Question 33325: A Kernel-Started Path Excludes Any Initial-Law Mass

Original question and givens

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

• G001 — Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. nonempty path=[0, 1, 0, 1, 0] is conditional on the specified initial state. Give (path,successive transition factors,their product,conditional_on_initial_state). Do not include an initial-law mass; none is supplied. A one-state path has no transitions and empty-product value 1.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-finite-markov-chain

Step-by-step derivation

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, 4], [], 1, 4]

derived = [1, 4, 1/4]

Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator := div(1, 4)

Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator = 1/4

source roles = [Markov RAW P row1 column1, RAW-pair, [3, 4], [], 3, 4]

derived = [3, 4, 3/4]

Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator := div(3, 4)

Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator = 3/4

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

derived = [1, 4, 1/4]

Markov RAW row1 normalization through column0 ORIGINAL exact rational sum := sum(0, 1/4)

Markov RAW row1 normalization through column0 ORIGINAL exact rational sum = 1/4

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

derived = [1, 1, 1]

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

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

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

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

source roles = [2, 4]

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

derived = [0, 1, 0]

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

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

derived = [0, 1, 0]

Markov power1 entry0,1 middle1 product ORIGINAL exact rational product := mul(0, 3/4)

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

derived = [1, 4, 1/4]

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

Markov power1 entry1,0 middle1 product ORIGINAL exact rational product = 1/4

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

derived = [1, 4, 1/4]

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

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

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

derived = [3, 4, 3/4]

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

Markov power1 entry1,1 middle1 product ORIGINAL exact rational product = 3/4

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

derived = [3, 4, 3/4]

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

Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum = 3/4

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

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

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

derived = [1, 5, 1/5]

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

Markov power2 entry0,0 middle1 product ORIGINAL exact rational product = 1/5

source roles = [Markov power2 entry0,0 through middle1, radd, [[1, 25], [1, 5]], [5], 6, 25]

derived = [6, 25, 6/25]

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

Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum = 6/25

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

derived = [3, 5, 3/5]

Markov power2 entry0,1 middle1 product ORIGINAL exact rational product := mul(4/5, 3/4)

Markov power2 entry0,1 middle1 product ORIGINAL exact rational product = 3/5

source roles = [Markov power2 entry0,1 through middle1, radd, [[4, 25], [3, 5]], [5], 19, 25]

derived = [19, 25, 19/25]

Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum := sum(4/25, 3/5)

Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum = 19/25

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

derived = [1, 20, 1/20]

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

Markov power2 entry1,0 middle0 product ORIGINAL exact rational product = 1/20

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

derived = [1, 20, 1/20]

Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 1/20)

Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum = 1/20

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

derived = [3, 16, 3/16]

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

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

source roles = [Markov power2 entry1,0 through middle1, radd, [[1, 20], [3, 16]], [4], 19, 80]

derived = [19, 80, 19/80]

Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum := sum(1/20, 3/16)

Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum = 19/80

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

derived = [1, 5, 1/5]

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

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

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

derived = [1, 5, 1/5]

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

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

source roles = [Markov power2 entry1,1 middle1 product, rmul, [[3, 4], [3, 4]], [1, 1], 9, 16]

derived = [9, 16, 9/16]

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

Markov power2 entry1,1 middle1 product ORIGINAL exact rational product = 9/16

source roles = [Markov power2 entry1,1 through middle1, radd, [[1, 5], [9, 16]], [1], 61, 80]

derived = [61, 80, 61/80]

Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum := sum(1/5, 9/16)

Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum = 61/80

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

derived = [[[6/25, 19/25], [19/80, 61/80]]]

source roles = [Markov power3 entry0,0 middle0 product, rmul, [[6, 25], [1, 5]], [1, 1], 6, 125]

derived = [6, 125, 6/125]

Markov power3 entry0,0 middle0 product ORIGINAL exact rational product := mul(6/25, 1/5)

Markov power3 entry0,0 middle0 product ORIGINAL exact rational product = 6/125

source roles = [Markov power3 entry0,0 through middle0, radd, [[0, 1], [6, 125]], [1], 6, 125]

derived = [6, 125, 6/125]

Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 6/125)

Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum = 6/125

source roles = [Markov power3 entry0,0 middle1 product, rmul, [[19, 25], [1, 4]], [1, 1], 19, 100]

derived = [19, 100, 19/100]

Markov power3 entry0,0 middle1 product ORIGINAL exact rational product := mul(19/25, 1/4)

Markov power3 entry0,0 middle1 product ORIGINAL exact rational product = 19/100

source roles = [Markov power3 entry0,0 through middle1, radd, [[6, 125], [19, 100]], [25], 119, 500]

derived = [119, 500, 119/500]

Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum := sum(6/125, 19/100)

Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum = 119/500

source roles = [Markov power3 entry0,1 middle0 product, rmul, [[6, 25], [4, 5]], [1, 1], 24, 125]

derived = [24, 125, 24/125]

Markov power3 entry0,1 middle0 product ORIGINAL exact rational product := mul(6/25, 4/5)

Markov power3 entry0,1 middle0 product ORIGINAL exact rational product = 24/125

source roles = [Markov power3 entry0,1 through middle0, radd, [[0, 1], [24, 125]], [1], 24, 125]

derived = [24, 125, 24/125]

Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 24/125)

Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum = 24/125

source roles = [Markov power3 entry0,1 middle1 product, rmul, [[19, 25], [3, 4]], [1, 1], 57, 100]

derived = [57, 100, 57/100]

Markov power3 entry0,1 middle1 product ORIGINAL exact rational product := mul(19/25, 3/4)

Markov power3 entry0,1 middle1 product ORIGINAL exact rational product = 57/100

source roles = [Markov power3 entry0,1 through middle1, radd, [[24, 125], [57, 100]], [25], 381, 500]

derived = [381, 500, 381/500]

Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum := sum(24/125, 57/100)

Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum = 381/500

source roles = [Markov power3 entry1,0 middle0 product, rmul, [[19, 80], [1, 5]], [1, 1], 19, 400]

derived = [19, 400, 19/400]

Markov power3 entry1,0 middle0 product ORIGINAL exact rational product := mul(19/80, 1/5)

Markov power3 entry1,0 middle0 product ORIGINAL exact rational product = 19/400

source roles = [Markov power3 entry1,0 through middle0, radd, [[0, 1], [19, 400]], [1], 19, 400]

derived = [19, 400, 19/400]

Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 19/400)

Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum = 19/400

source roles = [Markov power3 entry1,0 middle1 product, rmul, [[61, 80], [1, 4]], [1, 1], 61, 320]

derived = [61, 320, 61/320]

Markov power3 entry1,0 middle1 product ORIGINAL exact rational product := mul(61/80, 1/4)

Markov power3 entry1,0 middle1 product ORIGINAL exact rational product = 61/320

source roles = [Markov power3 entry1,0 through middle1, radd, [[19, 400], [61, 320]], [80], 381, 1600]

derived = [381, 1600, 381/1600]

Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum := sum(19/400, 61/320)

Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum = 381/1600

source roles = [Markov power3 entry1,1 middle0 product, rmul, [[19, 80], [4, 5]], [1, 4], 19, 100]

derived = [19, 100, 19/100]

Markov power3 entry1,1 middle0 product ORIGINAL exact rational product := mul(19/80, 4/5)

Markov power3 entry1,1 middle0 product ORIGINAL exact rational product = 19/100

source roles = [Markov power3 entry1,1 through middle0, radd, [[0, 1], [19, 100]], [1], 19, 100]

derived = [19, 100, 19/100]

Markov power3 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 19/100)

Markov power3 entry1,1 through middle0 ORIGINAL exact rational sum = 19/100

source roles = [Markov power3 entry1,1 middle1 product, rmul, [[61, 80], [3, 4]], [1, 1], 183, 320]

derived = [183, 320, 183/320]

Markov power3 entry1,1 middle1 product ORIGINAL exact rational product := mul(61/80, 3/4)

Markov power3 entry1,1 middle1 product ORIGINAL exact rational product = 183/320

source roles = [Markov power3 entry1,1 through middle1, radd, [[19, 100], [183, 320]], [20], 1219, 1600]

derived = [1219, 1600, 1219/1600]

Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum := sum(19/100, 183/320)

Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum = 1219/1600

source roles = [3, [[6/25, 19/25], [19/80, 61/80]], [[1/5, 4/5], [1/4, 3/4]]]

derived = [[[119/500, 381/500], [381/1600, 1219/1600]]]

source roles = [Markov power4 entry0,0 middle0 product, rmul, [[119, 500], [1, 5]], [1, 1], 119, 2500]

derived = [119, 2500, 119/2500]

Markov power4 entry0,0 middle0 product ORIGINAL exact rational product := mul(119/500, 1/5)

Markov power4 entry0,0 middle0 product ORIGINAL exact rational product = 119/2500

source roles = [Markov power4 entry0,0 through middle0, radd, [[0, 1], [119, 2500]], [1], 119, 2500]

derived = [119, 2500, 119/2500]

Markov power4 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 119/2500)

Markov power4 entry0,0 through middle0 ORIGINAL exact rational sum = 119/2500

source roles = [Markov power4 entry0,0 middle1 product, rmul, [[381, 500], [1, 4]], [1, 1], 381, 2000]

derived = [381, 2000, 381/2000]

Markov power4 entry0,0 middle1 product ORIGINAL exact rational product := mul(381/500, 1/4)

Markov power4 entry0,0 middle1 product ORIGINAL exact rational product = 381/2000

source roles = [Markov power4 entry0,0 through middle1, radd, [[119, 2500], [381, 2000]], [500], 2381, 10000]

derived = [2381, 10000, 2381/10000]

Markov power4 entry0,0 through middle1 ORIGINAL exact rational sum := sum(119/2500, 381/2000)

Markov power4 entry0,0 through middle1 ORIGINAL exact rational sum = 2381/10000

source roles = [Markov power4 entry0,1 middle0 product, rmul, [[119, 500], [4, 5]], [1, 4], 119, 625]

derived = [119, 625, 119/625]

Markov power4 entry0,1 middle0 product ORIGINAL exact rational product := mul(119/500, 4/5)

Markov power4 entry0,1 middle0 product ORIGINAL exact rational product = 119/625

source roles = [Markov power4 entry0,1 through middle0, radd, [[0, 1], [119, 625]], [1], 119, 625]

derived = [119, 625, 119/625]

Markov power4 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 119/625)

Markov power4 entry0,1 through middle0 ORIGINAL exact rational sum = 119/625

source roles = [Markov power4 entry0,1 middle1 product, rmul, [[381, 500], [3, 4]], [1, 1], 1143, 2000]

derived = [1143, 2000, 1143/2000]

Markov power4 entry0,1 middle1 product ORIGINAL exact rational product := mul(381/500, 3/4)

Markov power4 entry0,1 middle1 product ORIGINAL exact rational product = 1143/2000

source roles = [Markov power4 entry0,1 through middle1, radd, [[119, 625], [1143, 2000]], [125], 7619, 10000]

derived = [7619, 10000, 7619/10000]

Markov power4 entry0,1 through middle1 ORIGINAL exact rational sum := sum(119/625, 1143/2000)

Markov power4 entry0,1 through middle1 ORIGINAL exact rational sum = 7619/10000

source roles = [Markov power4 entry1,0 middle0 product, rmul, [[381, 1600], [1, 5]], [1, 1], 381, 8000]

derived = [381, 8000, 381/8000]

Markov power4 entry1,0 middle0 product ORIGINAL exact rational product := mul(381/1600, 1/5)

Markov power4 entry1,0 middle0 product ORIGINAL exact rational product = 381/8000

source roles = [Markov power4 entry1,0 through middle0, radd, [[0, 1], [381, 8000]], [1], 381, 8000]

derived = [381, 8000, 381/8000]

Markov power4 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 381/8000)

Markov power4 entry1,0 through middle0 ORIGINAL exact rational sum = 381/8000

source roles = [Markov power4 entry1,0 middle1 product, rmul, [[1219, 1600], [1, 4]], [1, 1], 1219, 6400]

derived = [1219, 6400, 1219/6400]

Markov power4 entry1,0 middle1 product ORIGINAL exact rational product := mul(1219/1600, 1/4)

Markov power4 entry1,0 middle1 product ORIGINAL exact rational product = 1219/6400

source roles = [Markov power4 entry1,0 through middle1, radd, [[381, 8000], [1219, 6400]], [1600], 7619, 32000]

derived = [7619, 32000, 7619/32000]

Markov power4 entry1,0 through middle1 ORIGINAL exact rational sum := sum(381/8000, 1219/6400)

Markov power4 entry1,0 through middle1 ORIGINAL exact rational sum = 7619/32000

source roles = [Markov power4 entry1,1 middle0 product, rmul, [[381, 1600], [4, 5]], [1, 4], 381, 2000]

derived = [381, 2000, 381/2000]

Markov power4 entry1,1 middle0 product ORIGINAL exact rational product := mul(381/1600, 4/5)

Markov power4 entry1,1 middle0 product ORIGINAL exact rational product = 381/2000

source roles = [Markov power4 entry1,1 through middle0, radd, [[0, 1], [381, 2000]], [1], 381, 2000]

derived = [381, 2000, 381/2000]

Markov power4 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 381/2000)

Markov power4 entry1,1 through middle0 ORIGINAL exact rational sum = 381/2000

source roles = [Markov power4 entry1,1 middle1 product, rmul, [[1219, 1600], [3, 4]], [1, 1], 3657, 6400]

derived = [3657, 6400, 3657/6400]

Markov power4 entry1,1 middle1 product ORIGINAL exact rational product := mul(1219/1600, 3/4)

Markov power4 entry1,1 middle1 product ORIGINAL exact rational product = 3657/6400

source roles = [Markov power4 entry1,1 through middle1, radd, [[381, 2000], [3657, 6400]], [400], 24381, 32000]

derived = [24381, 32000, 24381/32000]

Markov power4 entry1,1 through middle1 ORIGINAL exact rational sum := sum(381/2000, 3657/6400)

Markov power4 entry1,1 through middle1 ORIGINAL exact rational sum = 24381/32000

source roles = [4, [[119/500, 381/500], [381/1600, 1219/1600]], [[1/5, 4/5], [1/4, 3/4]]]

derived = [[[2381/10000, 7619/10000], [7619/32000, 24381/32000]]]

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

derived = [1, 5, 1/5]

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

Markov conditional path ORIGINAL transition2 ORIGINAL exact rational product = 1/5

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

derived = [4, 25, 4/25]

Markov conditional path ORIGINAL transition3 ORIGINAL exact rational product := mul(1/5, 4/5)

Markov conditional path ORIGINAL transition3 ORIGINAL exact rational product = 4/25

source roles = [Markov conditional path ORIGINAL transition4, rmul, [[4, 25], [1, 4]], [4, 1], 1, 25]

derived = [1, 25, 1/25]

Markov conditional path ORIGINAL transition4 ORIGINAL exact rational product := mul(4/25, 1/4)

Markov conditional path ORIGINAL transition4 ORIGINAL exact rational product = 1/25

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

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

Reverse checks

• S160 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S159 · premise_and_result_audit — Recompute Markov conditional path ORIGINAL transition4 ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S157 · premise_and_result_audit — Recompute Markov conditional path ORIGINAL transition3 ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S155 · premise_and_result_audit — Recompute Markov conditional path ORIGINAL transition2 ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S153 · premise_and_result_audit — Recompute Markov conditional path ORIGINAL transition1 ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S151 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S150 · premise_and_result_audit — Recompute Markov power4 entry1,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S148 · premise_and_result_audit — Recompute Markov power4 entry1,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S146 · premise_and_result_audit — Recompute Markov power4 entry1,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S144 · premise_and_result_audit — Recompute Markov power4 entry1,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S142 · premise_and_result_audit — Recompute Markov power4 entry1,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S140 · premise_and_result_audit — Recompute Markov power4 entry1,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S138 · premise_and_result_audit — Recompute Markov power4 entry1,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S136 · premise_and_result_audit — Recompute Markov power4 entry1,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S134 · premise_and_result_audit — Recompute Markov power4 entry0,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S132 · premise_and_result_audit — Recompute Markov power4 entry0,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S130 · premise_and_result_audit — Recompute Markov power4 entry0,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S128 · premise_and_result_audit — Recompute Markov power4 entry0,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S126 · premise_and_result_audit — Recompute Markov power4 entry0,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S124 · premise_and_result_audit — Recompute Markov power4 entry0,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S122 · premise_and_result_audit — Recompute Markov power4 entry0,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S120 · premise_and_result_audit — Recompute Markov power4 entry0,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S118 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S117 · premise_and_result_audit — Recompute Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S115 · premise_and_result_audit — Recompute Markov power3 entry1,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S113 · premise_and_result_audit — Recompute Markov power3 entry1,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S111 · premise_and_result_audit — Recompute Markov power3 entry1,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S109 · premise_and_result_audit — Recompute Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S107 · premise_and_result_audit — Recompute Markov power3 entry1,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S105 · premise_and_result_audit — Recompute Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S103 · premise_and_result_audit — Recompute Markov power3 entry1,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S101 · premise_and_result_audit — Recompute Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S099 · premise_and_result_audit — Recompute Markov power3 entry0,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S097 · premise_and_result_audit — Recompute Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S095 · premise_and_result_audit — Recompute Markov power3 entry0,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S093 · premise_and_result_audit — Recompute Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S091 · premise_and_result_audit — Recompute Markov power3 entry0,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S089 · premise_and_result_audit — Recompute Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S087 · premise_and_result_audit — Recompute Markov power3 entry0,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S085 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S084 · premise_and_result_audit — Recompute Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S082 · premise_and_result_audit — Recompute Markov power2 entry1,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S080 · premise_and_result_audit — Recompute Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S078 · premise_and_result_audit — Recompute Markov power2 entry1,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S076 · premise_and_result_audit — Recompute Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S074 · premise_and_result_audit — Recompute Markov power2 entry1,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S072 · premise_and_result_audit — Recompute Markov power2 entry1,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S070 · premise_and_result_audit — Recompute Markov power2 entry1,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S068 · premise_and_result_audit — Recompute Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S066 · premise_and_result_audit — Recompute Markov power2 entry0,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S064 · premise_and_result_audit — Recompute Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S062 · premise_and_result_audit — Recompute Markov power2 entry0,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S060 · premise_and_result_audit — Recompute Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S058 · premise_and_result_audit — Recompute Markov power2 entry0,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S056 · premise_and_result_audit — Recompute Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S054 · premise_and_result_audit — Recompute Markov power2 entry0,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S052 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S051 · premise_and_result_audit — Recompute Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S049 · premise_and_result_audit — Recompute Markov power1 entry1,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S047 · premise_and_result_audit — Recompute Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S045 · premise_and_result_audit — Recompute Markov power1 entry1,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S043 · premise_and_result_audit — Recompute Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S041 · premise_and_result_audit — Recompute Markov power1 entry1,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S039 · premise_and_result_audit — Recompute Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S037 · premise_and_result_audit — Recompute Markov power1 entry1,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S035 · premise_and_result_audit — Recompute Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S033 · premise_and_result_audit — Recompute Markov power1 entry0,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S031 · premise_and_result_audit — Recompute Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S029 · premise_and_result_audit — Recompute Markov power1 entry0,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S027 · premise_and_result_audit — Recompute Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S025 · premise_and_result_audit — Recompute Markov power1 entry0,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S023 · premise_and_result_audit — Recompute Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S021 · premise_and_result_audit — Recompute Markov power1 entry0,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S019 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S018 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S017 · premise_and_result_audit — Recompute Markov RAW row1 normalization through column1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S015 · premise_and_result_audit — Recompute Markov RAW row1 normalization through column0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S013 · premise_and_result_audit — Recompute Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S011 · premise_and_result_audit — Recompute Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S009 · premise_and_result_audit — Recompute Markov RAW row0 normalization through column1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S007 · premise_and_result_audit — Recompute Markov RAW row0 normalization through column0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S005 · premise_and_result_audit — Recompute Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S003 · premise_and_result_audit — Recompute Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.

Result

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

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.