第 33346 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-endpoint-joint-probability；資料={"initial":[[1,2],[1,2]],"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 欄序回答。

• G001 — 精確有限時間齊次row-stochastic kernel。操作=markovdeep-endpoint-joint-probability；資料={"initial":[[1,2],[1,2]],"transition":[[[1,5],[4,5]],[[3,4],[1,4]]],"from":0,"to":1,"steps":3}。請依 [initial,from,to,steps,initial_mass,n_step_entry,joint_probability] 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-endpoint-joint-probability, markovdeep-endpoint-joint-probability, Finite_exact_Markov_density, []]

derived = [[initial, from, to, steps, initial_mass, power_entry, joint]]

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, [3, 4], [], 3, 4]

derived = [3, 4, 3/4]

Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator := div(3, 4)

Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator = 3/4

source roles = [Markov RAW P row1 column1, RAW-pair, [1, 4], [], 1, 4]

derived = [1, 4, 1/4]

Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator := div(1, 4)

Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator = 1/4

source roles = [Markov RAW row1 normalization through column0, radd, [[0, 1], [3, 4]], [1], 3, 4]

derived = [3, 4, 3/4]

Markov RAW row1 normalization through column0 ORIGINAL exact rational sum := sum(0, 3/4)

Markov RAW row1 normalization through column0 ORIGINAL exact rational sum = 3/4

source roles = [Markov RAW row1 normalization through column1, radd, [[3, 4], [1, 4]], [4], 4, 4]

derived = [1, 1, 1]

Markov RAW row1 normalization through column1 ORIGINAL exact rational sum := sum(3/4, 1/4)

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

source roles = [[[[1, 5], [4, 5]], [[3, 4], [1, 4]]]]

derived = [2, [[1/5, 4/5], [3/4, 1/4]]]

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], [3, 4]], [4, 1], 0, 1]

derived = [0, 1, 0]

Markov power1 entry0,0 middle1 product ORIGINAL exact rational product := mul(0, 3/4)

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, 4]], [4, 1], 0, 1]

derived = [0, 1, 0]

Markov power1 entry0,1 middle1 product ORIGINAL exact rational product := mul(0, 1/4)

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], [3, 4]], [1, 1], 3, 4]

derived = [3, 4, 3/4]

Markov power1 entry1,0 middle1 product ORIGINAL exact rational product := mul(1, 3/4)

Markov power1 entry1,0 middle1 product ORIGINAL exact rational product = 3/4

source roles = [Markov power1 entry1,0 through middle1, radd, [[0, 1], [3, 4]], [1], 3, 4]

derived = [3, 4, 3/4]

Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum := sum(0, 3/4)

Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum = 3/4

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, 4]], [1, 1], 1, 4]

derived = [1, 4, 1/4]

Markov power1 entry1,1 middle1 product ORIGINAL exact rational product := mul(1, 1/4)

Markov power1 entry1,1 middle1 product ORIGINAL exact rational product = 1/4

source roles = [Markov power1 entry1,1 through middle1, radd, [[0, 1], [1, 4]], [1], 1, 4]

derived = [1, 4, 1/4]

Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum := sum(0, 1/4)

Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum = 1/4

source roles = [1, [[1, 0], [0, 1]], [[1/5, 4/5], [3/4, 1/4]]]

derived = [[[1/5, 4/5], [3/4, 1/4]]]

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], [3, 4]], [4, 1], 3, 5]

derived = [3, 5, 3/5]

Markov power2 entry0,0 middle1 product ORIGINAL exact rational product := mul(4/5, 3/4)

Markov power2 entry0,0 middle1 product ORIGINAL exact rational product = 3/5

source roles = [Markov power2 entry0,0 through middle1, radd, [[1, 25], [3, 5]], [5], 16, 25]

derived = [16, 25, 16/25]

Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum := sum(1/25, 3/5)

Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum = 16/25

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, 4]], [4, 1], 1, 5]

derived = [1, 5, 1/5]

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

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

source roles = [Markov power2 entry0,1 through middle1, radd, [[4, 25], [1, 5]], [5], 9, 25]

derived = [9, 25, 9/25]

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

Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum = 9/25

source roles = [Markov power2 entry1,0 middle0 product, rmul, [[3, 4], [1, 5]], [1, 1], 3, 20]

derived = [3, 20, 3/20]

Markov power2 entry1,0 middle0 product ORIGINAL exact rational product := mul(3/4, 1/5)

Markov power2 entry1,0 middle0 product ORIGINAL exact rational product = 3/20

source roles = [Markov power2 entry1,0 through middle0, radd, [[0, 1], [3, 20]], [1], 3, 20]

derived = [3, 20, 3/20]

Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 3/20)

Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum = 3/20

source roles = [Markov power2 entry1,0 middle1 product, rmul, [[1, 4], [3, 4]], [1, 1], 3, 16]

derived = [3, 16, 3/16]

Markov power2 entry1,0 middle1 product ORIGINAL exact rational product := mul(1/4, 3/4)

Markov power2 entry1,0 middle1 product ORIGINAL exact rational product = 3/16

source roles = [Markov power2 entry1,0 through middle1, radd, [[3, 20], [3, 16]], [4], 27, 80]

derived = [27, 80, 27/80]

Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum := sum(3/20, 3/16)

Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum = 27/80

source roles = [Markov power2 entry1,1 middle0 product, rmul, [[3, 4], [4, 5]], [1, 4], 3, 5]

derived = [3, 5, 3/5]

Markov power2 entry1,1 middle0 product ORIGINAL exact rational product := mul(3/4, 4/5)

Markov power2 entry1,1 middle0 product ORIGINAL exact rational product = 3/5

source roles = [Markov power2 entry1,1 through middle0, radd, [[0, 1], [3, 5]], [1], 3, 5]

derived = [3, 5, 3/5]

Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 3/5)

Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum = 3/5

source roles = [Markov power2 entry1,1 middle1 product, rmul, [[1, 4], [1, 4]], [1, 1], 1, 16]

derived = [1, 16, 1/16]

Markov power2 entry1,1 middle1 product ORIGINAL exact rational product := mul(1/4, 1/4)

Markov power2 entry1,1 middle1 product ORIGINAL exact rational product = 1/16

source roles = [Markov power2 entry1,1 through middle1, radd, [[3, 5], [1, 16]], [1], 53, 80]

derived = [53, 80, 53/80]

Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum := sum(3/5, 1/16)

Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum = 53/80

source roles = [2, [[1/5, 4/5], [3/4, 1/4]], [[1/5, 4/5], [3/4, 1/4]]]

derived = [[[16/25, 9/25], [27/80, 53/80]]]

source roles = [Markov power3 entry0,0 middle0 product, rmul, [[16, 25], [1, 5]], [1, 1], 16, 125]

derived = [16, 125, 16/125]

Markov power3 entry0,0 middle0 product ORIGINAL exact rational product := mul(16/25, 1/5)

Markov power3 entry0,0 middle0 product ORIGINAL exact rational product = 16/125

source roles = [Markov power3 entry0,0 through middle0, radd, [[0, 1], [16, 125]], [1], 16, 125]

derived = [16, 125, 16/125]

Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 16/125)

Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum = 16/125

source roles = [Markov power3 entry0,0 middle1 product, rmul, [[9, 25], [3, 4]], [1, 1], 27, 100]

derived = [27, 100, 27/100]

Markov power3 entry0,0 middle1 product ORIGINAL exact rational product := mul(9/25, 3/4)

Markov power3 entry0,0 middle1 product ORIGINAL exact rational product = 27/100

source roles = [Markov power3 entry0,0 through middle1, radd, [[16, 125], [27, 100]], [25], 199, 500]

derived = [199, 500, 199/500]

Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum := sum(16/125, 27/100)

Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum = 199/500

source roles = [Markov power3 entry0,1 middle0 product, rmul, [[16, 25], [4, 5]], [1, 1], 64, 125]

derived = [64, 125, 64/125]

Markov power3 entry0,1 middle0 product ORIGINAL exact rational product := mul(16/25, 4/5)

Markov power3 entry0,1 middle0 product ORIGINAL exact rational product = 64/125

source roles = [Markov power3 entry0,1 through middle0, radd, [[0, 1], [64, 125]], [1], 64, 125]

derived = [64, 125, 64/125]

Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 64/125)

Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum = 64/125

source roles = [Markov power3 entry0,1 middle1 product, rmul, [[9, 25], [1, 4]], [1, 1], 9, 100]

derived = [9, 100, 9/100]

Markov power3 entry0,1 middle1 product ORIGINAL exact rational product := mul(9/25, 1/4)

Markov power3 entry0,1 middle1 product ORIGINAL exact rational product = 9/100

source roles = [Markov power3 entry0,1 through middle1, radd, [[64, 125], [9, 100]], [25], 301, 500]

derived = [301, 500, 301/500]

Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum := sum(64/125, 9/100)

Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum = 301/500

source roles = [Markov power3 entry1,0 middle0 product, rmul, [[27, 80], [1, 5]], [1, 1], 27, 400]

derived = [27, 400, 27/400]

Markov power3 entry1,0 middle0 product ORIGINAL exact rational product := mul(27/80, 1/5)

Markov power3 entry1,0 middle0 product ORIGINAL exact rational product = 27/400

source roles = [Markov power3 entry1,0 through middle0, radd, [[0, 1], [27, 400]], [1], 27, 400]

derived = [27, 400, 27/400]

Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 27/400)

Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum = 27/400

source roles = [Markov power3 entry1,0 middle1 product, rmul, [[53, 80], [3, 4]], [1, 1], 159, 320]

derived = [159, 320, 159/320]

Markov power3 entry1,0 middle1 product ORIGINAL exact rational product := mul(53/80, 3/4)

Markov power3 entry1,0 middle1 product ORIGINAL exact rational product = 159/320

source roles = [Markov power3 entry1,0 through middle1, radd, [[27, 400], [159, 320]], [80], 903, 1600]

derived = [903, 1600, 903/1600]

Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum := sum(27/400, 159/320)

Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum = 903/1600

source roles = [Markov power3 entry1,1 middle0 product, rmul, [[27, 80], [4, 5]], [1, 4], 27, 100]

derived = [27, 100, 27/100]

Markov power3 entry1,1 middle0 product ORIGINAL exact rational product := mul(27/80, 4/5)

Markov power3 entry1,1 middle0 product ORIGINAL exact rational product = 27/100

source roles = [Markov power3 entry1,1 through middle0, radd, [[0, 1], [27, 100]], [1], 27, 100]

derived = [27, 100, 27/100]

Markov power3 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 27/100)

Markov power3 entry1,1 through middle0 ORIGINAL exact rational sum = 27/100

source roles = [Markov power3 entry1,1 middle1 product, rmul, [[53, 80], [1, 4]], [1, 1], 53, 320]

derived = [53, 320, 53/320]

Markov power3 entry1,1 middle1 product ORIGINAL exact rational product := mul(53/80, 1/4)

Markov power3 entry1,1 middle1 product ORIGINAL exact rational product = 53/320

source roles = [Markov power3 entry1,1 through middle1, radd, [[27, 100], [53, 320]], [20], 697, 1600]

derived = [697, 1600, 697/1600]

Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum := sum(27/100, 53/320)

Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum = 697/1600

source roles = [3, [[16/25, 9/25], [27/80, 53/80]], [[1/5, 4/5], [3/4, 1/4]]]

derived = [[[199/500, 301/500], [903/1600, 697/1600]]]

source roles = [Markov RAW initial state0, RAW-pair, [1, 2], [], 1, 2]

derived = [1, 2, 1/2]

Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator := div(1, 2)

Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator = 1/2

source roles = [Markov RAW initial state1, RAW-pair, [1, 2], [], 1, 2]

derived = [1, 2, 1/2]

Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator := div(1, 2)

Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator = 1/2

source roles = [Markov initial normalization through state0, radd, [[0, 1], [1, 2]], [1], 1, 2]

derived = [1, 2, 1/2]

Markov initial normalization through state0 ORIGINAL exact rational sum := sum(0, 1/2)

Markov initial normalization through state0 ORIGINAL exact rational sum = 1/2

source roles = [Markov initial normalization through state1, radd, [[1, 2], [1, 2]], [2], 2, 2]

derived = [1, 1, 1]

Markov initial normalization through state1 ORIGINAL exact rational sum := sum(1/2, 1/2)

Markov initial normalization through state1 ORIGINAL exact rational sum = 1

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

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

source roles = [Markov endpoint joint ORIGINAL initial-from mass times conditional entry, rmul, [[1, 2], [301, 500]], [1, 1], 301, 1000]

derived = [301, 1000, 301/1000]

Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product := mul(1/2, 301/500)

Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product = 301/1000

source roles = [[1/2, 1/2], 0, 1, 3]

derived = [1/2, 301/500, 301/1000]

逐步反向核對

• S130 · 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。
• S129 · premise_and_result_audit — 以精確有理數重算 Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product，不以小數近似替代。
• S128 · 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。
• S127 · 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。
• S126 · premise_and_result_audit — 以精確有理數重算 Markov initial normalization through state1 ORIGINAL exact rational sum，不以小數近似替代。
• 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 initial normalization through state0 ORIGINAL exact rational sum，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 RAW initial state1 ORIGINAL raw numerator over nonzero denominator，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 RAW initial state0 ORIGINAL raw numerator over nonzero denominator，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

([1/2, 1/2], 0, 1, 3, 1/2, 301/500, 301/1000)

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。