Question 33343: Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0065-33343 · q-20261013-advance-20261025-r06-finite-markov-chain-0068

Original question and givens

Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. 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. initial law=[1/2, 1/2], from=1, to=0, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

G001 — Use the fixed exact finite time-homogeneous row-stochastic transition table=[[2/5, 3/5], [1/2, 1/2]]. 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. initial law=[1/2, 1/2], from=1, to=0, steps=3. Give (initial,from,to,steps,initial[from],P^steps[from][to],product).

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

S001 · implies

Before

source roles = [qf-mod25-markovdeep-endpoint-joint-probability, markovdeep-endpoint-joint-probability, Finite_exact_Markov_density, []]

After

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

Conditions: none

MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status.

Step identifiers and dependencies

Rule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001

S002 · implies

Before

source roles = [Markov RAW P row0 column0, RAW-pair, [2, 5], [], 2, 5]

After

derived = [2, 5, 2/5]

Conditions: none

This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer.

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S001

S003 · calculation

Before

Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator := div(2, 5)

After

Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator = 2/5

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [Markov RAW P row0 column1, RAW-pair, [3, 5], [], 3, 5]

After

derived = [3, 5, 3/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S003

S005 · calculation

Before

Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator := div(3, 5)

After

Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator = 3/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [Markov RAW row0 normalization through column0, radd, [[0, 1], [2, 5]], [1], 2, 5]

After

derived = [2, 5, 2/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S005

S007 · calculation

Before

Markov RAW row0 normalization through column0 ORIGINAL exact rational sum := sum(0, 2/5)

After

Markov RAW row0 normalization through column0 ORIGINAL exact rational sum = 2/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [Markov RAW row0 normalization through column1, radd, [[2, 5], [3, 5]], [5], 5, 5]

After

derived = [1, 1, 1]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S007

S009 · calculation

Before

Markov RAW row0 normalization through column1 ORIGINAL exact rational sum := sum(2/5, 3/5)

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S009

S011 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S011

S013 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S013

S015 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

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

After

derived = [1, 1, 1]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S015

S017 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

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

After

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

Conditions: none

Validate EVERY raw rational/envelope,then nonnegativity after sign/gcd normalization and exact row sum1.Keep row=current/column=next order.Even steps0/path probability0 cannot waive unused cells.

Step identifiers and dependencies

Rule: Markov_FULL_RAW_kernel_ALL_cells_row_stochastic_fixed_state_order · Uses: G001, S017

S019 · implies

Before

source roles = [2, 3]

After

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

Conditions: none

P0 is the identity,then multiply RIGHT by P in original state order.Preserve every dot-product middle order and SOURCE rational envelope.This is not a stationarity or limit proof.

Step identifiers and dependencies

Rule: Markov_ZERO_step_identity_then_fixed_RIGHT_multiplication · Uses: G001, S018

S020 · implies

Before

source roles = [Markov power1 entry0,0 middle0 product, rmul, [[1, 1], [2, 5]], [1, 1], 2, 5]

After

derived = [2, 5, 2/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S019

S021 · calculation

Before

Markov power1 entry0,0 middle0 product ORIGINAL exact rational product := mul(1, 2/5)

After

Markov power1 entry0,0 middle0 product ORIGINAL exact rational product = 2/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [Markov power1 entry0,0 through middle0, radd, [[0, 1], [2, 5]], [1], 2, 5]

After

derived = [2, 5, 2/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S021

S023 · calculation

Before

Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 2/5)

After

Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum = 2/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

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

After

derived = [0, 1, 0]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S023

S025 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

source roles = [Markov power1 entry0,0 through middle1, radd, [[2, 5], [0, 1]], [1], 2, 5]

After

derived = [2, 5, 2/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S025

S027 · calculation

Before

Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum := sum(2/5, 0)

After

Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum = 2/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [Markov power1 entry0,1 middle0 product, rmul, [[1, 1], [3, 5]], [1, 1], 3, 5]

After

derived = [3, 5, 3/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S027

S029 · calculation

Before

Markov power1 entry0,1 middle0 product ORIGINAL exact rational product := mul(1, 3/5)

After

Markov power1 entry0,1 middle0 product ORIGINAL exact rational product = 3/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

source roles = [Markov power1 entry0,1 through middle0, radd, [[0, 1], [3, 5]], [1], 3, 5]

After

derived = [3, 5, 3/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S029

S031 · calculation

Before

Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 3/5)

After

Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum = 3/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

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

After

derived = [0, 1, 0]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S031

S033 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

source roles = [Markov power1 entry0,1 through middle1, radd, [[3, 5], [0, 1]], [1], 3, 5]

After

derived = [3, 5, 3/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S033

S035 · calculation

Before

Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum := sum(3/5, 0)

After

Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum = 3/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

source roles = [Markov power1 entry1,0 middle0 product, rmul, [[0, 1], [2, 5]], [5, 1], 0, 1]

After

derived = [0, 1, 0]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S035

S037 · calculation

Before

Markov power1 entry1,0 middle0 product ORIGINAL exact rational product := mul(0, 2/5)

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

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

After

derived = [0, 1, 0]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S037

S039 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S039

S041 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S041

S043 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

source roles = [Markov power1 entry1,1 middle0 product, rmul, [[0, 1], [3, 5]], [5, 1], 0, 1]

After

derived = [0, 1, 0]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S043

S045 · calculation

Before

Markov power1 entry1,1 middle0 product ORIGINAL exact rational product := mul(0, 3/5)

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

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

After

derived = [0, 1, 0]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S045

S047 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S047

S049 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S049

S051 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

source roles = [1, [[1, 0], [0, 1]], [[2/5, 3/5], [1/2, 1/2]]]

After

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

Conditions: none

Compute EVERY full-matrix entry by its COMPLETE dot product,not only a queried entry.The SOURCE also computes this power before a path branch;its envelope remains required even when not returned.

Step identifiers and dependencies

Rule: Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes · Uses: G001, S051

S053 · implies

Before

source roles = [Markov power2 entry0,0 middle0 product, rmul, [[2, 5], [2, 5]], [1, 1], 4, 25]

After

derived = [4, 25, 4/25]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S052

S054 · calculation

Before

Markov power2 entry0,0 middle0 product ORIGINAL exact rational product := mul(2/5, 2/5)

After

Markov power2 entry0,0 middle0 product ORIGINAL exact rational product = 4/25

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · implies

Before

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

After

derived = [4, 25, 4/25]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S054

S056 · calculation

Before

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

After

Markov power2 entry0,0 through middle0 ORIGINAL exact rational sum = 4/25

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · implies

Before

source roles = [Markov power2 entry0,0 middle1 product, rmul, [[3, 5], [1, 2]], [1, 1], 3, 10]

After

derived = [3, 10, 3/10]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S056

S058 · calculation

Before

Markov power2 entry0,0 middle1 product ORIGINAL exact rational product := mul(3/5, 1/2)

After

Markov power2 entry0,0 middle1 product ORIGINAL exact rational product = 3/10

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

source roles = [Markov power2 entry0,0 through middle1, radd, [[4, 25], [3, 10]], [5], 23, 50]

After

derived = [23, 50, 23/50]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S058

S060 · calculation

Before

Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum := sum(4/25, 3/10)

After

Markov power2 entry0,0 through middle1 ORIGINAL exact rational sum = 23/50

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · implies

Before

source roles = [Markov power2 entry0,1 middle0 product, rmul, [[2, 5], [3, 5]], [1, 1], 6, 25]

After

derived = [6, 25, 6/25]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S060

S062 · calculation

Before

Markov power2 entry0,1 middle0 product ORIGINAL exact rational product := mul(2/5, 3/5)

After

Markov power2 entry0,1 middle0 product ORIGINAL exact rational product = 6/25

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · implies

Before

source roles = [Markov power2 entry0,1 through middle0, radd, [[0, 1], [6, 25]], [1], 6, 25]

After

derived = [6, 25, 6/25]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S062

S064 · calculation

Before

Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 6/25)

After

Markov power2 entry0,1 through middle0 ORIGINAL exact rational sum = 6/25

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · implies

Before

source roles = [Markov power2 entry0,1 middle1 product, rmul, [[3, 5], [1, 2]], [1, 1], 3, 10]

After

derived = [3, 10, 3/10]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S064

S066 · calculation

Before

Markov power2 entry0,1 middle1 product ORIGINAL exact rational product := mul(3/5, 1/2)

After

Markov power2 entry0,1 middle1 product ORIGINAL exact rational product = 3/10

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · implies

Before

source roles = [Markov power2 entry0,1 through middle1, radd, [[6, 25], [3, 10]], [5], 27, 50]

After

derived = [27, 50, 27/50]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S066

S068 · calculation

Before

Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum := sum(6/25, 3/10)

After

Markov power2 entry0,1 through middle1 ORIGINAL exact rational sum = 27/50

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · implies

Before

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

After

derived = [1, 5, 1/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S068

S070 · calculation

Before

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

After

Markov power2 entry1,0 middle0 product ORIGINAL exact rational product = 1/5

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · implies

Before

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

After

derived = [1, 5, 1/5]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S070

S072 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · implies

Before

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

After

derived = [1, 4, 1/4]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S072

S074 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · implies

Before

source roles = [Markov power2 entry1,0 through middle1, radd, [[1, 5], [1, 4]], [1], 9, 20]

After

derived = [9, 20, 9/20]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S074

S076 · calculation

Before

Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum := sum(1/5, 1/4)

After

Markov power2 entry1,0 through middle1 ORIGINAL exact rational sum = 9/20

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · implies

Before

source roles = [Markov power2 entry1,1 middle0 product, rmul, [[1, 2], [3, 5]], [1, 1], 3, 10]

After

derived = [3, 10, 3/10]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S076

S078 · calculation

Before

Markov power2 entry1,1 middle0 product ORIGINAL exact rational product := mul(1/2, 3/5)

After

Markov power2 entry1,1 middle0 product ORIGINAL exact rational product = 3/10

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · implies

Before

source roles = [Markov power2 entry1,1 through middle0, radd, [[0, 1], [3, 10]], [1], 3, 10]

After

derived = [3, 10, 3/10]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S078

S080 · calculation

Before

Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum := sum(0, 3/10)

After

Markov power2 entry1,1 through middle0 ORIGINAL exact rational sum = 3/10

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · implies

Before

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

After

derived = [1, 4, 1/4]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S080

S082 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S081

S083 · implies

Before

source roles = [Markov power2 entry1,1 through middle1, radd, [[3, 10], [1, 4]], [2], 11, 20]

After

derived = [11, 20, 11/20]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S082

S084 · calculation

Before

Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum := sum(3/10, 1/4)

After

Markov power2 entry1,1 through middle1 ORIGINAL exact rational sum = 11/20

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · implies

Before

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

After

derived = [[[23/50, 27/50], [9/20, 11/20]]]

Conditions: none

Compute EVERY full-matrix entry by its COMPLETE dot product,not only a queried entry.The SOURCE also computes this power before a path branch;its envelope remains required even when not returned.

Step identifiers and dependencies

Rule: Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes · Uses: G001, S084

S086 · implies

Before

source roles = [Markov power3 entry0,0 middle0 product, rmul, [[23, 50], [2, 5]], [1, 2], 23, 125]

After

derived = [23, 125, 23/125]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S085

S087 · calculation

Before

Markov power3 entry0,0 middle0 product ORIGINAL exact rational product := mul(23/50, 2/5)

After

Markov power3 entry0,0 middle0 product ORIGINAL exact rational product = 23/125

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S086

S088 · implies

Before

source roles = [Markov power3 entry0,0 through middle0, radd, [[0, 1], [23, 125]], [1], 23, 125]

After

derived = [23, 125, 23/125]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S087

S089 · calculation

Before

Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 23/125)

After

Markov power3 entry0,0 through middle0 ORIGINAL exact rational sum = 23/125

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · implies

Before

source roles = [Markov power3 entry0,0 middle1 product, rmul, [[27, 50], [1, 2]], [1, 1], 27, 100]

After

derived = [27, 100, 27/100]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S089

S091 · calculation

Before

Markov power3 entry0,0 middle1 product ORIGINAL exact rational product := mul(27/50, 1/2)

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S090

S092 · implies

Before

source roles = [Markov power3 entry0,0 through middle1, radd, [[23, 125], [27, 100]], [25], 227, 500]

After

derived = [227, 500, 227/500]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S091

S093 · calculation

Before

Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum := sum(23/125, 27/100)

After

Markov power3 entry0,0 through middle1 ORIGINAL exact rational sum = 227/500

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · implies

Before

source roles = [Markov power3 entry0,1 middle0 product, rmul, [[23, 50], [3, 5]], [1, 1], 69, 250]

After

derived = [69, 250, 69/250]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S093

S095 · calculation

Before

Markov power3 entry0,1 middle0 product ORIGINAL exact rational product := mul(23/50, 3/5)

After

Markov power3 entry0,1 middle0 product ORIGINAL exact rational product = 69/250

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S094

S096 · implies

Before

source roles = [Markov power3 entry0,1 through middle0, radd, [[0, 1], [69, 250]], [1], 69, 250]

After

derived = [69, 250, 69/250]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S095

S097 · calculation

Before

Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum := sum(0, 69/250)

After

Markov power3 entry0,1 through middle0 ORIGINAL exact rational sum = 69/250

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · implies

Before

source roles = [Markov power3 entry0,1 middle1 product, rmul, [[27, 50], [1, 2]], [1, 1], 27, 100]

After

derived = [27, 100, 27/100]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S097

S099 · calculation

Before

Markov power3 entry0,1 middle1 product ORIGINAL exact rational product := mul(27/50, 1/2)

After

Markov power3 entry0,1 middle1 product ORIGINAL exact rational product = 27/100

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · implies

Before

source roles = [Markov power3 entry0,1 through middle1, radd, [[69, 250], [27, 100]], [50], 273, 500]

After

derived = [273, 500, 273/500]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S099

S101 · calculation

Before

Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum := sum(69/250, 27/100)

After

Markov power3 entry0,1 through middle1 ORIGINAL exact rational sum = 273/500

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S100

S102 · implies

Before

source roles = [Markov power3 entry1,0 middle0 product, rmul, [[9, 20], [2, 5]], [1, 2], 9, 50]

After

derived = [9, 50, 9/50]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S101

S103 · calculation

Before

Markov power3 entry1,0 middle0 product ORIGINAL exact rational product := mul(9/20, 2/5)

After

Markov power3 entry1,0 middle0 product ORIGINAL exact rational product = 9/50

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S102

S104 · implies

Before

source roles = [Markov power3 entry1,0 through middle0, radd, [[0, 1], [9, 50]], [1], 9, 50]

After

derived = [9, 50, 9/50]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S103

S105 · calculation

Before

Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum := sum(0, 9/50)

After

Markov power3 entry1,0 through middle0 ORIGINAL exact rational sum = 9/50

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · implies

Before

source roles = [Markov power3 entry1,0 middle1 product, rmul, [[11, 20], [1, 2]], [1, 1], 11, 40]

After

derived = [11, 40, 11/40]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S105

S107 · calculation

Before

Markov power3 entry1,0 middle1 product ORIGINAL exact rational product := mul(11/20, 1/2)

After

Markov power3 entry1,0 middle1 product ORIGINAL exact rational product = 11/40

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S106

S108 · implies

Before

source roles = [Markov power3 entry1,0 through middle1, radd, [[9, 50], [11, 40]], [10], 91, 200]

After

derived = [91, 200, 91/200]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S107

S109 · calculation

Before

Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum := sum(9/50, 11/40)

After

Markov power3 entry1,0 through middle1 ORIGINAL exact rational sum = 91/200

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S108

S110 · implies

Before

source roles = [Markov power3 entry1,1 middle0 product, rmul, [[9, 20], [3, 5]], [1, 1], 27, 100]

After

derived = [27, 100, 27/100]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S109

S111 · calculation

Before

Markov power3 entry1,1 middle0 product ORIGINAL exact rational product := mul(9/20, 3/5)

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S110

S112 · implies

Before

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

After

derived = [27, 100, 27/100]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S111

S113 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S112

S114 · implies

Before

source roles = [Markov power3 entry1,1 middle1 product, rmul, [[11, 20], [1, 2]], [1, 1], 11, 40]

After

derived = [11, 40, 11/40]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S113

S115 · calculation

Before

Markov power3 entry1,1 middle1 product ORIGINAL exact rational product := mul(11/20, 1/2)

After

Markov power3 entry1,1 middle1 product ORIGINAL exact rational product = 11/40

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S114

S116 · implies

Before

source roles = [Markov power3 entry1,1 through middle1, radd, [[27, 100], [11, 40]], [20], 109, 200]

After

derived = [109, 200, 109/200]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S115

S117 · calculation

Before

Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum := sum(27/100, 11/40)

After

Markov power3 entry1,1 through middle1 ORIGINAL exact rational sum = 109/200

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S116

S118 · implies

Before

source roles = [3, [[23/50, 27/50], [9/20, 11/20]], [[2/5, 3/5], [1/2, 1/2]]]

After

derived = [[[227/500, 273/500], [91/200, 109/200]]]

Conditions: none

Compute EVERY full-matrix entry by its COMPLETE dot product,not only a queried entry.The SOURCE also computes this power before a path branch;its envelope remains required even when not returned.

Step identifiers and dependencies

Rule: Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes · Uses: G001, S117

S119 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S118

S120 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S119

S121 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S120

S122 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S121

S123 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S122

S124 · calculation

Before

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

After

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

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S123

S125 · implies

Before

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

After

derived = [1, 1, 1]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S124

S126 · calculation

Before

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

After

Markov initial normalization through state1 ORIGINAL exact rational sum = 1

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S125

S127 · implies

Before

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

After

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

Conditions: none

The initial law retains ORIGINAL state order,nonnegativity and exact sum1.Do not uniformize or independently sort mu and the matrix.

Step identifiers and dependencies

Rule: Markov_RAW_initial_law_nonnegative_exact_sum1_ORDER_preserved · Uses: G001, S126

S128 · implies

Before

source roles = [Markov endpoint joint ORIGINAL initial-from mass times conditional entry, rmul, [[1, 2], [91, 200]], [1, 1], 91, 400]

After

derived = [91, 400, 91/400]

Conditions: none

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S127

S129 · calculation

Before

Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product := mul(1/2, 91/200)

After

Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product = 91/400

Conditions: none

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S128

S130 · implies

Before

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

After

derived = [1/2, 91/200, 91/400]

Conditions: none

The endpoint JOINT is mu[from]*Ppower[from,to],not the conditional entry alone or the later marginal.Zero initial mass gives a legal joint0.

Step identifiers and dependencies

Rule: Markov_ENDPOINT_joint_initial_mass_times_entry_NOT_marginal · Uses: G001, S129

Reverse checks

S130 · 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.
S129 · premise_and_result_audit — Recompute Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
S128 · 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.
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 initial normalization through state1 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 initial normalization through state0 ORIGINAL exact rational sum 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 RAW initial state1 ORIGINAL raw numerator over nonzero denominator 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 RAW initial state0 ORIGINAL raw numerator over nonzero denominator 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

([1/2, 1/2], 1, 0, 3, 1/2, 91/200, 91/400)

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.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: ba995155a5fb8989696f1b414d9753cd5cedc5e121935b4e105a0f207bff6f47
candidate record SHA-256: 1cf14b436ecfb1a125f4147a5438e91a88192ffff285f58deb139863a02be3b2
semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON