# Question 11327: A Row Initial Law Advances by mu*P [Return to catalogue page 57](https://ocme.evemisslab.com/en/questions/page/57/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r03-finite-markov-chain-foundations/#q-20261013-advance-20261024-r03-finite-markov-chain-0027) 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. R0021-11327 · q-20261013-advance-20261024-r03-finite-markov-chain-0027 ## Original question and givens Use one fixed finite time-homogeneous transition table=[[1/2, 1/2], [3/5, 2/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[2/3, 1/3] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j]. - G001 — Use one fixed finite time-homogeneous transition table=[[1/2, 1/2], [3/5, 2/5]]. Rows denote the current state and columns the next state; row entries must be nonnegative and sum exactly to 1 before stepping. State order stays fixed. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. No stationary, irreducibility, ergodicity or long-term convergence claim follows. Require a qualified kernel and initial law=[2/3, 1/3] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j]. ## 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 transition row0:raw rational pairs=[[1, 2], [1, 2]] After transition row0:normalized=[1/2, 1/2] Conditions: none Every original denominator must be nonzero.Reduce fractions to positive denominators without decimal approximation. Step identifiers and dependenciesRule: exact_weight_normalization · Uses: G001 S002 · calculationBefore transition row0 total := sum(1/2, 1/2) After transition row0 total = 1 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore transition row0:weights=[1/2, 1/2];sum=1 After nonnegative=Yes;total_one=Yes;qualified=Yes Conditions: none Check nonnegativity and exact total one separately.Zero weights are allowed;negative weights or near-one totals are not repaired. Step identifiers and dependenciesRule: separate_weight_guards · Uses: G001, S002 S004 · impliesBefore transition row1:raw rational pairs=[[3, 5], [2, 5]] After transition row1:normalized=[3/5, 2/5] Conditions: none Every original denominator must be nonzero.Reduce fractions to positive denominators without decimal approximation. Step identifiers and dependenciesRule: exact_weight_normalization · Uses: G001, S003 S005 · calculationBefore transition row1 total := sum(3/5, 2/5) After transition row1 total = 1 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore transition row1:weights=[3/5, 2/5];sum=1 After nonnegative=Yes;total_one=Yes;qualified=Yes Conditions: none Check nonnegativity and exact total one separately.Zero weights are allowed;negative weights or near-one totals are not repaired. Step identifiers and dependenciesRule: separate_weight_guards · Uses: G001, S005 S007 · impliesBefore complete transition matrix=[[1/2, 1/2], [3/5, 2/5]];row sums=[1, 1] After nonnegative=Yes;row_stochastic=Yes;rows=current state, columns=next state Conditions: none Every complete row must be nonnegative with exact sum1;column sums are not the convention.This fixed time-homogeneous Markov model gives no stationary,ergodic or long-term convergence theorem. Step identifiers and dependenciesRule: whole_row_stochastic_kernel_convention · Uses: G001, S006 S008 · impliesBefore initial law:raw rational pairs=[[2, 3], [1, 3]] After initial law:normalized=[2/3, 1/3] Conditions: none Every original denominator must be nonzero.Reduce fractions to positive denominators without decimal approximation. Step identifiers and dependenciesRule: exact_weight_normalization · Uses: G001, S007 S009 · calculationBefore initial law total := sum(2/3, 1/3) After initial law total = 1 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore initial law:weights=[2/3, 1/3];sum=1 After nonnegative=Yes;total_one=Yes;qualified=Yes Conditions: none Check nonnegativity and exact total one separately.Zero weights are allowed;negative weights or near-one totals are not repaired. Step identifiers and dependenciesRule: separate_weight_guards · Uses: G001, S009 S011 · impliesBefore initial row law=[2/3, 1/3];transition=[[1/2, 1/2], [3/5, 2/5]] After next[j]=sum_i initial[i]*P[i][j] over every current state Conditions: none Partition by current state and use total probability.The initial row law multiplies P on the right;do not transpose or assume a uniform law. Step identifiers and dependenciesRule: markov_total_probability_current_to_next · Uses: G001, S010 S012 · calculationBefore next state0 current0 term := mul(2/3, 1/2) After next state0 current0 term = 1/3 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore next state0 current1 term := mul(1/3, 3/5) After next state0 current1 term = 1/5 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore next state0 total := sum(1/3, 1/5) After next state0 total = 8/15 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore next state1 current0 term := mul(2/3, 1/2) After next state1 current0 term = 1/3 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore next state1 current1 term := mul(1/3, 2/5) After next state1 current1 term = 2/15 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore next state1 total := sum(1/3, 2/15) After next state1 total = 7/15 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 ## Reverse checks - S017 · premise_and_result_audit — Recompute next state1 total with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute next state1 current1 term with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute next state1 current0 term with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute next state0 total with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute next state0 current1 term with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute next state0 current0 term with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute initial law total with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute transition row1 total with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute transition row0 total with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([2/3, 1/3], [[1/2, 1/2], [3/5, 2/5]], [8/15, 7/15]) 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: 6b8da03841b87345951d1ec211ab74b385922042672b3d3333b7d087aa789969 - candidate record SHA-256: 995441152ee93e11a0f81a0585aca3a2eeed3ccbd3bde7eef3132e8d8bed9bc6 - semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258 - question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r03-finite-markov-chain-0027.json)