第 33317 題：有限Markov密度

原題與已知條件

精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"path":[0,1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 欄序回答。

• G001 — 精確有限時間齊次row-stochastic kernel。操作=markovdeep-conditional-path-probability；資料={"transition":[[[1,5],[4,5]],[[1,2],[1,2]]],"path":[0,1,0,1]}。請依 [path,transition_factors,conditional_probability,scope] 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-finite-markov-chain

逐步推導

source roles = [qf-mod25-markovdeep-conditional-path-probability, markovdeep-conditional-path-probability, Finite_exact_Markov_density, []]

derived = [[path, factors, conditional_product, scope]]

source roles = [Markov RAW P row0 column0, RAW-pair, [1, 5], [], 1, 5]

derived = [1, 5, 1/5]

Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator := div(1, 5)

Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator = 1/5

source roles = [Markov RAW P row0 column1, RAW-pair, [4, 5], [], 4, 5]

derived = [4, 5, 4/5]

Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator := div(4, 5)

Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator = 4/5

source roles = [Markov RAW row0 normalization through column0, radd, [[0, 1], [1, 5]], [1], 1, 5]

derived = [1, 5, 1/5]

Markov RAW row0 normalization through column0 ORIGINAL exact rational sum := sum(0, 1/5)

Markov RAW row0 normalization through column0 ORIGINAL exact rational sum = 1/5

source roles = [Markov RAW row0 normalization through column1, radd, [[1, 5], [4, 5]], [5], 5, 5]

derived = [1, 1, 1]

Markov RAW row0 normalization through column1 ORIGINAL exact rational sum := sum(1/5, 4/5)

Markov RAW row0 normalization through column1 ORIGINAL exact rational sum = 1

source roles = [Markov RAW P row1 column0, RAW-pair, [1, 2], [], 1, 2]

derived = [1, 2, 1/2]

Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator := div(1, 2)

Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator = 1/2

source roles = [Markov RAW P row1 column1, RAW-pair, [1, 2], [], 1, 2]

derived = [1, 2, 1/2]

Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator := div(1, 2)

Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator = 1/2

source roles = [Markov RAW row1 normalization through column0, radd, [[0, 1], [1, 2]], [1], 1, 2]

derived = [1, 2, 1/2]

Markov RAW row1 normalization through column0 ORIGINAL exact rational sum := sum(0, 1/2)

Markov RAW row1 normalization through column0 ORIGINAL exact rational sum = 1/2

source roles = [Markov RAW row1 normalization through column1, radd, [[1, 2], [1, 2]], [2], 2, 2]

derived = [1, 1, 1]

Markov RAW row1 normalization through column1 ORIGINAL exact rational sum := sum(1/2, 1/2)

Markov RAW row1 normalization through column1 ORIGINAL exact rational sum = 1

source roles = [[[[1, 5], [4, 5]], [[1, 2], [1, 2]]]]

derived = [2, [[1/5, 4/5], [1/2, 1/2]]]

source roles = [2, 3]

derived = [[[1, 0], [0, 1]]]

source roles = [Markov power1 entry0,0 middle0 product, rmul, [[1, 1], [1, 5]], [1, 1], 1, 5]

derived = [1, 5, 1/5]

Markov power1 entry0,0 middle0 product ORIGINAL exact rational product := mul(1, 1/5)

Markov power1 entry0,0 middle0 product ORIGINAL exact rational product = 1/5

source roles = [Markov power1 entry0,0 through middle0, radd, [[0, 1], [1, 5]], [1], 1, 5]

derived = [1, 5, 1/5]

Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 1/5)

Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum = 1/5

source roles = [Markov power1 entry0,0 middle1 product, rmul, [[0, 1], [1, 2]], [2, 1], 0, 1]

derived = [0, 1, 0]

Markov power1 entry0,0 middle1 product ORIGINAL exact rational product := mul(0, 1/2)

Markov power1 entry0,0 middle1 product ORIGINAL exact rational product = 0

source roles = [Markov power1 entry0,0 through middle1, radd, [[1, 5], [0, 1]], [1], 1, 5]

derived = [1, 5, 1/5]

Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum := sum(1/5, 0)

Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum = 1/5

source roles = [Markov power1 entry0,1 middle0 product, rmul, [[1, 1], [4, 5]], [1, 1], 4, 5]

derived = [4, 5, 4/5]

Markov power1 entry0,1 middle0 product ORIGINAL exact rational product := mul(1, 4/5)

Markov power1 entry0,1 middle0 product ORIGINAL exact rational product = 4/5

source roles = [Markov power1 entry0,1 through middle0, radd, [[0, 1], [4, 5]], [1], 4, 5]

derived = [4, 5, 4/5]

Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 4/5)

Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum = 4/5

source roles = [Markov power1 entry0,1 middle1 product, rmul, [[0, 1], [1, 2]], [2, 1], 0, 1]

derived = [0, 1, 0]

Markov power1 entry0,1 middle1 product ORIGINAL exact rational product := mul(0, 1/2)

Markov power1 entry0,1 middle1 product ORIGINAL exact rational product = 0

source roles = [Markov power1 entry0,1 through middle1, radd, [[4, 5], [0, 1]], [1], 4, 5]

derived = [4, 5, 4/5]

Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum := sum(4/5, 0)

Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum = 4/5

source roles = [Markov power1 entry1,0 middle0 product, rmul, [[0, 1], [1, 5]], [5, 1], 0, 1]

derived = [0, 1, 0]

Markov power1 entry1,0 middle0 product ORIGINAL exact rational product := mul(0, 1/5)

Markov power1 entry1,0 middle0 product ORIGINAL exact rational product = 0

source roles = [Markov power1 entry1,0 through middle0, radd, [[0, 1], [0, 1]], [1], 0, 1]

derived = [0, 1, 0]

Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 0)

Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum = 0

source roles = [Markov power1 entry1,0 middle1 product, rmul, [[1, 1], [1, 2]], [1, 1], 1, 2]

derived = [1, 2, 1/2]

Markov power1 entry1,0 middle1 product ORIGINAL exact rational product := mul(1, 1/2)

Markov power1 entry1,0 middle1 product ORIGINAL exact rational product = 1/2

source roles = [Markov power1 entry1,0 through middle1, radd, [[0, 1], [1, 2]], [1], 1, 2]

derived = [1, 2, 1/2]

Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum := sum(0, 1/2)

Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum = 1/2

source roles = [Markov power1 entry1,1 middle0 product, rmul, [[0, 1], [4, 5]], [5, 1], 0, 1]

derived = [0, 1, 0]

Markov power1 entry1,1 middle0 product ORIGINAL exact rational product := mul(0, 4/5)

Markov power1 entry1,1 middle0 product ORIGINAL exact rational product = 0

source roles = [Markov power1 entry1,1 through middle0, radd, [[0, 1], [0, 1]], [1], 0, 1]

derived = [0, 1, 0]

Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 0)

Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum = 0

source roles = [Markov power1 entry1,1 middle1 product, rmul, [[1, 1], [1, 2]], [1, 1], 1, 2]

derived = [1, 2, 1/2]

Markov power1 entry1,1 middle1 product ORIGINAL exact rational product := mul(1, 1/2)

Markov power1 entry1,1 middle1 product ORIGINAL exact rational product = 1/2

source roles = [Markov power1 entry1,1 through middle1, radd, [[0, 1], [1, 2]], [1], 1, 2]

derived = [1, 2, 1/2]

Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum := sum(0, 1/2)

Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum = 1/2

source roles = [1, [[1, 0], [0, 1]], [[1/5, 4/5], [1/2, 1/2]]]

derived = [[[1/5, 4/5], [1/2, 1/2]]]

source roles = [Markov power2 entry0,0 middle0 product, rmul, [[1, 5], [1, 5]], [1, 1], 1, 25]

derived = [1, 25, 1/25]

Markov power2 entry0,0 middle0 product ORIGINAL exact rational product := mul(1/5, 1/5)

Markov power2 entry0,0 middle0 product ORIGINAL exact rational product = 1/25

source roles = [Markov power2 entry0,0 through middle0, radd, [[0, 1], [1, 25]], [1], 1, 25]

derived = [1, 25, 1/25]

Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 1/25)

Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum = 1/25

source roles = [Markov power2 entry0,0 middle1 product, rmul, [[4, 5], [1, 2]], [2, 1], 2, 5]

derived = [2, 5, 2/5]

Markov power2 entry0,0 middle1 product ORIGINAL exact rational product := mul(4/5, 1/2)

Markov power2 entry0,0 middle1 product ORIGINAL exact rational product = 2/5

source roles = [Markov power2 entry0,0 through middle1, radd, [[1, 25], [2, 5]], [5], 11, 25]

derived = [11, 25, 11/25]

Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum := sum(1/25, 2/5)

Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum = 11/25

source roles = [Markov power2 entry0,1 middle0 product, rmul, [[1, 5], [4, 5]], [1, 1], 4, 25]

derived = [4, 25, 4/25]

Markov power2 entry0,1 middle0 product ORIGINAL exact rational product := mul(1/5, 4/5)

Markov power2 entry0,1 middle0 product ORIGINAL exact rational product = 4/25

source roles = [Markov power2 entry0,1 through middle0, radd, [[0, 1], [4, 25]], [1], 4, 25]

derived = [4, 25, 4/25]

Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 4/25)

Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum = 4/25

source roles = [Markov power2 entry0,1 middle1 product, rmul, [[4, 5], [1, 2]], [2, 1], 2, 5]

derived = [2, 5, 2/5]

Markov power2 entry0,1 middle1 product ORIGINAL exact rational product := mul(4/5, 1/2)

Markov power2 entry0,1 middle1 product ORIGINAL exact rational product = 2/5

source roles = [Markov power2 entry0,1 through middle1, radd, [[4, 25], [2, 5]], [5], 14, 25]

derived = [14, 25, 14/25]

Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum := sum(4/25, 2/5)

Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum = 14/25

source roles = [Markov power2 entry1,0 middle0 product, rmul, [[1, 2], [1, 5]], [1, 1], 1, 10]

derived = [1, 10, 1/10]

Markov power2 entry1,0 middle0 product ORIGINAL exact rational product := mul(1/2, 1/5)

Markov power2 entry1,0 middle0 product ORIGINAL exact rational product = 1/10

source roles = [Markov power2 entry1,0 through middle0, radd, [[0, 1], [1, 10]], [1], 1, 10]

derived = [1, 10, 1/10]

Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 1/10)

Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum = 1/10

source roles = [Markov power2 entry1,0 middle1 product, rmul, [[1, 2], [1, 2]], [1, 1], 1, 4]

derived = [1, 4, 1/4]

Markov power2 entry1,0 middle1 product ORIGINAL exact rational product := mul(1/2, 1/2)

Markov power2 entry1,0 middle1 product ORIGINAL exact rational product = 1/4

source roles = [Markov power2 entry1,0 through middle1, radd, [[1, 10], [1, 4]], [2], 7, 20]

derived = [7, 20, 7/20]

Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum := sum(1/10, 1/4)

Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum = 7/20

source roles = [Markov power2 entry1,1 middle0 product, rmul, [[1, 2], [4, 5]], [1, 2], 2, 5]

derived = [2, 5, 2/5]

Markov power2 entry1,1 middle0 product ORIGINAL exact rational product := mul(1/2, 4/5)

Markov power2 entry1,1 middle0 product ORIGINAL exact rational product = 2/5

source roles = [Markov power2 entry1,1 through middle0, radd, [[0, 1], [2, 5]], [1], 2, 5]

derived = [2, 5, 2/5]

Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 2/5)

Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum = 2/5

source roles = [Markov power2 entry1,1 middle1 product, rmul, [[1, 2], [1, 2]], [1, 1], 1, 4]

derived = [1, 4, 1/4]

Markov power2 entry1,1 middle1 product ORIGINAL exact rational product := mul(1/2, 1/2)

Markov power2 entry1,1 middle1 product ORIGINAL exact rational product = 1/4

source roles = [Markov power2 entry1,1 through middle1, radd, [[2, 5], [1, 4]], [1], 13, 20]

derived = [13, 20, 13/20]

Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum := sum(2/5, 1/4)

Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum = 13/20

source roles = [2, [[1/5, 4/5], [1/2, 1/2]], [[1/5, 4/5], [1/2, 1/2]]]

derived = [[[11/25, 14/25], [7/20, 13/20]]]

source roles = [Markov power3 entry0,0 middle0 product, rmul, [[11, 25], [1, 5]], [1, 1], 11, 125]

derived = [11, 125, 11/125]

Markov power3 entry0,0 middle0 product ORIGINAL exact rational product := mul(11/25, 1/5)

Markov power3 entry0,0 middle0 product ORIGINAL exact rational product = 11/125

source roles = [Markov power3 entry0,0 through middle0, radd, [[0, 1], [11, 125]], [1], 11, 125]

derived = [11, 125, 11/125]

Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 11/125)

Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum = 11/125

source roles = [Markov power3 entry0,0 middle1 product, rmul, [[14, 25], [1, 2]], [2, 1], 7, 25]

derived = [7, 25, 7/25]

Markov power3 entry0,0 middle1 product ORIGINAL exact rational product := mul(14/25, 1/2)

Markov power3 entry0,0 middle1 product ORIGINAL exact rational product = 7/25

source roles = [Markov power3 entry0,0 through middle1, radd, [[11, 125], [7, 25]], [25], 46, 125]

derived = [46, 125, 46/125]

Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum := sum(11/125, 7/25)

Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum = 46/125

source roles = [Markov power3 entry0,1 middle0 product, rmul, [[11, 25], [4, 5]], [1, 1], 44, 125]

derived = [44, 125, 44/125]

Markov power3 entry0,1 middle0 product ORIGINAL exact rational product := mul(11/25, 4/5)

Markov power3 entry0,1 middle0 product ORIGINAL exact rational product = 44/125

source roles = [Markov power3 entry0,1 through middle0, radd, [[0, 1], [44, 125]], [1], 44, 125]

derived = [44, 125, 44/125]

Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 44/125)

Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum = 44/125

source roles = [Markov power3 entry0,1 middle1 product, rmul, [[14, 25], [1, 2]], [2, 1], 7, 25]

derived = [7, 25, 7/25]

Markov power3 entry0,1 middle1 product ORIGINAL exact rational product := mul(14/25, 1/2)

Markov power3 entry0,1 middle1 product ORIGINAL exact rational product = 7/25

source roles = [Markov power3 entry0,1 through middle1, radd, [[44, 125], [7, 25]], [25], 79, 125]

derived = [79, 125, 79/125]

Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum := sum(44/125, 7/25)

Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum = 79/125

source roles = [Markov power3 entry1,0 middle0 product, rmul, [[7, 20], [1, 5]], [1, 1], 7, 100]

derived = [7, 100, 7/100]

Markov power3 entry1,0 middle0 product ORIGINAL exact rational product := mul(7/20, 1/5)

Markov power3 entry1,0 middle0 product ORIGINAL exact rational product = 7/100

source roles = [Markov power3 entry1,0 through middle0, radd, [[0, 1], [7, 100]], [1], 7, 100]

derived = [7, 100, 7/100]

Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 7/100)

Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum = 7/100

source roles = [Markov power3 entry1,0 middle1 product, rmul, [[13, 20], [1, 2]], [1, 1], 13, 40]

derived = [13, 40, 13/40]

Markov power3 entry1,0 middle1 product ORIGINAL exact rational product := mul(13/20, 1/2)

Markov power3 entry1,0 middle1 product ORIGINAL exact rational product = 13/40

source roles = [Markov power3 entry1,0 through middle1, radd, [[7, 100], [13, 40]], [20], 79, 200]

derived = [79, 200, 79/200]

Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum := sum(7/100, 13/40)

Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum = 79/200

source roles = [Markov power3 entry1,1 middle0 product, rmul, [[7, 20], [4, 5]], [1, 4], 7, 25]

derived = [7, 25, 7/25]

Markov power3 entry1,1 middle0 product ORIGINAL exact rational product := mul(7/20, 4/5)

Markov power3 entry1,1 middle0 product ORIGINAL exact rational product = 7/25

source roles = [Markov power3 entry1,1 through middle0, radd, [[0, 1], [7, 25]], [1], 7, 25]

derived = [7, 25, 7/25]

Markov power3 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 7/25)

Markov power3 entry1,1 through middle0 ORIGINAL exact rational sum = 7/25

source roles = [Markov power3 entry1,1 middle1 product, rmul, [[13, 20], [1, 2]], [1, 1], 13, 40]

derived = [13, 40, 13/40]

Markov power3 entry1,1 middle1 product ORIGINAL exact rational product := mul(13/20, 1/2)

Markov power3 entry1,1 middle1 product ORIGINAL exact rational product = 13/40

source roles = [Markov power3 entry1,1 through middle1, radd, [[7, 25], [13, 40]], [5], 121, 200]

derived = [121, 200, 121/200]

Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum := sum(7/25, 13/40)

Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum = 121/200

source roles = [3, [[11/25, 14/25], [7/20, 13/20]], [[1/5, 4/5], [1/2, 1/2]]]

derived = [[[46/125, 79/125], [79/200, 121/200]]]

source roles = [Markov conditional path ORIGINAL transition1, rmul, [[1, 1], [4, 5]], [1, 1], 4, 5]

derived = [4, 5, 4/5]

Markov conditional path ORIGINAL transition1 ORIGINAL exact rational product := mul(1, 4/5)

Markov conditional path ORIGINAL transition1 ORIGINAL exact rational product = 4/5

source roles = [Markov conditional path ORIGINAL transition2, rmul, [[4, 5], [1, 2]], [2, 1], 2, 5]

derived = [2, 5, 2/5]

Markov conditional path ORIGINAL transition2 ORIGINAL exact rational product := mul(4/5, 1/2)

Markov conditional path ORIGINAL transition2 ORIGINAL exact rational product = 2/5

source roles = [Markov conditional path ORIGINAL transition3, rmul, [[2, 5], [4, 5]], [1, 1], 8, 25]

derived = [8, 25, 8/25]

Markov conditional path ORIGINAL transition3 ORIGINAL exact rational product := mul(2/5, 4/5)

Markov conditional path ORIGINAL transition3 ORIGINAL exact rational product = 8/25

source roles = [[0, 1, 0, 1]]

derived = [[4/5, 1/2, 4/5], 8/25, conditional_on_initial_state]

逐步反向核對

• S125 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S124 · premise_and_result_audit — 以精確有理數重算 Markov conditional path ORIGINAL transition3 ORIGINAL exact rational product，不以小數近似替代。
• S123 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S122 · premise_and_result_audit — 以精確有理數重算 Markov conditional path ORIGINAL transition2 ORIGINAL exact rational product，不以小數近似替代。
• S121 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S120 · premise_and_result_audit — 以精確有理數重算 Markov conditional path ORIGINAL transition1 ORIGINAL exact rational product，不以小數近似替代。
• S119 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S118 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S117 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S116 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S115 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry1,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S114 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S113 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry1,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S112 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S111 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry1,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S110 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S109 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S108 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S107 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry1,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S106 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S105 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S104 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S103 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry1,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S102 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S101 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S100 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S099 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry0,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S098 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S097 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S096 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S095 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry0,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S094 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S093 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S092 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S091 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry0,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S090 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S089 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S088 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S087 · premise_and_result_audit — 以精確有理數重算 Markov power3 entry0,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S086 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S085 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S084 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S083 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S082 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry1,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S081 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S080 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S079 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S078 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry1,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S077 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S076 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S075 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S074 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry1,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S073 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S072 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S071 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S070 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry1,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S069 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S068 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S067 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S066 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry0,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S065 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S064 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S063 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S062 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry0,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S061 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S060 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S059 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S058 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry0,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S057 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S056 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S055 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S054 · premise_and_result_audit — 以精確有理數重算 Markov power2 entry0,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S053 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S052 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S051 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S050 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S049 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry1,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S048 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S047 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S046 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S045 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry1,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S044 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S043 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S042 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S041 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry1,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S040 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S039 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S038 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S037 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry1,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S036 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S035 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S034 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S033 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry0,1 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S032 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S031 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S030 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S029 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry0,1 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S028 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S027 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum，不以小數近似替代。
• S026 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S025 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry0,0 middle1 product ORIGINAL exact rational product，不以小數近似替代。
• S024 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S023 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum，不以小數近似替代。
• S022 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S021 · premise_and_result_audit — 以精確有理數重算 Markov power1 entry0,0 middle0 product ORIGINAL exact rational product，不以小數近似替代。
• S020 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S019 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S018 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S017 · premise_and_result_audit — 以精確有理數重算 Markov RAW row1 normalization through column1 ORIGINAL exact rational sum，不以小數近似替代。
• S016 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S015 · premise_and_result_audit — 以精確有理數重算 Markov RAW row1 normalization through column0 ORIGINAL exact rational sum，不以小數近似替代。
• S014 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S013 · premise_and_result_audit — 以精確有理數重算 Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。
• S012 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S011 · premise_and_result_audit — 以精確有理數重算 Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。
• S010 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S009 · premise_and_result_audit — 以精確有理數重算 Markov RAW row0 normalization through column1 ORIGINAL exact rational sum，不以小數近似替代。
• S008 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S007 · premise_and_result_audit — 以精確有理數重算 Markov RAW row0 normalization through column0 ORIGINAL exact rational sum，不以小數近似替代。
• S006 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S005 · premise_and_result_audit — 以精確有理數重算 Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。
• S004 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S003 · premise_and_result_audit — 以精確有理數重算 Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。
• S002 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。
• S001 · premise_and_result_audit — 以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。

結論

([0, 1, 0, 1], [4/5, 1/2, 4/5], 8/25, conditional_on_initial_state)

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。