(2310, 5)

Fair ORIGINAL binomial recurrence division k=5 = 462

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, 11)

Fair ORIGINAL positive minus negative count k=5 = -1

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

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

source roles = [11, 2, 5]

derived = [-2, 462]

Fair ORIGINAL binomial previous times remaining k=6 := mul(462, 6)

Fair ORIGINAL binomial previous times remaining k=6 = 2772

Fair ORIGINAL binomial recurrence division k=6 := div(2772, 6)

Fair ORIGINAL binomial recurrence division k=6 = 462

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, 11)

Fair ORIGINAL positive minus negative count k=6 = 1

Fair ORIGINAL scale times signed count k=6 := mul(2, 1)

Fair ORIGINAL scale times signed count k=6 = 2

source roles = [11, 2, 6]

derived = [2, 462]

Fair ORIGINAL binomial previous times remaining k=7 := mul(462, 5)

Fair ORIGINAL binomial previous times remaining k=7 = 2310

Fair ORIGINAL binomial recurrence division k=7 := div(2310, 7)

Fair ORIGINAL binomial recurrence division k=7 = 330

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, 11)

Fair ORIGINAL positive minus negative count k=7 = 3

Fair ORIGINAL scale times signed count k=7 := mul(2, 3)

Fair ORIGINAL scale times signed count k=7 = 6

source roles = [11, 2, 7]

derived = [6, 330]

Fair ORIGINAL binomial previous times remaining k=8 := mul(330, 4)

Fair ORIGINAL binomial previous times remaining k=8 = 1320

Fair ORIGINAL binomial recurrence division k=8 := div(1320, 8)

Fair ORIGINAL binomial recurrence division k=8 = 165

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, 11)

Fair ORIGINAL positive minus negative count k=8 = 5

Fair ORIGINAL scale times signed count k=8 := mul(2, 5)

Fair ORIGINAL scale times signed count k=8 = 10

source roles = [11, 2, 8]

derived = [10, 165]

Fair ORIGINAL binomial previous times remaining k=9 := mul(165, 3)

Fair ORIGINAL binomial previous times remaining k=9 = 495

Fair ORIGINAL binomial recurrence division k=9 := div(495, 9)

Fair ORIGINAL binomial recurrence division k=9 = 55

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, 11)

Fair ORIGINAL positive minus negative count k=9 = 7

Fair ORIGINAL scale times signed count k=9 := mul(2, 7)

Fair ORIGINAL scale times signed count k=9 = 14

source roles = [11, 2, 9]

derived = [14, 55]

Fair ORIGINAL binomial previous times remaining k=10 := mul(55, 2)

Fair ORIGINAL binomial previous times remaining k=10 = 110

Fair ORIGINAL binomial recurrence division k=10 := div(110, 10)

Fair ORIGINAL binomial recurrence division k=10 = 11

Fair ORIGINAL two positive signs k=10 := mul(2, 10)

Fair ORIGINAL two positive signs k=10 = 20

Fair ORIGINAL positive minus negative count k=10 := sub(20, 11)

Fair ORIGINAL positive minus negative count k=10 = 9

Fair ORIGINAL scale times signed count k=10 := mul(2, 9)

Fair ORIGINAL scale times signed count k=10 = 18

source roles = [11, 2, 10]

derived = [18, 11]

Fair ORIGINAL binomial previous times remaining k=11 := mul(11, 1)

Fair ORIGINAL binomial previous times remaining k=11 = 11

Fair ORIGINAL binomial recurrence division k=11 := div(11, 11)

Fair ORIGINAL binomial recurrence division k=11 = 1

Fair ORIGINAL two positive signs k=11 := mul(2, 11)

Fair ORIGINAL two positive signs k=11 = 22

Fair ORIGINAL positive minus negative count k=11 := sub(22, 11)

Fair ORIGINAL positive minus negative count k=11 = 11

Fair ORIGINAL scale times signed count k=11 := mul(2, 11)

Fair ORIGINAL scale times signed count k=11 = 22

source roles = [11, 2, 11]

derived = [22, 1]

Fair ORIGINAL all binomial multiplicities := sum(1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1)

Fair ORIGINAL all binomial multiplicities = 2048

source roles = [11, 2, 2048]

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], [time=9, prefixes=512, next_signs=+s,-s equiprobable, conditional_increment_mean=0], [time=10, prefixes=1024, next_signs=+s,-s equiprobable, conditional_increment_mean=0]]

逐步反向核對

• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=11，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=11，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=11，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=11，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=11，不以小數近似替代。
• S065 · 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。
• S064 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL scale times signed count k=10，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL positive minus negative count k=10，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL two positive signs k=10，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial recurrence division k=10，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Fair ORIGINAL binomial previous times remaining k=10，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

(11, 2, [-22, -18, -14, -10, -6, -2, 2, 6, 10, 14, 18, 22], [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1], 2^11, 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。