# Question 33350: Initial Mass Times an n-Step Entry Gives an Endpoint Joint Probability [Return to catalogue page 167](https://ocme.evemisslab.com/en/questions/page/167/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261025-r06-finite-markov-chain-foundations/#q-20261013-advance-20261025-r06-finite-markov-chain-0075) 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. R0065-33350 · q-20261013-advance-20261025-r06-finite-markov-chain-0075 ## Original question and givens Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[3/4, 1/4], from=0, to=1, steps=1. Give (initial,from,to,steps,initial[from],P^steps[from][to],product). - G001 — Use the fixed exact finite time-homogeneous row-stochastic transition table=[[1/5, 4/5], [1/4, 3/4]]. Rows=current state; columns=next state. Matrix powers keep state order and P^0 is the identity. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Finite-step laws and conditional paths imply no stationarity, irreducibility, ergodicity or limit theorem. initial law=[3/4, 1/4], from=0, to=1, steps=1. 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, [1, 5], [], 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, S001 S003 · calculationBefore Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator := div(1, 5) After Markov RAW P row0 column0 ORIGINAL raw numerator over nonzero denominator = 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, S002 S004 · impliesBefore source roles = [Markov RAW P row0 column1, RAW-pair, [4, 5], [], 4, 5] After derived = [4, 5, 4/5] 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(4, 5) After Markov RAW P row0 column1 ORIGINAL raw numerator over nonzero denominator = 4/5 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], [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, S005 S007 · calculationBefore Markov RAW row0 normalization through column0 ORIGINAL exact rational sum := sum(0, 1/5) After Markov RAW row0 normalization through column0 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, S006 S008 · impliesBefore source roles = [Markov RAW row0 normalization through column1, radd, [[1, 5], [4, 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(1/5, 4/5) 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, 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, S009 S011 · calculationBefore Markov RAW P row1 column0 ORIGINAL raw numerator over nonzero denominator := div(1, 4) After Markov RAW P row1 column0 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, S010 S012 · impliesBefore source roles = [Markov RAW P row1 column1, 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, S011 S013 · calculationBefore Markov RAW P row1 column1 ORIGINAL raw numerator over nonzero denominator := div(3, 4) After Markov RAW P row1 column1 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, S012 S014 · impliesBefore source roles = [Markov RAW row1 normalization through column0, 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, S013 S015 · calculationBefore Markov RAW row1 normalization through column0 ORIGINAL exact rational sum := sum(0, 1/4) After Markov RAW row1 normalization through column0 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, S014 S016 · impliesBefore source roles = [Markov RAW row1 normalization through column1, 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, S015 S017 · calculationBefore Markov RAW row1 normalization through column1 ORIGINAL exact rational sum := sum(1/4, 3/4) 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 = [[[[1, 5], [4, 5]], [[1, 4], [3, 4]]]] After derived = [2, [[1/5, 4/5], [1/4, 3/4]]] 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, 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 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], [1, 5]], [1, 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, S019 S021 · calculationBefore Markov power1 entry0,0 middle0 product ORIGINAL exact rational product := mul(1, 1/5) After Markov power1 entry0,0 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, S020 S022 · impliesBefore source roles = [Markov power1 entry0,0 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, S021 S023 · calculationBefore Markov power1 entry0,0 through middle0 ORIGINAL exact rational sum := sum(0, 1/5) After Markov power1 entry0,0 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, S022 S024 · impliesBefore source roles = [Markov power1 entry0,0 middle1 product, rmul, [[0, 1], [1, 4]], [4, 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, 1/4) 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, [[1, 5], [0, 1]], [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, S025 S027 · calculationBefore Markov power1 entry0,0 through middle1 ORIGINAL exact rational sum := sum(1/5, 0) After Markov power1 entry0,0 through middle1 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, S026 S028 · impliesBefore source roles = [Markov power1 entry0,1 middle0 product, rmul, [[1, 1], [4, 5]], [1, 1], 4, 5] After derived = [4, 5, 4/5] 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, 4/5) After Markov power1 entry0,1 middle0 product ORIGINAL exact rational product = 4/5 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], [4, 5]], [1], 4, 5] After derived = [4, 5, 4/5] 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, 4/5) After Markov power1 entry0,1 through middle0 ORIGINAL exact rational sum = 4/5 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], [3, 4]], [4, 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, 3/4) 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, [[4, 5], [0, 1]], [1], 4, 5] After derived = [4, 5, 4/5] 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(4/5, 0) After Markov power1 entry0,1 through middle1 ORIGINAL exact rational sum = 4/5 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], [1, 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, 1/5) 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, 4]], [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, S039 S041 · calculationBefore Markov power1 entry1,0 middle1 product ORIGINAL exact rational product := mul(1, 1/4) After Markov power1 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, S040 S042 · impliesBefore source roles = [Markov power1 entry1,0 through middle1, 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, S041 S043 · calculationBefore Markov power1 entry1,0 through middle1 ORIGINAL exact rational sum := sum(0, 1/4) After Markov power1 entry1,0 through middle1 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, S042 S044 · impliesBefore source roles = [Markov power1 entry1,1 middle0 product, rmul, [[0, 1], [4, 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, 4/5) 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], [3, 4]], [1, 1], 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, S047 S049 · calculationBefore Markov power1 entry1,1 middle1 product ORIGINAL exact rational product := mul(1, 3/4) After Markov power1 entry1,1 middle1 product ORIGINAL exact rational product = 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, S048 S050 · impliesBefore source roles = [Markov power1 entry1,1 through middle1, radd, [[0, 1], [3, 4]], [1], 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, S049 S051 · calculationBefore Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum := sum(0, 3/4) After Markov power1 entry1,1 through middle1 ORIGINAL exact rational sum = 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, S050 S052 · impliesBefore source roles = [1, [[1, 0], [0, 1]], [[1/5, 4/5], [1/4, 3/4]]] After derived = [[[1/5, 4/5], [1/4, 3/4]]] 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 RAW initial state0, 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, S052 S054 · calculationBefore Markov RAW initial state0 ORIGINAL raw numerator over nonzero denominator := div(3, 4) After Markov RAW initial state0 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, S053 S055 · impliesBefore source roles = [Markov RAW initial state1, 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, S054 S056 · calculationBefore Markov RAW initial state1 ORIGINAL raw numerator over nonzero denominator := div(1, 4) After Markov RAW initial state1 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, S055 S057 · impliesBefore source roles = [Markov initial normalization through state0, radd, [[0, 1], [3, 4]], [1], 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, S056 S058 · calculationBefore Markov initial normalization through state0 ORIGINAL exact rational sum := sum(0, 3/4) After Markov initial normalization through state0 ORIGINAL exact rational sum = 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, S057 S059 · impliesBefore source roles = [Markov initial normalization through state1, radd, [[3, 4], [1, 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, S058 S060 · calculationBefore Markov initial normalization through state1 ORIGINAL exact rational sum := sum(3/4, 1/4) 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, S059 S061 · impliesBefore source roles = [[[3, 4], [1, 4]]] After derived = [[3/4, 1/4]] 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, S060 S062 · impliesBefore source roles = [Markov endpoint joint ORIGINAL initial-from mass times conditional entry, rmul, [[3, 4], [4, 5]], [1, 4], 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, S061 S063 · calculationBefore Markov endpoint joint ORIGINAL initial-from mass times conditional entry ORIGINAL exact rational product := mul(3/4, 4/5) After Markov endpoint joint ORIGINAL initial-from mass times conditional entry 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, S062 S064 · impliesBefore source roles = [[3/4, 1/4], 0, 1, 1] After derived = [3/4, 4/5, 3/5] 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, S063 ## Reverse checks - S064 · 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. - S063 · 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. - S062 · 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. - 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 initial normalization through state1 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 initial normalization through state0 ORIGINAL exact rational sum 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 RAW initial state1 ORIGINAL raw numerator over nonzero denominator 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 RAW initial state0 ORIGINAL raw numerator over nonzero denominator 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 ([3/4, 1/4], 0, 1, 1, 3/4, 4/5, 3/5) 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: a9ba196a1c63c760847891f993818f97f5ce67860b66f2015c6a9d351e380b01 - candidate record SHA-256: ec4609adfafad4fe412ff18b219356b2baf00225c4bc9468dc8f172c7fdc6bce - 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-0075.json)