# Question 11361: A Finite Path Includes Its Initial-State Mass [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-0086) 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. R0021-11361 · q-20261013-advance-20261024-r03-finite-markov-chain-0086 ## Original question and givens Use one fixed finite time-homogeneous transition table=[[11/13, 2/13], [6/7, 1/7]]. 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 initial law=[11/12, 1/12] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor. - G001 — Use one fixed finite time-homogeneous transition table=[[11/13, 2/13], [6/7, 1/7]]. 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 initial law=[11/12, 1/12] to be a qualified PMF. Nonempty path=[0, 1, 0]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor. ## 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=[[11, 13], [2, 13]] After transition row0:normalized=[11/13, 2/13] 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(11/13, 2/13) 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=[11/13, 2/13];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=[[6, 7], [1, 7]] After transition row1:normalized=[6/7, 1/7] 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(6/7, 1/7) 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=[6/7, 1/7];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=[[11/13, 2/13], [6/7, 1/7]];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=[[11, 12], [1, 12]] After initial law:normalized=[11/12, 1/12] 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(11/12, 1/12) 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=[11/12, 1/12];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 path=[0, 1, 0];initial law=[11/12, 1/12] After factor0=initial[0]=11/12 Conditions: none The path event includes its first state.Start with its initial mass;a one-state path has only this factor. Step identifiers and dependenciesRule: path_includes_initial_state_mass · Uses: G001, S010 S012 · impliesBefore path step=1;current=0;next=1 After factor1=P[0][1]=2/13 Conditions: none Read current row,next-state column in original path order.Zero probabilities may be product factors;do not divide by a zero-probability event. Step identifiers and dependenciesRule: path_ordered_transition_factor · Uses: G001, S011 S013 · impliesBefore path step=2;current=1;next=0 After factor2=P[1][0]=6/7 Conditions: none Read current row,next-state column in original path order.Zero probabilities may be product factors;do not divide by a zero-probability event. Step identifiers and dependenciesRule: path_ordered_transition_factor · Uses: G001, S012 S014 · calculationBefore path prefix product1 := mul(11/12, 2/13) After path prefix product1 = 11/78 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 path prefix product2 := mul(11/78, 6/7) After path prefix product2 = 11/91 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 · impliesBefore ordered path factors=[11/12, 2/13, 6/7] After finite path probability=11/91 Conditions: none Use this model's finite-path product:initial mass times every transition.This does not assign the Markov property to arbitrary stochastic processes. Step identifiers and dependenciesRule: finite_markov_path_joint_probability · Uses: G001, S015 ## Reverse checks - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute path prefix product2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute path prefix product1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 ([0, 1, 0], [11/12, 2/13, 6/7], 11/91) 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: 6d39ce46fe368c77cf6eb1ae87ffd20df1ff65d00f60328f052f31e6bc68941f - candidate record SHA-256: 54dd1ef5b8a43adc324f15407f36893859d770ea26e4242a91a559c0193274ae - 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-0086.json)