# Question 33331: Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0065-33331 · q-20261013-advance-20261025-r06-finite-markov-chain-0056



## 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=[1/4, 3/4], 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=[1/4, 3/4], 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](https://ocme.evemisslab.com/en/mko/mko-finite-markov-chain/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001





S002 · impliesBefore

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S001





S003 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

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

After

derived = [3, 5, 3/5]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S003





S005 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

source roles = [Markov RAW row0 normalization through column0, radd, [[0, 1], [2, 5]], [1], 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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S005





S007 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

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

After

derived = [1, 1, 1]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S007





S009 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

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

After

derived = [1, 2, 1/2]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S009





S011 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · impliesBefore

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

After

derived = [1, 2, 1/2]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S011





S013 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

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

After

derived = [1, 2, 1/2]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S013





S015 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · impliesBefore

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

After

derived = [1, 1, 1]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S015





S017 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

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 dependenciesRule: Markov_FULL_RAW_kernel_ALL_cells_row_stochastic_fixed_state_order · Uses: G001, S017





S019 · impliesBefore

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 dependenciesRule: Markov_ZERO_step_identity_then_fixed_RIGHT_multiplication · Uses: G001, S018





S020 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S019





S021 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · impliesBefore

source roles = [Markov power1 entry0,0 through middle0, radd, [[0, 1], [2, 5]], [1], 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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S021





S023 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · impliesBefore

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

After

derived = [0, 1, 0]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S023





S025 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

source roles = [Markov power1 entry0,0 through middle1, radd, [[2, 5], [0, 1]], [1], 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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S025





S027 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S027





S029 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · impliesBefore

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

After

derived = [3, 5, 3/5]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S029





S031 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · impliesBefore

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

After

derived = [0, 1, 0]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S031





S033 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · impliesBefore

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

After

derived = [3, 5, 3/5]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S033





S035 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · impliesBefore

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

After

derived = [0, 1, 0]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S035





S037 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

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

After

derived = [0, 1, 0]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S037





S039 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S039





S041 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · impliesBefore

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

After

derived = [1, 2, 1/2]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S041





S043 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · impliesBefore

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

After

derived = [0, 1, 0]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S043





S045 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · impliesBefore

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

After

derived = [0, 1, 0]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S045





S047 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S047





S049 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · impliesBefore

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

After

derived = [1, 2, 1/2]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S049





S051 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S050





S052 · impliesBefore

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 dependenciesRule: Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes · Uses: G001, S051





S053 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S052





S054 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S053





S055 · impliesBefore

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

After

derived = [4, 25, 4/25]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S054





S056 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S055





S057 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S056





S058 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S057





S059 · impliesBefore

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

After

derived = [23, 50, 23/50]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S058





S060 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S059





S061 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S060





S062 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S061





S063 · impliesBefore

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

After

derived = [6, 25, 6/25]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S062





S064 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S063





S065 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S064





S066 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S065





S067 · impliesBefore

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

After

derived = [27, 50, 27/50]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S066





S068 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S067





S069 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S068





S070 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S069





S071 · impliesBefore

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

After

derived = [1, 5, 1/5]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S070





S072 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S071





S073 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S072





S074 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S073





S075 · impliesBefore

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

After

derived = [9, 20, 9/20]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S074





S076 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S075





S077 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S076





S078 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S077





S079 · impliesBefore

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

After

derived = [3, 10, 3/10]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S078





S080 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S079





S081 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S080





S082 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S081





S083 · impliesBefore

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

After

derived = [11, 20, 11/20]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S082





S084 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S083





S085 · impliesBefore

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 dependenciesRule: Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes · Uses: G001, S084





S086 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S085





S087 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S086





S088 · impliesBefore

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

After

derived = [23, 125, 23/125]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S087





S089 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S088





S090 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S089





S091 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S090





S092 · impliesBefore

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

After

derived = [227, 500, 227/500]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S091





S093 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S092





S094 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S093





S095 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S094





S096 · impliesBefore

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

After

derived = [69, 250, 69/250]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S095





S097 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S096





S098 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S097





S099 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S098





S100 · impliesBefore

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

After

derived = [273, 500, 273/500]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S099





S101 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S100





S102 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S101





S103 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S102





S104 · impliesBefore

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

After

derived = [9, 50, 9/50]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S103





S105 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S104





S106 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S105





S107 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S106





S108 · impliesBefore

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

After

derived = [91, 200, 91/200]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S107





S109 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S108





S110 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S109





S111 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S110





S112 · impliesBefore

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

After

derived = [27, 100, 27/100]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S111





S113 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S112





S114 · impliesBefore

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

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S113





S115 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S114





S116 · impliesBefore

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

After

derived = [109, 200, 109/200]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S115





S117 · calculationBefore

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

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S116





S118 · impliesBefore

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 dependenciesRule: Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes · Uses: G001, S117





S119 · impliesBefore

source roles = [Markov RAW initial state0, RAW-pair, [1, 4], [], 1, 4]

After

derived = [1, 4, 1/4]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S118





S120 · calculationBefore

Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator := div(1, 4)

After

Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator = 1/4

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S119





S121 · impliesBefore

source roles = [Markov RAW initial state1, RAW-pair, [3, 4], [], 3, 4]

After

derived = [3, 4, 3/4]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S120





S122 · calculationBefore

Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator := div(3, 4)

After

Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator = 3/4

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S121





S123 · impliesBefore

source roles = [Markov initial normalization through state0, radd, [[0, 1], [1, 4]], [1], 1, 4]

After

derived = [1, 4, 1/4]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S122





S124 · calculationBefore

Markov initial normalization through state0 ORIGINAL exact rational sum := sum(0, 1/4)

After

Markov initial normalization through state0 ORIGINAL exact rational sum = 1/4

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S123





S125 · impliesBefore

source roles = [Markov initial normalization through state1, radd, [[1, 4], [3, 4]], [4], 4, 4]

After

derived = [1, 1, 1]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S124





S126 · calculationBefore

Markov initial normalization through state1 ORIGINAL exact rational sum := sum(1/4, 3/4)

After

Markov initial normalization through state1 ORIGINAL exact rational sum = 1

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S125





S127 · impliesBefore

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

After

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

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 dependenciesRule: Markov_RAW_initial_law_nonnegative_exact_sum1_ORDER_preserved · Uses: G001, S126





S128 · impliesBefore

source roles = [Markov endpoint joint ORIGINAL initial-from mass times conditional entry, rmul, [[3, 4], [91, 200]], [1, 1], 273, 800]

After

derived = [273, 800, 273/800]

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 dependenciesRule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S127





S129 · calculationBefore

Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product := mul(3/4, 91/200)

After

Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product = 273/800

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S128





S130 · impliesBefore

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

After

derived = [3/4, 91/200, 273/800]

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 dependenciesRule: 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

([1/4, 3/4], 1, 0, 3, 3/4, 91/200, 273/800)

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: 12f1a726d3af71f13944c19cea6bdc63cb29c97f7c934760c5e1d83aa7491e51
- candidate record SHA-256: f2035e806b5378b78352520a777b93a4eb778c3684930fd64a04d7f4c11cbfcf
- semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
- question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261025-r06-finite-markov-chain-0056.json)