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Question 33357: A Conditional n-Step Entry without Inventing an Initial Law

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0065-33357 · q-20261013-advance-20261025-r06-finite-markov-chain-0032

Original question and givens

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

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

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-mod25-markovdeep-n-step-entry, markovdeep-n-step-entry, Finite_exact_Markov_density, []]

After

derived = [[from, to, steps, power_matrix, entry]]

Conditions: none

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

Step identifiers and dependencies

Rule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001

S002 · implies

Before

source roles = [Markov RAW P row0 column0, RAW-pair, [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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S001

S003 · calculation

Before

Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator := div(3, 5)

After

Markov RAW P row0 column0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [Markov RAW P row0 column1, RAW-pair, [2, 5], [], 2, 5]

After

derived = [2, 5, 2/5]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S003

S005 · calculation

Before

Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator := div(2, 5)

After

Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator = 2/5

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [Markov RAW row0 normalization through column0, 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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S005

S007 · calculation

Before

Markov RAW row0 normalization through column0 ORIGINAL exact rational sum := sum(0, 3/5)

After

Markov RAW row0 normalization through column0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

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

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S007

S009 · calculation

Before

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

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S009

S011 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S011

S013 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S013

S015 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

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

After

derived = [1, 1, 1]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S015

S017 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Markov_FULL_RAW_kernel_ALL_cells_row_stochastic_fixed_state_order · Uses: G001, S017

S019 · implies

Before

source roles = [2, 1]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Markov_ZERO_step_identity_then_fixed_RIGHT_multiplication · Uses: G001, S018

S020 · implies

Before

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

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S019

S021 · calculation

Before

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

After

Markov power1 entry0,0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [Markov power1 entry0,0 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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S021

S023 · calculation

Before

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

After

Markov power1 entry0,0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

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

After

derived = [0, 1, 0]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S023

S025 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

source roles = [Markov power1 entry0,0 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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S025

S027 · calculation

Before

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

After

Markov power1 entry0,0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [Markov power1 entry0,1 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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S027

S029 · calculation

Before

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

After

Markov power1 entry0,1 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

source roles = [Markov power1 entry0,1 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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S029

S031 · calculation

Before

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

After

Markov power1 entry0,1 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

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

After

derived = [0, 1, 0]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S031

S033 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

source roles = [Markov power1 entry0,1 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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S033

S035 · calculation

Before

Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum := sum(2/5, 0)

After

Markov power1 entry0,1 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

source roles = [Markov power1 entry1,0 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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S035

S037 · calculation

Before

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

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

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

After

derived = [0, 1, 0]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S037

S039 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S039

S041 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S041

S043 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

source roles = [Markov power1 entry1,1 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 dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S043

S045 · calculation

Before

Markov power1 entry1,1 middle0 product ORIGINAL exact rational product := mul(0, 2/5)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

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

After

derived = [0, 1, 0]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S045

S047 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S047

S049 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

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

After

derived = [1, 2, 1/2]

Conditions: none

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

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S049

S051 · calculation

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Markov_EACH_full_power_matrix_ALL_entries_and_source_envelopes · Uses: G001, S051

S053 · implies

Before

source roles = [0, 1, 1]

After

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

Conditions: none

Read FROM row TO column;at zero steps the entry is1 only for equal state labels.Do not invent an initial law.

Step identifiers and dependencies

Rule: Markov_ENTRY_current_row_to_next_column_NO_invented_initial_law · Uses: G001, S052

Reverse checks

  • S053 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S052 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S051 · premise_and_result_audit — Recompute Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S049 · premise_and_result_audit — Recompute Markov power1 entry1,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
  • S048 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S047 · premise_and_result_audit — Recompute Markov power1 entry1,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S045 · premise_and_result_audit — Recompute Markov power1 entry1,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
  • S044 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S043 · premise_and_result_audit — Recompute Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S041 · premise_and_result_audit — Recompute Markov power1 entry1,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S039 · premise_and_result_audit — Recompute Markov power1 entry1,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S037 · premise_and_result_audit — Recompute Markov power1 entry1,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S035 · premise_and_result_audit — Recompute Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S033 · premise_and_result_audit — Recompute Markov power1 entry0,1 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S031 · premise_and_result_audit — Recompute Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S029 · premise_and_result_audit — Recompute Markov power1 entry0,1 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S027 · premise_and_result_audit — Recompute Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S025 · premise_and_result_audit — Recompute Markov power1 entry0,0 middle1 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S023 · premise_and_result_audit — Recompute Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S021 · premise_and_result_audit — Recompute Markov power1 entry0,0 middle0 product ORIGINAL exact rational product with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S019 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S018 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S017 · premise_and_result_audit — Recompute Markov RAW row1 normalization through column1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S015 · premise_and_result_audit — Recompute Markov RAW row1 normalization through column0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Recompute Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S011 · premise_and_result_audit — Recompute Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Recompute Markov RAW row0 normalization through column1 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S007 · premise_and_result_audit — Recompute Markov RAW row0 normalization through column0 ORIGINAL exact rational sum with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Recompute Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S003 · premise_and_result_audit — Recompute Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.

Result

(0, 1, 1, [[3/5, 2/5], [1/2, 1/2]], 2/5)

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: 2c743f6d77f672161ca96356fb90de96025613489113a3183694da0906c79e37
  • candidate record SHA-256: d9cb523069cc2b7d596a989630a318d02ea2dc9d7345ae9bebe665e6898c03a9
  • semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
  • question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON