{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0001","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e0ebfae3fc679a648efefc29f3261cf217ee3bcf0a111420f8ffe71852151eb","family_id":"qf-mod24-markov-kernel-audit","seed":202610243001,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[1,3],[2,3]],[[1,2],[1,2]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[1/3, 2/3], [1/2, 1/2]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[1/3, 2/3], [1/2, 1/2]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0002","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30a0eab2544479f1e559df0e2866687bd83aa12afd74d9243cf3010f4f7eece0","family_id":"qf-mod24-markov-kernel-audit","seed":202610243002,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[1/2, 1/2], [3/5, 2/5]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[1/2, 1/2], [3/5, 2/5]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0003","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3b7f7ead25cab9a938c0cf4bb120c64ab4f3f52bf7649e3606ee273e1bedf3c","family_id":"qf-mod24-markov-kernel-audit","seed":202610243003,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[3/5, 2/5], [2/3, 1/3]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[3/5, 2/5], [2/3, 1/3]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[3/5, 2/5], [2/3, 1/3]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0004","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6670bfc0db9d9c5e0935353656bdc5312388b0b5a0c9c1b7a42345f1fc3abdc1","family_id":"qf-mod24-markov-kernel-audit","seed":202610243004,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[2/3, 1/3], [5/7, 2/7]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[2/3, 1/3], [5/7, 2/7]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[2/3, 1/3], [5/7, 2/7]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0005","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e137864192d8d213cb3046f21e535c334074b17a5569b5e70be4767112a6a84","family_id":"qf-mod24-markov-kernel-audit","seed":202610243005,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[5,7],[2,7]],[[3,4],[1,4]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[5/7, 2/7], [3/4, 1/4]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[5/7, 2/7], [3/4, 1/4]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[5/7, 2/7], [3/4, 1/4]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0006","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c6d83ad91462ed79c4d3459330cee69906e290155b3f02210d23a667ca93c6e","family_id":"qf-mod24-markov-kernel-audit","seed":202610243006,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[3/4, 1/4], [7/9, 2/9]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[3/4, 1/4], [7/9, 2/9]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0007","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20527d4f0882aac1a8a91c413480bc5c5dbd04cedf425561ab7ae987454eeb3a","family_id":"qf-mod24-markov-kernel-audit","seed":202610243007,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[7/9, 2/9], [4/5, 1/5]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[7/9, 2/9], [4/5, 1/5]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[7/9, 2/9], [4/5, 1/5]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0008","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a888cd0623c8594f8a87751c0d9e9bc2beae54ba53f5c99dd6b959add6f96f17","family_id":"qf-mod24-markov-kernel-audit","seed":202610243008,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[4/5, 1/5], [9/11, 2/11]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[4/5, 1/5], [9/11, 2/11]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[4/5, 1/5], [9/11, 2/11]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0009","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c08773db6f4fa87b676898c5de68a4e19a61dc7b29380902cd12a18f1dfc9d50","family_id":"qf-mod24-markov-kernel-audit","seed":202610243009,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[9/11, 2/11], [5/6, 1/6]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[9/11, 2/11], [5/6, 1/6]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[9/11, 2/11], [5/6, 1/6]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0010","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02c2d9fe2f5acd5877c48fb1f17a38f09314b9f623c57920d2cc455836bed137","family_id":"qf-mod24-markov-kernel-audit","seed":202610243010,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[5/6, 1/6], [11/13, 2/13]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[5/6, 1/6], [11/13, 2/13]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[5/6, 1/6], [11/13, 2/13]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0011","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24e625cae45ccb56ce3e2edc691ba3a745141d2f2ae94be4c2f153b18c863f3c","family_id":"qf-mod24-markov-kernel-audit","seed":202610243011,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[11/13, 2/13], [6/7, 1/7]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[11/13, 2/13], [6/7, 1/7]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0012","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1596d2e4e0a345eeaad4f1445ee3a522c388ef7457327000f7a4b160cfefb707","family_id":"qf-mod24-markov-kernel-audit","seed":202610243012,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[6/7, 1/7], [13/15, 2/15]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[6/7, 1/7], [13/15, 2/15]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[6/7, 1/7], [13/15, 2/15]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0013","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f575c5b79f62ec932fe1f14941b89d162a51d680e0c2421ae7aab8801b1daa57","family_id":"qf-mod24-markov-kernel-audit","seed":202610243013,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[13/15, 2/15], [7/8, 1/8]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[13/15, 2/15], [7/8, 1/8]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0014","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2150385594147a3440b2b1cd0191af53b65dd3688c68c0012f6d5446c0a26627","family_id":"qf-mod24-markov-kernel-audit","seed":202610243014,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[7/8, 1/8], [15/17, 2/17]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[7/8, 1/8], [15/17, 2/17]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[7/8, 1/8], [15/17, 2/17]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0015","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92b70f57ad6a885e21c6c9ae32fa106ec8df3a84562411fc6c9d432f7d5fd36a","family_id":"qf-mod24-markov-kernel-audit","seed":202610243015,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[15/17, 2/17], [8/9, 1/9]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[15/17, 2/17], [8/9, 1/9]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[15/17, 2/17], [8/9, 1/9]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0016","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63eff3785802a0e1f9fd252ee626b0ef5aa5e0bbea43763a6d51a81cdb330ee0","family_id":"qf-mod24-markov-kernel-audit","seed":202610243016,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[8/9, 1/9], [17/19, 2/19]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[8/9, 1/9], [17/19, 2/19]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[8/9, 1/9], [17/19, 2/19]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0017","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45c53d192defd47f52f60c2bac5f27912c575319223961d5e5f361a610fd8a9f","family_id":"qf-mod24-markov-kernel-audit","seed":202610243017,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[17/19, 2/19], [9/10, 1/10]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[17/19, 2/19], [9/10, 1/10]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[17/19, 2/19], [9/10, 1/10]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0018","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76b1b61bbe725b2f8597a81c1d87b07b97cb93fe72722cd38b5af7222a1ecca7","family_id":"qf-mod24-markov-kernel-audit","seed":202610243018,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[9/10, 1/10], [19/21, 2/21]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[9/10, 1/10], [19/21, 2/21]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[9/10, 1/10], [19/21, 2/21]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0019","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"900d9c2a63f4627af16d4735993978199edba1b50fc294d2d9daa33d2e3362ba","family_id":"qf-mod24-markov-kernel-audit","seed":202610243019,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[19/21, 2/21], [10/11, 1/11]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[19/21, 2/21], [10/11, 1/11]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0020","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2ad0af8fdae2421d0a189fdfbd68ff163b221d7826765cfe8c5698f7370e692","family_id":"qf-mod24-markov-kernel-audit","seed":202610243020,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[10/11, 1/11], [21/23, 2/23]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[10/11, 1/11], [21/23, 2/23]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[10/11, 1/11], [21/23, 2/23]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0021","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a4c9b10efc1fae24ad594da1449465a468e71d04ab3bd76366bc591cb1a72d3","family_id":"qf-mod24-markov-kernel-audit","seed":202610243021,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[21/23, 2/23], [11/12, 1/12]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[21/23, 2/23], [11/12, 1/12]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0022","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37e23c524d95fe30bb8296a195635a351bc3871c856a71558c6e773f5b707075","family_id":"qf-mod24-markov-kernel-audit","seed":202610243022,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[11/12, 1/12], [23/25, 2/25]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[11/12, 1/12], [23/25, 2/25]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[11/12, 1/12], [23/25, 2/25]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0023","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a212e3e7e4613c8ddb24259dddf22a9fd3c20af364422b94afb13c170c24e43b","family_id":"qf-mod24-markov-kernel-audit","seed":202610243023,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[23/25, 2/25], [12/13, 1/13]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[23/25, 2/25], [12/13, 1/13]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0024","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68cb5f60b1cb5c0cbdf9438e6d31250d32c8e977e138406f32430abc67ba3d59","family_id":"qf-mod24-markov-kernel-audit","seed":202610243024,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[12/13, 1/13], [25/27, 2/27]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[12/13, 1/13], [25/27, 2/27]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[12/13, 1/13], [25/27, 2/27]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0025","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f2c3ac8fac302c382f4281aa554688bc5b054fb537169a4e27cffad112152ca","family_id":"qf-mod24-markov-kernel-audit","seed":202610243025,"semantic_template":"qf-mod24-markov-kernel-audit/{\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}"},"fields":{"title":"Row-Stochastic Qualification before Any Markov Step","stem":"Use one fixed finite time-homogeneous transition table=[[25/27, 2/27], [13/14, 1/14]]. 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. This qualification family also admits negative or non-unit-row candidate entries for rejection. Give (size,normalized transition table,all entries nonnegative?,exact row sums,row-stochastic?,rows=current state, columns=next state).","answer":{"kind":"expression","value":"(2, [[25/27, 2/27], [13/14, 1/14]], Yes, [1, 1], Yes, rows=current state, columns=next state)","explanation":"Check rows rather than columns, and keep nonnegativity separate from total-one checks. A kernel alone does not specify an initial law. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (2, [[25/27, 2/27], [13/14, 1/14]], Yes, [1, 1], Yes, rows=current state, columns=next state)."},"skills":["Row-Stochastic Qualification before Any Markov Step"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0026","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aa539b088c3fc04135e6cbe85e292184c87f52733964ab71c33616e7ad5ce87","family_id":"qf-mod24-markov-one-step-law","seed":202610243026,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[1,2],[1,2]],\"transition\":[[[1,3],[2,3]],[[1,2],[1,2]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. 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=[1/2, 1/2] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([1/2, 1/2], [[1/3, 2/3], [1/2, 1/2]], [5/12, 7/12])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1/2, 1/2], [[1/3, 2/3], [1/2, 1/2]], [5/12, 7/12])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0027","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b8da03841b87345951d1ec211ab74b385922042672b3d3333b7d087aa789969","family_id":"qf-mod24-markov-one-step-law","seed":202610243027,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[2,3],[1,3]],\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"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].","answer":{"kind":"expression","value":"([2/3, 1/3], [[1/2, 1/2], [3/5, 2/5]], [8/15, 7/15])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2/3, 1/3], [[1/2, 1/2], [3/5, 2/5]], [8/15, 7/15])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0028","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"360ccef7cac64b7a1ffbee9cf4292ea25b7fe71d7c00d434082e3bf9c7337667","family_id":"qf-mod24-markov-one-step-law","seed":202610243028,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[3,4],[1,4]],\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[3/5, 2/5], [2/3, 1/3]]. 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=[3/4, 1/4] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([3/4, 1/4], [[3/5, 2/5], [2/3, 1/3]], [37/60, 23/60])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3/4, 1/4], [[3/5, 2/5], [2/3, 1/3]], [37/60, 23/60])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0029","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbf32fcbb46286cb0f8b2a84eaccf6648c12bde28c22d68bc5466b761cf682b6","family_id":"qf-mod24-markov-one-step-law","seed":202610243029,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[4,5],[1,5]],\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[2/3, 1/3], [5/7, 2/7]]. 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 a qualified kernel and initial law=[4/5, 1/5] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([4/5, 1/5], [[2/3, 1/3], [5/7, 2/7]], [71/105, 34/105])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4/5, 1/5], [[2/3, 1/3], [5/7, 2/7]], [71/105, 34/105])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0030","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e40c0724af63f5b662dd3a0d85be93da12cde5f205b4c00108224d2837bda0c9","family_id":"qf-mod24-markov-one-step-law","seed":202610243030,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[5,6],[1,6]],\"transition\":[[[5,7],[2,7]],[[3,4],[1,4]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[5/7, 2/7], [3/4, 1/4]]. 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=[5/6, 1/6] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([5/6, 1/6], [[5/7, 2/7], [3/4, 1/4]], [121/168, 47/168])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5/6, 1/6], [[5/7, 2/7], [3/4, 1/4]], [121/168, 47/168])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0031","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9aaa8aa7fd3f3eef02579abe1056ec498d8f518828ec9ae1e98a92b7ffe8573","family_id":"qf-mod24-markov-one-step-law","seed":202610243031,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[6,7],[1,7]],\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. 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=[6/7, 1/7] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([6/7, 1/7], [[3/4, 1/4], [7/9, 2/9]], [95/126, 31/126])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6/7, 1/7], [[3/4, 1/4], [7/9, 2/9]], [95/126, 31/126])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0032","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1659490eae5a9b24c341da5382518a1cc96cda2aa78e6880c5af6d89a905de0d","family_id":"qf-mod24-markov-one-step-law","seed":202610243032,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[7,8],[1,8]],\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[7/9, 2/9], [4/5, 1/5]]. 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=[7/8, 1/8] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([7/8, 1/8], [[7/9, 2/9], [4/5, 1/5]], [281/360, 79/360])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7/8, 1/8], [[7/9, 2/9], [4/5, 1/5]], [281/360, 79/360])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0033","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07d6b208417419dd924a8073ce4c14981e41b2cc96913fd76d7f22d07acbae27","family_id":"qf-mod24-markov-one-step-law","seed":202610243033,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[8,9],[1,9]],\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[4/5, 1/5], [9/11, 2/11]]. 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=[8/9, 1/9] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([8/9, 1/9], [[4/5, 1/5], [9/11, 2/11]], [397/495, 98/495])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8/9, 1/9], [[4/5, 1/5], [9/11, 2/11]], [397/495, 98/495])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0034","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"087dcb27f2ecc78351cd31657dffb078094ba0847b30581e8fe39e78de0d1566","family_id":"qf-mod24-markov-one-step-law","seed":202610243034,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[9,10],[1,10]],\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[9/11, 2/11], [5/6, 1/6]]. 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=[9/10, 1/10] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([9/10, 1/10], [[9/11, 2/11], [5/6, 1/6]], [541/660, 119/660])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9/10, 1/10], [[9/11, 2/11], [5/6, 1/6]], [541/660, 119/660])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0035","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ed93e902ae48c6ad24e141f1c2df677bd8dcdd7ac6a2ca74d947eee37e5db70","family_id":"qf-mod24-markov-one-step-law","seed":202610243035,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[10,11],[1,11]],\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[5/6, 1/6], [11/13, 2/13]]. 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=[10/11, 1/11] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([10/11, 1/11], [[5/6, 1/6], [11/13, 2/13]], [358/429, 71/429])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10/11, 1/11], [[5/6, 1/6], [11/13, 2/13]], [358/429, 71/429])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0036","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c171502e7f2f6fc1b331ee7a79af687249a975a472cc604ebe10b17964b2b442","family_id":"qf-mod24-markov-one-step-law","seed":202610243036,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[11,12],[1,12]],\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"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 a qualified kernel and initial law=[11/12, 1/12] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([11/12, 1/12], [[11/13, 2/13], [6/7, 1/7]], [925/1092, 167/1092])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11/12, 1/12], [[11/13, 2/13], [6/7, 1/7]], [925/1092, 167/1092])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0037","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67569b8225ec8f153a7ee0b028d55c2f5f0235765282d04c071682b6569577e9","family_id":"qf-mod24-markov-one-step-law","seed":202610243037,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[12,13],[1,13]],\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[6/7, 1/7], [13/15, 2/15]]. 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=[12/13, 1/13] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([12/13, 1/13], [[6/7, 1/7], [13/15, 2/15]], [1171/1365, 194/1365])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12/13, 1/13], [[6/7, 1/7], [13/15, 2/15]], [1171/1365, 194/1365])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0038","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba43768bb7217b079ea9b097c462549f4baf6bbb8668c13b5fbd4efe3a5ba90e","family_id":"qf-mod24-markov-one-step-law","seed":202610243038,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[13,14],[1,14]],\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. 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=[13/14, 1/14] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([13/14, 1/14], [[13/15, 2/15], [7/8, 1/8]], [1457/1680, 223/1680])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([13/14, 1/14], [[13/15, 2/15], [7/8, 1/8]], [1457/1680, 223/1680])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0039","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9f7126b7969f705961368158411429a703abb7b63d240f98f9b48805a719ab9","family_id":"qf-mod24-markov-one-step-law","seed":202610243039,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[14,15],[1,15]],\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[7/8, 1/8], [15/17, 2/17]]. 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=[14/15, 1/15] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([14/15, 1/15], [[7/8, 1/8], [15/17, 2/17]], [893/1020, 127/1020])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([14/15, 1/15], [[7/8, 1/8], [15/17, 2/17]], [893/1020, 127/1020])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0040","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"980874391226d5c0128731674ad2c64c50f9df4ad4a4d123ffd3bf8a027030df","family_id":"qf-mod24-markov-one-step-law","seed":202610243040,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[15,16],[1,16]],\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[15/17, 2/17], [8/9, 1/9]]. 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=[15/16, 1/16] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([15/16, 1/16], [[15/17, 2/17], [8/9, 1/9]], [2161/2448, 287/2448])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([15/16, 1/16], [[15/17, 2/17], [8/9, 1/9]], [2161/2448, 287/2448])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0041","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbe0e10db6b348ee4e07a882a3c7779060fbcb839c21dc6506a9d2ac834a67dc","family_id":"qf-mod24-markov-one-step-law","seed":202610243041,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[16,17],[1,17]],\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[8/9, 1/9], [17/19, 2/19]]. 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=[16/17, 1/17] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([16/17, 1/17], [[8/9, 1/9], [17/19, 2/19]], [2585/2907, 322/2907])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([16/17, 1/17], [[8/9, 1/9], [17/19, 2/19]], [2585/2907, 322/2907])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0042","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"252d8ef387f909def048ba366c91657f2cdd619d164255842580b0507d006fa0","family_id":"qf-mod24-markov-one-step-law","seed":202610243042,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[17,18],[1,18]],\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[17/19, 2/19], [9/10, 1/10]]. 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=[17/18, 1/18] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([17/18, 1/18], [[17/19, 2/19], [9/10, 1/10]], [3061/3420, 359/3420])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([17/18, 1/18], [[17/19, 2/19], [9/10, 1/10]], [3061/3420, 359/3420])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0043","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05c3ad8fe5876cd8c565b3a8b22b5ebcab730c548d63eb9d36643488bc7e8035","family_id":"qf-mod24-markov-one-step-law","seed":202610243043,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[18,19],[1,19]],\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[9/10, 1/10], [19/21, 2/21]]. 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=[18/19, 1/19] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([18/19, 1/19], [[9/10, 1/10], [19/21, 2/21]], [1796/1995, 199/1995])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([18/19, 1/19], [[9/10, 1/10], [19/21, 2/21]], [1796/1995, 199/1995])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0044","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33cd219f5c638ef6572bba3138f8efab69055a2ae014c32e769c11a6d5bd627d","family_id":"qf-mod24-markov-one-step-law","seed":202610243044,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[19,20],[1,20]],\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. 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=[19/20, 1/20] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([19/20, 1/20], [[19/21, 2/21], [10/11, 1/11]], [4181/4620, 439/4620])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([19/20, 1/20], [[19/21, 2/21], [10/11, 1/11]], [4181/4620, 439/4620])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0045","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"019c1307b31b7f02c9927b3b4997df5c37657017651586f1e3532c3bb0ece9c5","family_id":"qf-mod24-markov-one-step-law","seed":202610243045,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[20,21],[1,21]],\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[10/11, 1/11], [21/23, 2/23]]. 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=[20/21, 1/21] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([20/21, 1/21], [[10/11, 1/11], [21/23, 2/23]], [4831/5313, 482/5313])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([20/21, 1/21], [[10/11, 1/11], [21/23, 2/23]], [4831/5313, 482/5313])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0046","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebd4977a7273c5f5aca745d2c72abce471baf3f0a2bdd84b34809fac15073d60","family_id":"qf-mod24-markov-one-step-law","seed":202610243046,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[21,22],[1,22]],\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. 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=[21/22, 1/22] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([21/22, 1/22], [[21/23, 2/23], [11/12, 1/12]], [5545/6072, 527/6072])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([21/22, 1/22], [[21/23, 2/23], [11/12, 1/12]], [5545/6072, 527/6072])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0047","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72194460f49db5171edfd6d5dbb3858aeabfdeacd1d8ff2bf757470affd7a2ba","family_id":"qf-mod24-markov-one-step-law","seed":202610243047,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[22,23],[1,23]],\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[11/12, 1/12], [23/25, 2/25]]. 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=[22/23, 1/23] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([22/23, 1/23], [[11/12, 1/12], [23/25, 2/25]], [3163/3450, 287/3450])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([22/23, 1/23], [[11/12, 1/12], [23/25, 2/25]], [3163/3450, 287/3450])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0048","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8da1953122db495e60a5afdfa4e0d5c266285bd5031ff53a3cb8f9326f15d474","family_id":"qf-mod24-markov-one-step-law","seed":202610243048,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[23,24],[1,24]],\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. 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=[23/24, 1/24] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([23/24, 1/24], [[23/25, 2/25], [12/13, 1/13]], [7177/7800, 623/7800])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([23/24, 1/24], [[23/25, 2/25], [12/13, 1/13]], [7177/7800, 623/7800])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0049","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f2202c21afa3b04d60e990b07d073a1748bf210178fe0a2d4bd41b03b44eefc","family_id":"qf-mod24-markov-one-step-law","seed":202610243049,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[24,25],[1,25]],\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[12/13, 1/13], [25/27, 2/27]]. 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=[24/25, 1/25] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([24/25, 1/25], [[12/13, 1/13], [25/27, 2/27]], [8101/8775, 674/8775])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([24/25, 1/25], [[12/13, 1/13], [25/27, 2/27]], [8101/8775, 674/8775])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0050","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0be635f770af2b68a6bef8aa37bcd9b99eb7bc5bd181aa6dcff71d85f2238e3","family_id":"qf-mod24-markov-one-step-law","seed":202610243050,"semantic_template":"qf-mod24-markov-one-step-law/{\"initial\":[[25,26],[1,26]],\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}"},"fields":{"title":"A Row Initial Law Advances by mu*P","stem":"Use one fixed finite time-homogeneous transition table=[[25/27, 2/27], [13/14, 1/14]]. 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=[25/26, 1/26] with nonnegative entries and exact total 1. Give (initial law,transition table,next law), where next[j]=sum_i initial[i]*P[i][j].","answer":{"kind":"expression","value":"([25/26, 1/26], [[25/27, 2/27], [13/14, 1/14]], [9101/9828, 727/9828])","explanation":"Multiply the row-vector initial law on the right by P. Do not transpose the transition convention or infer a law without initial data. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([25/26, 1/26], [[25/27, 2/27], [13/14, 1/14]], [9101/9828, 727/9828])."},"skills":["A Row Initial Law Advances by mu*P"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0051","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afdc68b2ca949731842c2182c855275cd7e2a47ff90ca89a12501c9e1bc0bb98","family_id":"qf-mod24-markov-two-step-entry","seed":202610243051,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[1,3],[2,3]],[[1,2],[1,2]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [2/9, 1/3], 5/9)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [2/9, 1/3], 5/9)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0052","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0c8a7b1dd54e69cf5c95e0243e533858ccde60cc06998f4572a6f410c707619","family_id":"qf-mod24-markov-two-step-entry","seed":202610243052,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [3/10, 6/25], 27/50)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [3/10, 6/25], 27/50)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0053","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e8e19f955e0c5461aa5f3d78053dffb3bed3b76f057378e1b3cd995365236eb","family_id":"qf-mod24-markov-two-step-entry","seed":202610243053,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[3/5, 2/5], [2/3, 1/3]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [6/25, 2/15], 28/75)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [6/25, 2/15], 28/75)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0054","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb7afa5f683fdd1a7526f35f2b0544dea76f2f076f3365d1a6a8ddcc9f4fcd38","family_id":"qf-mod24-markov-two-step-entry","seed":202610243054,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[2/3, 1/3], [5/7, 2/7]]. 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 a qualified kernel. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [10/21, 10/49], 100/147)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [10/21, 10/49], 100/147)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0055","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3e67f6dd381dd7534051806259156bfedbca183457cfa45a5d88f4f9080f989","family_id":"qf-mod24-markov-two-step-entry","seed":202610243055,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[5,7],[2,7]],[[3,4],[1,4]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[5/7, 2/7], [3/4, 1/4]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [10/49, 1/14], 27/98)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [10/49, 1/14], 27/98)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0056","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e148c2e5af2ff3ef072bb58cfeefc7eae41fcc6838931cc48116985013a928c0","family_id":"qf-mod24-markov-two-step-entry","seed":202610243056,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [7/12, 14/81], 245/324)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [7/12, 14/81], 245/324)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0057","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc1038982b48f0425261e7b571843f86ca337fd053638babe64fd3e7a0b1d67d","family_id":"qf-mod24-markov-two-step-entry","seed":202610243057,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[7/9, 2/9], [4/5, 1/5]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [14/81, 2/45], 88/405)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [14/81, 2/45], 88/405)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0058","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68ea9b9aaea671521293a68c0ddbc99677ef3abca2f6244e67cd9607d3fded8a","family_id":"qf-mod24-markov-two-step-entry","seed":202610243058,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[4/5, 1/5], [9/11, 2/11]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [36/55, 18/121], 486/605)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [36/55, 18/121], 486/605)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0059","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"519857f3966498816c582c23168623d19966b9641571f01e970b1adb1ac792d2","family_id":"qf-mod24-markov-two-step-entry","seed":202610243059,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[9/11, 2/11], [5/6, 1/6]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [18/121, 1/33], 65/363)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [18/121, 1/33], 65/363)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0060","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df90bafd8476578ecaec648e3426bf96fbac65e47cd6cb6dcc919d2cd01a002b","family_id":"qf-mod24-markov-two-step-entry","seed":202610243060,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[5/6, 1/6], [11/13, 2/13]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [55/78, 22/169], 847/1014)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [55/78, 22/169], 847/1014)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0061","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e437538aa24a457326a4817d4ec894808f5f714b74f96a055412d3283afdeee","family_id":"qf-mod24-markov-two-step-entry","seed":202610243061,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"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 a qualified kernel. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [22/169, 2/91], 180/1183)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [22/169, 2/91], 180/1183)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0062","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d91506622254a02f828b9cdc3b50e6d932ce074155f5723f0d47913407dc8338","family_id":"qf-mod24-markov-two-step-entry","seed":202610243062,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[6/7, 1/7], [13/15, 2/15]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [26/35, 26/225], 1352/1575)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [26/35, 26/225], 1352/1575)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0063","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5076cffe07cd1cf197e5e85e544cc3556a484634432be668f78d7e19683518bf","family_id":"qf-mod24-markov-two-step-entry","seed":202610243063,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [26/225, 1/60], 119/900)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [26/225, 1/60], 119/900)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0064","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30c6885d64f05ab63ebea70abd09089b65cf4da9142ae8e5cf55ec2045fc67c5","family_id":"qf-mod24-markov-two-step-entry","seed":202610243064,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[7/8, 1/8], [15/17, 2/17]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [105/136, 30/289], 2025/2312)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [105/136, 30/289], 2025/2312)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0065","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19bcaac6657808239308083336477bbec59a24d21f9a8a8bb01acca01e6148a1","family_id":"qf-mod24-markov-two-step-entry","seed":202610243065,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[15/17, 2/17], [8/9, 1/9]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [30/289, 2/153], 304/2601)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [30/289, 2/153], 304/2601)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0066","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2863e1e869264f13613ac2cc0647429060581c21227638acb1b568ac18a5e0ce","family_id":"qf-mod24-markov-two-step-entry","seed":202610243066,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[8/9, 1/9], [17/19, 2/19]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [136/171, 34/361], 2890/3249)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [136/171, 34/361], 2890/3249)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0067","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"643112113867d76fdbcdfd9d45203c90e0d04a6c6a2115261dbac931b8fd5c4d","family_id":"qf-mod24-markov-two-step-entry","seed":202610243067,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[17/19, 2/19], [9/10, 1/10]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [34/361, 1/95], 189/1805)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [34/361, 1/95], 189/1805)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0068","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81963d92eef4ae0331ac397c6a3193d6ba4dceb6cae52d401c0b0707e68d792b","family_id":"qf-mod24-markov-two-step-entry","seed":202610243068,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[9/10, 1/10], [19/21, 2/21]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [57/70, 38/441], 3971/4410)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [57/70, 38/441], 3971/4410)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0069","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"376fdebb690f9192e55c06e446ea2f3533e0b470c23a0afb4f890a78ae7b9388","family_id":"qf-mod24-markov-two-step-entry","seed":202610243069,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [38/441, 2/231], 460/4851)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [38/441, 2/231], 460/4851)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0070","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c097da362dc6a591f9af0e143778af8af965cec04b9efe85f0b434ada05b935a","family_id":"qf-mod24-markov-two-step-entry","seed":202610243070,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[10/11, 1/11], [21/23, 2/23]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [210/253, 42/529], 5292/5819)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [210/253, 42/529], 5292/5819)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0071","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38f2be5f1eaed47a28a3a349885b7cd64c44e5dfef318959a10ec812e4545ab3","family_id":"qf-mod24-markov-two-step-entry","seed":202610243071,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [42/529, 1/138], 275/3174)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [42/529, 1/138], 275/3174)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0072","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85bbe261331ac22d1a56d3126b1a114f9b7e31065a341e5fcaf4b16b3ab31aa3","family_id":"qf-mod24-markov-two-step-entry","seed":202610243072,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[11/12, 1/12], [23/25, 2/25]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [253/300, 46/625], 6877/7500)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [253/300, 46/625], 6877/7500)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0073","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1277878afad536a88690fa591487c8a0d047704618b33bf5fcb58037c3263847","family_id":"qf-mod24-markov-two-step-entry","seed":202610243073,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [46/625, 2/325], 648/8125)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [46/625, 2/325], 648/8125)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0074","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10f0cc63b20e5ea5cfdb7cd7096b1b2a9b38f1b8486ff5c8eb57f2803f73c108","family_id":"qf-mod24-markov-two-step-entry","seed":202610243074,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":1,\"to\":0,\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[12/13, 1/13], [25/27, 2/27]]. 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. from=1, to=0. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(1, 0, [100/117, 50/729], 8750/9477)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (1, 0, [100/117, 50/729], 8750/9477)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0075","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"582e76542f3aba69002222257b8759c67647194ca6b31dc2cec9c3f591dc781f","family_id":"qf-mod24-markov-two-step-entry","seed":202610243075,"semantic_template":"qf-mod24-markov-two-step-entry/{\"from\":0,\"to\":1,\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}"},"fields":{"title":"Two-Step Probability by Summing Every Intermediate State","stem":"Use one fixed finite time-homogeneous transition table=[[25/27, 2/27], [13/14, 1/14]]. 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. from=0, to=1. Give (from,to,every P[from][middle]*P[middle][to] term in middle-state order,sum). This conditional transition entry does not require an initial law.","answer":{"kind":"expression","value":"(0, 1, [50/729, 1/189], 377/5103)","explanation":"Sum over all intermediate states using exact products, rather than multiplying two unrelated marginal probabilities. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [50/729, 1/189], 377/5103)."},"skills":["Two-Step Probability by Summing Every Intermediate State"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0076","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2265ebb934330ad84ac3a062e3cc8517e5f16deee088215e1eacbba58bf598d9","family_id":"qf-mod24-markov-path-probability","seed":202610243076,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[1,2],[1,2]],\"path\":[0,1,0],\"transition\":[[[1,3],[2,3]],[[1,2],[1,2]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[1/3, 2/3], [1/2, 1/2]]. 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=[1/2, 1/2] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [1/2, 2/3, 1/2], 1/6)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [1/2, 2/3, 1/2], 1/6)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0077","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"984ac69a1ca7d33e5ef1a77d1fded9e8151fe0e25c14d5625c5d674442ddfbb1","family_id":"qf-mod24-markov-path-probability","seed":202610243077,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[2,3],[1,3]],\"path\":[1,0,1],\"transition\":[[[1,2],[1,2]],[[3,5],[2,5]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"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 initial law=[2/3, 1/3] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/3, 3/5, 1/2], 1/10)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/3, 3/5, 1/2], 1/10)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0078","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"212582490f710b9d339dee07f7ad78d89a716b3919cb6be9cea7133f7fdaecee","family_id":"qf-mod24-markov-path-probability","seed":202610243078,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[3,4],[1,4]],\"path\":[0,1,0],\"transition\":[[[3,5],[2,5]],[[2,3],[1,3]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[3/5, 2/5], [2/3, 1/3]]. 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=[3/4, 1/4] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [3/4, 2/5, 2/3], 1/5)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [3/4, 2/5, 2/3], 1/5)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0079","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a22e2bfb0eb9d11a96547d72df92081748670bf59ce41b4ef06959df42b8daf2","family_id":"qf-mod24-markov-path-probability","seed":202610243079,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[4,5],[1,5]],\"path\":[1,0,1],\"transition\":[[[2,3],[1,3]],[[5,7],[2,7]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[2/3, 1/3], [5/7, 2/7]]. 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=[4/5, 1/5] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/5, 5/7, 1/3], 1/21)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/5, 5/7, 1/3], 1/21)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0080","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f095168b9b775a8de26407ba5b3f5589867eb946cc00313bf6eb6df17dd42909","family_id":"qf-mod24-markov-path-probability","seed":202610243080,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[5,6],[1,6]],\"path\":[0,1,0],\"transition\":[[[5,7],[2,7]],[[3,4],[1,4]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[5/7, 2/7], [3/4, 1/4]]. 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=[5/6, 1/6] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [5/6, 2/7, 3/4], 5/28)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [5/6, 2/7, 3/4], 5/28)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0081","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c788f70eda348a5a580b2d9db229e5a785d444abed06ae9aecf4c1955d7469b5","family_id":"qf-mod24-markov-path-probability","seed":202610243081,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[6,7],[1,7]],\"path\":[1,0,1],\"transition\":[[[3,4],[1,4]],[[7,9],[2,9]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[3/4, 1/4], [7/9, 2/9]]. 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=[6/7, 1/7] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/7, 7/9, 1/4], 1/36)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/7, 7/9, 1/4], 1/36)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0082","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a00fee459a9a095c208c2990994756561a3b6fb5413e5485bf021f141bdf3639","family_id":"qf-mod24-markov-path-probability","seed":202610243082,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[7,8],[1,8]],\"path\":[0,1,0],\"transition\":[[[7,9],[2,9]],[[4,5],[1,5]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[7/9, 2/9], [4/5, 1/5]]. 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 initial law=[7/8, 1/8] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [7/8, 2/9, 4/5], 7/45)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [7/8, 2/9, 4/5], 7/45)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0083","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08bdce974d20d0d64a7d1dbc5ed703a40f29ef72d5c86e63d77ed88f3970be7c","family_id":"qf-mod24-markov-path-probability","seed":202610243083,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[8,9],[1,9]],\"path\":[1,0,1],\"transition\":[[[4,5],[1,5]],[[9,11],[2,11]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[4/5, 1/5], [9/11, 2/11]]. 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=[8/9, 1/9] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/9, 9/11, 1/5], 1/55)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/9, 9/11, 1/5], 1/55)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0084","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b64a7ebd5c30329faf4b61a5ba291a8979fea5d5e400db4ee4b722e6cc10d2f9","family_id":"qf-mod24-markov-path-probability","seed":202610243084,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[9,10],[1,10]],\"path\":[0,1,0],\"transition\":[[[9,11],[2,11]],[[5,6],[1,6]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[9/11, 2/11], [5/6, 1/6]]. 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=[9/10, 1/10] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [9/10, 2/11, 5/6], 3/22)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [9/10, 2/11, 5/6], 3/22)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0085","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4657d8fdb85513cf7c61023aaa5c0c74ae4527f2eacd9ee46ff75ba834184f48","family_id":"qf-mod24-markov-path-probability","seed":202610243085,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[10,11],[1,11]],\"path\":[1,0,1],\"transition\":[[[5,6],[1,6]],[[11,13],[2,13]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[5/6, 1/6], [11/13, 2/13]]. 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=[10/11, 1/11] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/11, 11/13, 1/6], 1/78)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/11, 11/13, 1/6], 1/78)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0086","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d39ce46fe368c77cf6eb1ae87ffd20df1ff65d00f60328f052f31e6bc68941f","family_id":"qf-mod24-markov-path-probability","seed":202610243086,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[11,12],[1,12]],\"path\":[0,1,0],\"transition\":[[[11,13],[2,13]],[[6,7],[1,7]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"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.","answer":{"kind":"expression","value":"([0, 1, 0], [11/12, 2/13, 6/7], 11/91)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [11/12, 2/13, 6/7], 11/91)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0087","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a877571d6a851e54c56738400a0f540f0ec1fd770ae8f12d7e58f1a8853e5d61","family_id":"qf-mod24-markov-path-probability","seed":202610243087,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[12,13],[1,13]],\"path\":[1,0,1],\"transition\":[[[6,7],[1,7]],[[13,15],[2,15]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[6/7, 1/7], [13/15, 2/15]]. 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=[12/13, 1/13] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/13, 13/15, 1/7], 1/105)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/13, 13/15, 1/7], 1/105)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0088","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efacaea6e4f913fa3b0e6497da98057140f02eefc134fee12c39a5b62935fdf8","family_id":"qf-mod24-markov-path-probability","seed":202610243088,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[13,14],[1,14]],\"path\":[0,1,0],\"transition\":[[[13,15],[2,15]],[[7,8],[1,8]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[13/15, 2/15], [7/8, 1/8]]. 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=[13/14, 1/14] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [13/14, 2/15, 7/8], 13/120)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [13/14, 2/15, 7/8], 13/120)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0089","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a980b4ec4e8f2d960a16047f9a60dd265d8de17da69b108d5c6fa44b4bdc160","family_id":"qf-mod24-markov-path-probability","seed":202610243089,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[14,15],[1,15]],\"path\":[1,0,1],\"transition\":[[[7,8],[1,8]],[[15,17],[2,17]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[7/8, 1/8], [15/17, 2/17]]. 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=[14/15, 1/15] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/15, 15/17, 1/8], 1/136)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/15, 15/17, 1/8], 1/136)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0090","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d696890a1a31dc4ace8f46d16209c32f66403a7c78d21bd90e352a77ebcbe60","family_id":"qf-mod24-markov-path-probability","seed":202610243090,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[15,16],[1,16]],\"path\":[0,1,0],\"transition\":[[[15,17],[2,17]],[[8,9],[1,9]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[15/17, 2/17], [8/9, 1/9]]. 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=[15/16, 1/16] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [15/16, 2/17, 8/9], 5/51)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [15/16, 2/17, 8/9], 5/51)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0091","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"434a89e34345572d46242c15df084ab6c3651dd23f48f4c1893cca30e2ca1977","family_id":"qf-mod24-markov-path-probability","seed":202610243091,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[16,17],[1,17]],\"path\":[1,0,1],\"transition\":[[[8,9],[1,9]],[[17,19],[2,19]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[8/9, 1/9], [17/19, 2/19]]. 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=[16/17, 1/17] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/17, 17/19, 1/9], 1/171)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/17, 17/19, 1/9], 1/171)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0092","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ac3a2377be41f337eab0a7baa1b218ad9360a65ea6a35f93e921895c63c68d1","family_id":"qf-mod24-markov-path-probability","seed":202610243092,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[17,18],[1,18]],\"path\":[0,1,0],\"transition\":[[[17,19],[2,19]],[[9,10],[1,10]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[17/19, 2/19], [9/10, 1/10]]. 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=[17/18, 1/18] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [17/18, 2/19, 9/10], 17/190)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [17/18, 2/19, 9/10], 17/190)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0093","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"699ec70440fef5ed7a33e4153a3cf380b55b7716f9a49cbd8b2f280d0d6b93d8","family_id":"qf-mod24-markov-path-probability","seed":202610243093,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[18,19],[1,19]],\"path\":[1,0,1],\"transition\":[[[9,10],[1,10]],[[19,21],[2,21]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[9/10, 1/10], [19/21, 2/21]]. 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=[18/19, 1/19] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/19, 19/21, 1/10], 1/210)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/19, 19/21, 1/10], 1/210)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0094","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"244f440db093e9bbdc354862b1a78b2ac393450ca661dec377734af40c3e11d2","family_id":"qf-mod24-markov-path-probability","seed":202610243094,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[19,20],[1,20]],\"path\":[0,1,0],\"transition\":[[[19,21],[2,21]],[[10,11],[1,11]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[19/21, 2/21], [10/11, 1/11]]. 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=[19/20, 1/20] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [19/20, 2/21, 10/11], 19/231)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [19/20, 2/21, 10/11], 19/231)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0095","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"221f2587a5651a92104070088876f2abc7cceb84f787edfe803aac4045c54fee","family_id":"qf-mod24-markov-path-probability","seed":202610243095,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[20,21],[1,21]],\"path\":[1,0,1],\"transition\":[[[10,11],[1,11]],[[21,23],[2,23]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[10/11, 1/11], [21/23, 2/23]]. 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=[20/21, 1/21] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/21, 21/23, 1/11], 1/253)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/21, 21/23, 1/11], 1/253)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0096","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f09bae7d31b496837505a6fbb081ab0b4b0da98a3299e14bdb3b7c69765c32a","family_id":"qf-mod24-markov-path-probability","seed":202610243096,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[21,22],[1,22]],\"path\":[0,1,0],\"transition\":[[[21,23],[2,23]],[[11,12],[1,12]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[21/23, 2/23], [11/12, 1/12]]. 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=[21/22, 1/22] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [21/22, 2/23, 11/12], 7/92)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [21/22, 2/23, 11/12], 7/92)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0097","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"660cb88b45a3aad359be7db5d984847c576c5227e90b0ce3796eba51e7d47b17","family_id":"qf-mod24-markov-path-probability","seed":202610243097,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[22,23],[1,23]],\"path\":[1,0,1],\"transition\":[[[11,12],[1,12]],[[23,25],[2,25]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[11/12, 1/12], [23/25, 2/25]]. 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=[22/23, 1/23] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/23, 23/25, 1/12], 1/300)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/23, 23/25, 1/12], 1/300)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0098","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c142940e2a5ba179cf6516dd64d522973978d06aa5d0e9f9dc7564ec34feac43","family_id":"qf-mod24-markov-path-probability","seed":202610243098,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[23,24],[1,24]],\"path\":[0,1,0],\"transition\":[[[23,25],[2,25]],[[12,13],[1,13]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[23/25, 2/25], [12/13, 1/13]]. 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=[23/24, 1/24] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [23/24, 2/25, 12/13], 23/325)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [23/24, 2/25, 12/13], 23/325)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0099","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b01f0dbc30a591ccf4f4839a1fea61654c169d47053c1ff3ed339eda737e564","family_id":"qf-mod24-markov-path-probability","seed":202610243099,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[24,25],[1,25]],\"path\":[1,0,1],\"transition\":[[[12,13],[1,13]],[[25,27],[2,27]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[12/13, 1/13], [25/27, 2/27]]. 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=[24/25, 1/25] to be a qualified PMF. Nonempty path=[1, 0, 1]. Give (path,[initial mass,each successive transition factor],product). A one-state path has only its initial-mass factor.","answer":{"kind":"expression","value":"([1, 0, 1], [1/25, 25/27, 1/13], 1/351)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], [1/25, 25/27, 1/13], 1/351)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r03-finite-markov-chain-0100","batch_id":"batch-20261013-advance-20261024-r03-finite-markov-chain-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7206a08c4310a5be801e1d7f86ed240b45fb9571e1d64a874912b9d8a809bff1","family_id":"qf-mod24-markov-path-probability","seed":202610243100,"semantic_template":"qf-mod24-markov-path-probability/{\"initial\":[[25,26],[1,26]],\"path\":[0,1,0],\"transition\":[[[25,27],[2,27]],[[13,14],[1,14]]]}"},"fields":{"title":"A Finite Path Includes Its Initial-State Mass","stem":"Use one fixed finite time-homogeneous transition table=[[25/27, 2/27], [13/14, 1/14]]. 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=[25/26, 1/26] 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.","answer":{"kind":"expression","value":"([0, 1, 0], [25/26, 2/27, 13/14], 25/378)","explanation":"Multiply the initial-state mass by the transitions in their original path order. This is a finite-path result, not a long-term theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0], [25/26, 2/27, 13/14], 25/378)."},"skills":["A Finite Path Includes Its Initial-State Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}