{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r06-discrete-stochastic-process-0001","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a295f3713c3d9492ed23fc49a267c0792d9fab01756b80091427cd77fdd7d9c","family_id":"qf-mod23-process-path-law-qualification","seed":202610236001,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[0, 0, 1], [0, 1, 1], [1, 1, 0]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0, 1], [0, 1, 1], [1, 1, 0]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0002","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b516c6af0893deecf02a5e6c4f739d7c236b59d881e0ba01f342ea64a340767","family_id":"qf-mod23-process-path-law-qualification","seed":202610236002,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[1,1,2],[1,2,2],[2,2,1]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[1, 1, 2], [1, 2, 2], [2, 2, 1]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[1, 1, 2], [1, 2, 2], [2, 2, 1]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1, 2], [1, 2, 2], [2, 2, 1]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0003","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4979185e9f3e6948c10290efb2b392ba7980089a62d9be6bf5f6a4eb4fb4219","family_id":"qf-mod23-process-path-law-qualification","seed":202610236003,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[2,2,3],[2,3,3],[3,3,2]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[2, 2, 3], [2, 3, 3], [3, 3, 2]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[2, 2, 3], [2, 3, 3], [3, 3, 2]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[2, 2, 3], [2, 3, 3], [3, 3, 2]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0004","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d9529f3a807c1c257302832b1f412c3dda35ce9e4429d802d497581571dfd4d","family_id":"qf-mod23-process-path-law-qualification","seed":202610236004,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[3,3,4],[3,4,4],[4,4,3]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[3, 3, 4], [3, 4, 4], [4, 4, 3]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[3, 3, 4], [3, 4, 4], [4, 4, 3]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0005","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b0b3956f7e6af7305c910cffddcd545c1c1c6642b1a3d2eb7a73b07d6b90d3b","family_id":"qf-mod23-process-path-law-qualification","seed":202610236005,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[4,4,5],[4,5,5],[5,5,4]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[4, 4, 5], [4, 5, 5], [5, 5, 4]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[4, 4, 5], [4, 5, 5], [5, 5, 4]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[4, 4, 5], [4, 5, 5], [5, 5, 4]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0006","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf758ee7675862383675379ff606d75961f7933872047ed2301bfa1c6050e8cf","family_id":"qf-mod23-process-path-law-qualification","seed":202610236006,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[5,5,6],[5,6,6],[6,6,5]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[5, 5, 6], [5, 6, 6], [6, 6, 5]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[5, 5, 6], [5, 6, 6], [6, 6, 5]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[5, 5, 6], [5, 6, 6], [6, 6, 5]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0007","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"806205331bd550dbf707df4065acb3b00d420ccd291fa676e4e06e47781ea5a6","family_id":"qf-mod23-process-path-law-qualification","seed":202610236007,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[6,6,7],[6,7,7],[7,7,6]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[6, 6, 7], [6, 7, 7], [7, 7, 6]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[6, 6, 7], [6, 7, 7], [7, 7, 6]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[6, 6, 7], [6, 7, 7], [7, 7, 6]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0008","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a524d601928a99a46a6fe49998634cfd4013453f97f8a28c5f183f0b11cc72e","family_id":"qf-mod23-process-path-law-qualification","seed":202610236008,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[7,7,8],[7,8,8],[8,8,7]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[7, 7, 8], [7, 8, 8], [8, 8, 7]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[7, 7, 8], [7, 8, 8], [8, 8, 7]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[7, 7, 8], [7, 8, 8], [8, 8, 7]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0009","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64e84fa2cf6844bdfcba1a6b317ce75b9bb44a6721b2c2b0f142cf63474572d2","family_id":"qf-mod23-process-path-law-qualification","seed":202610236009,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[8,8,9],[8,9,9],[9,9,8]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[8, 8, 9], [8, 9, 9], [9, 9, 8]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[8, 8, 9], [8, 9, 9], [9, 9, 8]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[8, 8, 9], [8, 9, 9], [9, 9, 8]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0010","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6605b28c85e399ee2ca68b20e1e441f60094e264db5e080e74ff789bf920633d","family_id":"qf-mod23-process-path-law-qualification","seed":202610236010,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,9],[9,10,10],[9,9,10]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[9, 9, 10], [9, 10, 10], [10, 10, 9]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[9, 9, 10], [9, 10, 10], [10, 10, 9]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0011","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fd4a60e92ff8315a72331c9d4f3f7cafe64f856263c20e06fcbec30ee818a14","family_id":"qf-mod23-process-path-law-qualification","seed":202610236011,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,11],[10,11,11],[11,11,10]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[10, 10, 11], [10, 11, 11], [11, 11, 10]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[10, 10, 11], [10, 11, 11], [11, 11, 10]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[10, 10, 11], [10, 11, 11], [11, 11, 10]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0012","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7b4531e3b3bb647ccff3008aaf8bd372e2966850c4f7156bf9f177406b7e091","family_id":"qf-mod23-process-path-law-qualification","seed":202610236012,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[11,11,12],[11,12,12],[12,12,11]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[11, 11, 12], [11, 12, 12], [12, 12, 11]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[11, 11, 12], [11, 12, 12], [12, 12, 11]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[11, 11, 12], [11, 12, 12], [12, 12, 11]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0013","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56037780b03717f205aec510e6ff18d60e621f44deea0a07fe4cefd4aca53206","family_id":"qf-mod23-process-path-law-qualification","seed":202610236013,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[12,12,13],[12,13,13],[13,13,12]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[12, 12, 13], [12, 13, 13], [13, 13, 12]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[12, 12, 13], [12, 13, 13], [13, 13, 12]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[12, 12, 13], [12, 13, 13], [13, 13, 12]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0014","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b311fb5cc1f33711aa4004d09656f40979edab0178f71ed4c818f1b64ecaf1f","family_id":"qf-mod23-process-path-law-qualification","seed":202610236014,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[13,13,14],[13,14,14],[14,14,13]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[13, 13, 14], [13, 14, 14], [14, 14, 13]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[13, 13, 14], [13, 14, 14], [14, 14, 13]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[13, 13, 14], [13, 14, 14], [14, 14, 13]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0015","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"904e2f345d20fcbdd33bf6def7003f9a1b7da8534214797678c2c5e59289961d","family_id":"qf-mod23-process-path-law-qualification","seed":202610236015,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[14,14,15],[14,15,15],[15,15,14]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[14, 14, 15], [14, 15, 15], [15, 15, 14]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[14, 14, 15], [14, 15, 15], [15, 15, 14]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[14, 14, 15], [14, 15, 15], [15, 15, 14]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0016","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65c7c4f68b36bc15fb845e794cd9a665774265493b0288acdcb634a733b5f6b1","family_id":"qf-mod23-process-path-law-qualification","seed":202610236016,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[15,15,16],[15,16,16],[16,16,15]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[15, 15, 16], [15, 16, 16], [16, 16, 15]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[15, 15, 16], [15, 16, 16], [16, 16, 15]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[15, 15, 16], [15, 16, 16], [16, 16, 15]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0017","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b6844cef698c649b2d85dc7a54360acbeb1d17d9f329fa3259d838451c0c34d","family_id":"qf-mod23-process-path-law-qualification","seed":202610236017,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[16,16,17],[16,17,17],[17,17,16]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[16, 16, 17], [16, 17, 17], [17, 17, 16]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[16, 16, 17], [16, 17, 17], [17, 17, 16]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[16, 16, 17], [16, 17, 17], [17, 17, 16]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0018","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"515ebed48bea0c80caeda8dbbdf56242d61ab2e5282266e52df9ed628da193a0","family_id":"qf-mod23-process-path-law-qualification","seed":202610236018,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[17,17,18],[17,18,18],[18,18,17]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[17, 17, 18], [17, 18, 18], [18, 18, 17]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[17, 17, 18], [17, 18, 18], [18, 18, 17]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[17, 17, 18], [17, 18, 18], [18, 18, 17]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0019","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd5fe96b096c88b3e9a7bf6ba425243942a37a8583326842e5ccdc291f0eeed2","family_id":"qf-mod23-process-path-law-qualification","seed":202610236019,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[18,18,19],[18,19,19],[19,19,18]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[18, 18, 19], [18, 19, 19], [19, 19, 18]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[18, 18, 19], [18, 19, 19], [19, 19, 18]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[18, 18, 19], [18, 19, 19], [19, 19, 18]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0020","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b18c0e5073c5c8151f97e38af9d2cbc5c4f71768df1dde796219e121f98ca88","family_id":"qf-mod23-process-path-law-qualification","seed":202610236020,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[19,19,20],[19,20,20],[20,20,19]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[19, 19, 20], [19, 20, 20], [20, 20, 19]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[19, 19, 20], [19, 20, 20], [20, 20, 19]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[19, 19, 20], [19, 20, 20], [20, 20, 19]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0021","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdacf6bdef94afc3ecbaea4abbf6b5996925506b7a95c93f0d78700ceab81b21","family_id":"qf-mod23-process-path-law-qualification","seed":202610236021,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[20,20,21],[20,21,21],[21,21,20]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[20, 20, 21], [20, 21, 21], [21, 21, 20]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[20, 20, 21], [20, 21, 21], [21, 21, 20]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[20, 20, 21], [20, 21, 21], [21, 21, 20]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0022","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcb54c6959bada1b2430aec44bbf356caf732f057062c18c4b1f9ebdcfa9b2fe","family_id":"qf-mod23-process-path-law-qualification","seed":202610236022,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[21,21,22],[21,22,22],[22,22,21]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[21, 21, 22], [21, 22, 22], [22, 22, 21]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[21, 21, 22], [21, 22, 22], [22, 22, 21]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[21, 21, 22], [21, 22, 22], [22, 22, 21]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0023","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c7f90bf0a650248de938ef904c3d2a492d77e118be733f7baee19736ccde474","family_id":"qf-mod23-process-path-law-qualification","seed":202610236023,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[22,22,23],[22,23,23],[23,23,22]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[22, 22, 23], [22, 23, 23], [23, 23, 22]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[22, 22, 23], [22, 23, 23], [23, 23, 22]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[22, 22, 23], [22, 23, 23], [23, 23, 22]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0024","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91f3cd4d97f405019b8180abef3fbd0c866b709cc21a685dbce115f9371ae860","family_id":"qf-mod23-process-path-law-qualification","seed":202610236024,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[23,23,24],[23,24,24],[24,24,23]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[23, 23, 24], [23, 24, 24], [24, 24, 23]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[23, 23, 24], [23, 24, 24], [24, 24, 23]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[23, 23, 24], [23, 24, 24], [24, 24, 23]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0025","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bd1cdb4596f83d81b637df85c9466d1e837da6f8cb593155920934a4d1d3cf8","family_id":"qf-mod23-process-path-law-qualification","seed":202610236025,"semantic_template":"qf-mod23-process-path-law-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[24,24,25],[24,25,25],[25,25,24]]}"},"fields":{"title":"Qualify a Complete Joint Path Law before Taking Marginals","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[24, 24, 25], [24, 25, 25], [25, 25, 24]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. This qualification family also permits negative or unnormalized candidate masses for rejection. Give (original paths,normalized masses,horizon,all masses nonnegative?,exact total,qualified joint path law?).","answer":{"kind":"expression","value":"([[24, 24, 25], [24, 25, 25], [25, 25, 24]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)","explanation":"Check distinct equal-length nonempty paths and the exact probability conditions without sorting paths away from their masses. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[24, 24, 25], [24, 25, 25], [25, 25, 24]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)."},"skills":["Qualify a Complete Joint Path Law before Taking Marginals"],"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-20261023-r06-discrete-stochastic-process-0026","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32716da528d08a29b2dd2f6375d64ff8c4521ea5b253f8ed8b100e23c9a1c540","family_id":"qf-mod23-process-time-marginal","seed":202610236026,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[0, 1/2], [1, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[0, 1/2], [1, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0027","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4592ea2d5c1991254c2b1053111d9a5a96f4f83cd8daab7ee9b91c11bd119919","family_id":"qf-mod23-process-time-marginal","seed":202610236027,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"time\":1}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[1, 1, 2], [1, 2, 2], [2, 2, 1]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(1, [[1, 1/4], [2, 3/4]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[1, 1/4], [2, 3/4]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0028","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"832781055162b287f5ce4eb68d47ae2f41f11dfb5380873f79cd28778444263a","family_id":"qf-mod23-process-time-marginal","seed":202610236028,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"time\":2}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[2, 2, 3], [2, 3, 3], [3, 3, 2]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(2, [[2, 1/2], [3, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[2, 1/2], [3, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0029","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2757f433f1cf31f2edf2b4de7eb82daf78d1f1f448983eef6f517b53d1cc23bc","family_id":"qf-mod23-process-time-marginal","seed":202610236029,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[3, 1/2], [4, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[3, 1/2], [4, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0030","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"004d5fe89b7e7799e1089eafcb315d5e931590c3fc60c28b24f65f78404ba270","family_id":"qf-mod23-process-time-marginal","seed":202610236030,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"time\":1}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[4, 4, 5], [4, 5, 5], [5, 5, 4]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(1, [[4, 1/4], [5, 3/4]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[4, 1/4], [5, 3/4]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0031","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"790aff4c24a1861c79a98dbc1e6ca0aadf2768632fe2dccf659d697c298351b5","family_id":"qf-mod23-process-time-marginal","seed":202610236031,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"time\":2}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[5, 5, 6], [5, 6, 6], [6, 6, 5]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(2, [[5, 1/2], [6, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[5, 1/2], [6, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0032","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a83a9f6b3b2b7d61c413412ee5abb3592b54783a1d91c1c309fe05725772957d","family_id":"qf-mod23-process-time-marginal","seed":202610236032,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[6, 6, 7], [6, 7, 7], [7, 7, 6]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[6, 1/2], [7, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[6, 1/2], [7, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0033","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4726bb00b5cadfc897248170486773cd00b11d11416b7a30df7ea5bf0d3d482","family_id":"qf-mod23-process-time-marginal","seed":202610236033,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"time\":1}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[7, 7, 8], [7, 8, 8], [8, 8, 7]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(1, [[7, 1/4], [8, 3/4]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[7, 1/4], [8, 3/4]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0034","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b9687da1c362d3eaf13a823c5a091c441cc4b3099d887633eac42bb7da647df","family_id":"qf-mod23-process-time-marginal","seed":202610236034,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"time\":2}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[8, 8, 9], [8, 9, 9], [9, 9, 8]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(2, [[8, 1/2], [9, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[8, 1/2], [9, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0035","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7986c577fc82015c0f913b4b9768b44cbac6041a36113640566b7e17485df1a7","family_id":"qf-mod23-process-time-marginal","seed":202610236035,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,9],[9,10,10],[9,9,10]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[9, 1/2], [10, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[9, 1/2], [10, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0036","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a70399207da565e803a659ece1092c94c201169163dbe9267399d4e4eaba6ffe","family_id":"qf-mod23-process-time-marginal","seed":202610236036,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"time\":1}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[10, 10, 11], [10, 11, 11], [11, 11, 10]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(1, [[10, 1/4], [11, 3/4]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[10, 1/4], [11, 3/4]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0037","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c66cb0de968a357590cd6c58544d0a801245df962ef562c3a88c55834e583369","family_id":"qf-mod23-process-time-marginal","seed":202610236037,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"time\":2}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[11, 11, 12], [11, 12, 12], [12, 12, 11]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(2, [[11, 1/2], [12, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[11, 1/2], [12, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0038","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f5f1cf88832386499bda30994882d91af1aaafe43942365c5eb43ca7b458414","family_id":"qf-mod23-process-time-marginal","seed":202610236038,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[12, 12, 13], [12, 13, 13], [13, 13, 12]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[12, 1/2], [13, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[12, 1/2], [13, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0039","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d627da09c65f8d5bfd5203f7adef52a852643fba9521800d01feb4720c34712e","family_id":"qf-mod23-process-time-marginal","seed":202610236039,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"time\":1}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[13, 13, 14], [13, 14, 14], [14, 14, 13]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(1, [[13, 1/4], [14, 3/4]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[13, 1/4], [14, 3/4]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0040","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c221df071794753f106ce9135647e89f0b86982ed3b8355291f5b14db24856cc","family_id":"qf-mod23-process-time-marginal","seed":202610236040,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"time\":2}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[14, 14, 15], [14, 15, 15], [15, 15, 14]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(2, [[14, 1/2], [15, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[14, 1/2], [15, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0041","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9b1475f079d4b1b609df8689682ef8e29ae6b6703467c4e3333fa2367d7e99e","family_id":"qf-mod23-process-time-marginal","seed":202610236041,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[15, 15, 16], [15, 16, 16], [16, 16, 15]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[15, 1/2], [16, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[15, 1/2], [16, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0042","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af71cd770b7a356c9ef73326fb6398dfc6f07ada5a5f02d5d1a3b7e91ba0ba7d","family_id":"qf-mod23-process-time-marginal","seed":202610236042,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"time\":1}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[16, 16, 17], [16, 17, 17], [17, 17, 16]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(1, [[16, 1/4], [17, 3/4]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[16, 1/4], [17, 3/4]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0043","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c11a24a3ef234691d1feb24bf64f85c569e5ad149133cee4da1345cbada17938","family_id":"qf-mod23-process-time-marginal","seed":202610236043,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"time\":2}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[17, 17, 18], [17, 18, 18], [18, 18, 17]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(2, [[17, 1/2], [18, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[17, 1/2], [18, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0044","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bd5b359614965c85ccbdc0e772a6419ce8430306599edb819b1243cbe7c6cf7","family_id":"qf-mod23-process-time-marginal","seed":202610236044,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[18, 18, 19], [18, 19, 19], [19, 19, 18]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[18, 1/2], [19, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[18, 1/2], [19, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0045","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9be8fe5d78ac0055fbb4ce935eec781c266df9799d55adbdc64f9ffa0043129a","family_id":"qf-mod23-process-time-marginal","seed":202610236045,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"time\":1}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[19, 19, 20], [19, 20, 20], [20, 20, 19]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(1, [[19, 1/4], [20, 3/4]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[19, 1/4], [20, 3/4]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0046","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c88a3400392fe96f9cf42fb5a977250e9d1fd5a5606a150bdef6878d09989090","family_id":"qf-mod23-process-time-marginal","seed":202610236046,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"time\":2}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[20, 20, 21], [20, 21, 21], [21, 21, 20]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(2, [[20, 1/2], [21, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[20, 1/2], [21, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0047","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b162f4ceac173f9547181996cf73c532eec45ce93088f7d42e7cd2ab65337f93","family_id":"qf-mod23-process-time-marginal","seed":202610236047,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[21, 21, 22], [21, 22, 22], [22, 22, 21]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[21, 1/2], [22, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[21, 1/2], [22, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0048","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"603d3f59815bb9cbf43ef16a3c41f7088930b171ebe17b2ce527a348566dfda3","family_id":"qf-mod23-process-time-marginal","seed":202610236048,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"time\":1}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[22, 22, 23], [22, 23, 23], [23, 23, 22]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(1, [[22, 1/4], [23, 3/4]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [[22, 1/4], [23, 3/4]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0049","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef9d381d32a6aea2934d85e9611c84fbf737fda405da466e846e34104e3b1c3e","family_id":"qf-mod23-process-time-marginal","seed":202610236049,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"time\":2}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[23, 23, 24], [23, 24, 24], [24, 24, 23]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(2, [[23, 1/2], [24, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [[23, 1/2], [24, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0050","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db6ed17f0263f965cf619a433c663dbdfbde133f5b2ca12cd5e03beb2746024e","family_id":"qf-mod23-process-time-marginal","seed":202610236050,"semantic_template":"qf-mod23-process-time-marginal/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"time\":0}"},"fields":{"title":"Aggregate a Marginal from the Complete Joint Path Law","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[24, 24, 25], [24, 25, 25], [25, 25, 24]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0. Give (time,every [state,aggregated mass] row in increasing numeric state order), retaining listed state values even when their aggregated mass is zero.","answer":{"kind":"expression","value":"(0, [[24, 1/2], [25, 1/2]])","explanation":"At the requested time group path masses by state. The marginal does not reconstruct the joint law or establish Markov behavior. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [[24, 1/2], [25, 1/2]])."},"skills":["Aggregate a Marginal from the Complete Joint Path Law"],"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-20261023-r06-discrete-stochastic-process-0051","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a034097278946d36e9e0851d25518d4ceff2f833791d7e19ff32f44d17d061b5","family_id":"qf-mod23-process-time-event-probability","seed":202610236051,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[1],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[1]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [1], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [1], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0052","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a206ebeefbd4f214f56ce37548352ce84aa7e60ea9392d72caeeb246b277081c","family_id":"qf-mod23-process-time-event-probability","seed":202610236052,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[2],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"time\":1}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[1, 1, 2], [1, 2, 2], [2, 2, 1]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1, event input=[2]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(1, [2], 3/4)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [2], 3/4)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0053","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"069548256dfd182997661f89b90f1c7f775a6d060499f98e16dd7042985b5bd8","family_id":"qf-mod23-process-time-event-probability","seed":202610236053,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[3],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"time\":2}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[2, 2, 3], [2, 3, 3], [3, 3, 2]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2, event input=[3]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(2, [3], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [3], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0054","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64948723de474cec64b97a848fbd2eeb2b7d0332336c685b3cab6eee1f8e3e55","family_id":"qf-mod23-process-time-event-probability","seed":202610236054,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[4],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[4]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [4], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [4], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0055","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6cd99df8cd85c46bf1bbb9ca2996b52b2af262ea833c423c15a571c6f3741d1","family_id":"qf-mod23-process-time-event-probability","seed":202610236055,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[5],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"time\":1}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[4, 4, 5], [4, 5, 5], [5, 5, 4]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1, event input=[5]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(1, [5], 3/4)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [5], 3/4)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0056","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bf9c0032c75cd593b27e241b35facae3a98d34c5fd0ee2e0361e3f0103d892d","family_id":"qf-mod23-process-time-event-probability","seed":202610236056,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[6],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"time\":2}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[5, 5, 6], [5, 6, 6], [6, 6, 5]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2, event input=[6]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(2, [6], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [6], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0057","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"309e395bc72de501cd6985ce73a04341ee7d1f8d3cea95e9436303df72afb387","family_id":"qf-mod23-process-time-event-probability","seed":202610236057,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[7],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[6, 6, 7], [6, 7, 7], [7, 7, 6]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[7]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [7], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [7], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0058","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c0800c6bf3e61775580649542d614d776746a253c4e514928e569b1e0b38f7","family_id":"qf-mod23-process-time-event-probability","seed":202610236058,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[8],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"time\":1}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[7, 7, 8], [7, 8, 8], [8, 8, 7]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1, event input=[8]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(1, [8], 3/4)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [8], 3/4)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0059","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91463f1d5802bdd14d92c8cfbcdc91bff67e66fa35ea98e5ff786ae7d2032242","family_id":"qf-mod23-process-time-event-probability","seed":202610236059,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[9],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"time\":2}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[8, 8, 9], [8, 9, 9], [9, 9, 8]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2, event input=[9]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(2, [9], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [9], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0060","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b417cc58daef6b9f42ada2ba85619377dbc0fe1dd2cfe127a0a52d7920b7f24","family_id":"qf-mod23-process-time-event-probability","seed":202610236060,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[10],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,9],[9,10,10],[9,9,10]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[10]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [10], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [10], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0061","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"649282c1b109ff72462c101af40bf31e0306993022cb8b564660d65a5d7b637b","family_id":"qf-mod23-process-time-event-probability","seed":202610236061,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[11],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"time\":1}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[10, 10, 11], [10, 11, 11], [11, 11, 10]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1, event input=[11]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(1, [11], 3/4)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [11], 3/4)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0062","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c29bed080fe5e75f48f04a817f6c041fe74b94cc48d43055b308bb034a6b2b2d","family_id":"qf-mod23-process-time-event-probability","seed":202610236062,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[12],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"time\":2}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[11, 11, 12], [11, 12, 12], [12, 12, 11]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2, event input=[12]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(2, [12], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [12], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0063","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6f2be7cb259e62e526e3a3c9758492c5109c6c42bd6035bc3746047e0c2bd1c","family_id":"qf-mod23-process-time-event-probability","seed":202610236063,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[13],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[12, 12, 13], [12, 13, 13], [13, 13, 12]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[13]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [13], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [13], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0064","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2de07ccb252c979824c88d3163bde49112dd751b1e146721c7bd47e85bdc5eb8","family_id":"qf-mod23-process-time-event-probability","seed":202610236064,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[14],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"time\":1}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[13, 13, 14], [13, 14, 14], [14, 14, 13]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1, event input=[14]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(1, [14], 3/4)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [14], 3/4)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0065","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc4a6054c4bd7cf167ccdf7d6c15b3bc5895b8e1720af98dc75b8144cf5ba798","family_id":"qf-mod23-process-time-event-probability","seed":202610236065,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[15],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"time\":2}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[14, 14, 15], [14, 15, 15], [15, 15, 14]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2, event input=[15]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(2, [15], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [15], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0066","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"add9b7cc8fb25fa5195a6a0b6c3c05453e139a722803e374d73d63fdb8d0e676","family_id":"qf-mod23-process-time-event-probability","seed":202610236066,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[16],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[15, 15, 16], [15, 16, 16], [16, 16, 15]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[16]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [16], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [16], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0067","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64fc0e286696a73e154219395e2d86dea36712af7e38fd2906fc3075b5af1986","family_id":"qf-mod23-process-time-event-probability","seed":202610236067,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[17],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"time\":1}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[16, 16, 17], [16, 17, 17], [17, 17, 16]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1, event input=[17]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(1, [17], 3/4)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [17], 3/4)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0068","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34db58a3780ff6b34bf6ea8f67f01681077a4f28074778162fb640de432fe040","family_id":"qf-mod23-process-time-event-probability","seed":202610236068,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[18],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"time\":2}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[17, 17, 18], [17, 18, 18], [18, 18, 17]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2, event input=[18]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(2, [18], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [18], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0069","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20fc293bc00536db1a51004ffb95ef885b5995ec0b1d6efe1a6a5c1a17d287ac","family_id":"qf-mod23-process-time-event-probability","seed":202610236069,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[19],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[18, 18, 19], [18, 19, 19], [19, 19, 18]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[19]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [19], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [19], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0070","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49727fb3800a3ffd1be475925a6d64dd563c4f4ceab8f55178e83039aa1bb9d2","family_id":"qf-mod23-process-time-event-probability","seed":202610236070,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[20],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"time\":1}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[19, 19, 20], [19, 20, 20], [20, 20, 19]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1, event input=[20]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(1, [20], 3/4)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [20], 3/4)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0071","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25581bb6d4b0cb9c029c6c0dbcee7c2e0a5bd5a37659c9416737a270a4c7d1a0","family_id":"qf-mod23-process-time-event-probability","seed":202610236071,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[21],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"time\":2}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[20, 20, 21], [20, 21, 21], [21, 21, 20]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2, event input=[21]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(2, [21], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [21], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0072","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1b2e2b3513721228f9bd7ee15b470ab76fdce0881c06a60db4490f51449263d","family_id":"qf-mod23-process-time-event-probability","seed":202610236072,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[22],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[21, 21, 22], [21, 22, 22], [22, 22, 21]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[22]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [22], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [22], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0073","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1693d2deb332dce80ace2c79f9ca759b92fc8bc25f32f29ecee534be89da726b","family_id":"qf-mod23-process-time-event-probability","seed":202610236073,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[23],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"time\":1}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[22, 22, 23], [22, 23, 23], [23, 23, 22]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=1, event input=[23]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(1, [23], 3/4)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, [23], 3/4)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0074","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"586130dddb3e989c1941768eaf05c7455f956ed8d59b17953d09324bfcfb1224","family_id":"qf-mod23-process-time-event-probability","seed":202610236074,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[24],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"time\":2}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[23, 23, 24], [23, 24, 24], [24, 24, 23]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=2, event input=[24]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(2, [24], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, [24], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0075","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3242c655a040b6daa56791dcf19694d48f0d3010840eab90156495cd2851d273","family_id":"qf-mod23-process-time-event-probability","seed":202610236075,"semantic_template":"qf-mod23-process-time-event-probability/{\"event\":[25],\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"time\":0}"},"fields":{"title":"One Time-Event Probability with Duplicate-Free Event Semantics","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[24, 24, 25], [24, 25, 25], [25, 25, 24]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time=0, event input=[25]. Give (time,sorted distinct event,probability that the path state at this time lies in it). Event labels outside the support remain in the event field but contribute no additional mass.","answer":{"kind":"expression","value":"(0, [25], 1/2)","explanation":"Sum the aligned path masses once for each path whose requested state is in the event; do not multiply contributions for repeated event entries. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, [25], 1/2)."},"skills":["One Time-Event Probability with Duplicate-Free Event Semantics"],"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-20261023-r06-discrete-stochastic-process-0076","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"811636c23b1a43f5cdf77b35f75f98a6eb29b955ab5f62bc6027fabdc7d2ba0d","family_id":"qf-mod23-process-two-time-joint","seed":202610236076,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[0,0,1],[0,1,1],[1,1,0]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[0, 0, 1], [0, 1, 1], [1, 1, 0]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[0, 0], 1/4], [[0, 1], 1/4], [[1, 1], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[0, 0], 1/4], [[0, 1], 1/4], [[1, 1], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0077","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53b3e6d9a0b5db3adea31ad77624aae4c5b29d682fbc40848c537d2861acb916","family_id":"qf-mod23-process-two-time-joint","seed":202610236077,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[1,1,2],[1,2,2],[2,2,1]],\"time_a\":1,\"time_b\":2}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[1, 1, 2], [1, 2, 2], [2, 2, 1]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(1, 2, [[[1, 2], 1/4], [[2, 1], 1/2], [[2, 2], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[1, 2], 1/4], [[2, 1], 1/2], [[2, 2], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0078","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0649df0f76e4fa1da73d1977b91eb9fd39da6cadb7fa83ba041e4a2a53adb4a9","family_id":"qf-mod23-process-two-time-joint","seed":202610236078,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[2,2,3],[2,3,3],[3,3,2]],\"time_a\":2,\"time_b\":0}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[2, 2, 3], [2, 3, 3], [3, 3, 2]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(2, 0, [[[2, 3], 1/2], [[3, 2], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[2, 3], 1/2], [[3, 2], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0079","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eb117b767121373f06d6f965503e9cdb0b300b09bb9f392e3e514b10a9a7894","family_id":"qf-mod23-process-two-time-joint","seed":202610236079,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[3,3,4],[3,4,4],[4,4,3]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[3, 3, 4], [3, 4, 4], [4, 4, 3]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[3, 3], 1/4], [[3, 4], 1/4], [[4, 4], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[3, 3], 1/4], [[3, 4], 1/4], [[4, 4], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0080","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c20fb73fedadfc5e7cc34bc7d213cc9ef22ecce5ee8adb65fb928af849766c00","family_id":"qf-mod23-process-two-time-joint","seed":202610236080,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[4,4,5],[4,5,5],[5,5,4]],\"time_a\":1,\"time_b\":2}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[4, 4, 5], [4, 5, 5], [5, 5, 4]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(1, 2, [[[4, 5], 1/4], [[5, 4], 1/2], [[5, 5], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[4, 5], 1/4], [[5, 4], 1/2], [[5, 5], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0081","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b84e7435b9e11d6190fc35431f08ecba254aec4db28093480c6a73413ba6eda","family_id":"qf-mod23-process-two-time-joint","seed":202610236081,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[5,5,6],[5,6,6],[6,6,5]],\"time_a\":2,\"time_b\":0}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[5, 5, 6], [5, 6, 6], [6, 6, 5]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(2, 0, [[[5, 6], 1/2], [[6, 5], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[5, 6], 1/2], [[6, 5], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0082","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"195b8e88c8cd036f3f901c96a5518e716509f83f28e498eafc09357462ab7ed2","family_id":"qf-mod23-process-two-time-joint","seed":202610236082,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[6,6,7],[6,7,7],[7,7,6]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[6, 6, 7], [6, 7, 7], [7, 7, 6]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[6, 6], 1/4], [[6, 7], 1/4], [[7, 7], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[6, 6], 1/4], [[6, 7], 1/4], [[7, 7], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0083","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c072f6e139784904cdc00cb23ab7b65419cc5bae337cfd71de94fd633d57c03b","family_id":"qf-mod23-process-two-time-joint","seed":202610236083,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[7,7,8],[7,8,8],[8,8,7]],\"time_a\":1,\"time_b\":2}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[7, 7, 8], [7, 8, 8], [8, 8, 7]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(1, 2, [[[7, 8], 1/4], [[8, 7], 1/2], [[8, 8], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[7, 8], 1/4], [[8, 7], 1/2], [[8, 8], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0084","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45545fb3e9bb1af32878bc2a7af5531e7edff1d8d73fa4d082d12e4b7a06b550","family_id":"qf-mod23-process-two-time-joint","seed":202610236084,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[8,8,9],[8,9,9],[9,9,8]],\"time_a\":2,\"time_b\":0}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[8, 8, 9], [8, 9, 9], [9, 9, 8]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(2, 0, [[[8, 9], 1/2], [[9, 8], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[8, 9], 1/2], [[9, 8], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0085","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e836a9209d0df6c9acfeb4302b531486e02fc796e95bd6b3be4fb168aac5d84","family_id":"qf-mod23-process-two-time-joint","seed":202610236085,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,9],[9,10,10],[9,9,10]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[9, 9, 10], [9, 10, 10], [10, 10, 9]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[10, 10], 1/2], [[9, 10], 1/4], [[9, 9], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[10, 10], 1/2], [[9, 10], 1/4], [[9, 9], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0086","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32d9543c663b6a6033e7481465d6059f87e2db503e57868c7b5daf158d69817f","family_id":"qf-mod23-process-two-time-joint","seed":202610236086,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[10,10,11],[10,11,11],[11,11,10]],\"time_a\":1,\"time_b\":2}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[10, 10, 11], [10, 11, 11], [11, 11, 10]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(1, 2, [[[10, 11], 1/4], [[11, 10], 1/2], [[11, 11], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[10, 11], 1/4], [[11, 10], 1/2], [[11, 11], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0087","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18372f50df8f8887728660be0756a30e201259804aa2f9d70aacec2c38893011","family_id":"qf-mod23-process-two-time-joint","seed":202610236087,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[11,11,12],[11,12,12],[12,12,11]],\"time_a\":2,\"time_b\":0}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[11, 11, 12], [11, 12, 12], [12, 12, 11]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(2, 0, [[[11, 12], 1/2], [[12, 11], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[11, 12], 1/2], [[12, 11], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0088","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8802728057d16870881b9a0e7e23a25df9df7f9dc80c5c51ee5bddafe6920546","family_id":"qf-mod23-process-two-time-joint","seed":202610236088,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[12,12,13],[12,13,13],[13,13,12]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[12, 12, 13], [12, 13, 13], [13, 13, 12]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[12, 12], 1/4], [[12, 13], 1/4], [[13, 13], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[12, 12], 1/4], [[12, 13], 1/4], [[13, 13], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0089","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb7b14bd3bcb1f8e182cf5cda5cc343a035b8f73912cfe53f0506b76c70dd10e","family_id":"qf-mod23-process-two-time-joint","seed":202610236089,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[13,13,14],[13,14,14],[14,14,13]],\"time_a\":1,\"time_b\":2}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[13, 13, 14], [13, 14, 14], [14, 14, 13]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(1, 2, [[[13, 14], 1/4], [[14, 13], 1/2], [[14, 14], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[13, 14], 1/4], [[14, 13], 1/2], [[14, 14], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0090","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c3f1cd6ae009f263a2fc76bfefe19dbf392330e279ddc312b7929b86dfd6087","family_id":"qf-mod23-process-two-time-joint","seed":202610236090,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[14,14,15],[14,15,15],[15,15,14]],\"time_a\":2,\"time_b\":0}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[14, 14, 15], [14, 15, 15], [15, 15, 14]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(2, 0, [[[14, 15], 1/2], [[15, 14], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[14, 15], 1/2], [[15, 14], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0091","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"243b31b4d6d10337dbf745fc28ce535247d1017c8918fdf93a2e5ffad98b10c0","family_id":"qf-mod23-process-two-time-joint","seed":202610236091,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[15,15,16],[15,16,16],[16,16,15]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[15, 15, 16], [15, 16, 16], [16, 16, 15]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[15, 15], 1/4], [[15, 16], 1/4], [[16, 16], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[15, 15], 1/4], [[15, 16], 1/4], [[16, 16], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0092","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63a608a864854cbf348cdc1ecf0e2f7b6435cda588b739dd3012d6b342989955","family_id":"qf-mod23-process-two-time-joint","seed":202610236092,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[16,16,17],[16,17,17],[17,17,16]],\"time_a\":1,\"time_b\":2}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[16, 16, 17], [16, 17, 17], [17, 17, 16]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(1, 2, [[[16, 17], 1/4], [[17, 16], 1/2], [[17, 17], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[16, 17], 1/4], [[17, 16], 1/2], [[17, 17], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0093","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1feb91f18ba811d2876a68789bcf886a645ffda044bd324a4bdc2ca92041f18","family_id":"qf-mod23-process-two-time-joint","seed":202610236093,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[17,17,18],[17,18,18],[18,18,17]],\"time_a\":2,\"time_b\":0}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[17, 17, 18], [17, 18, 18], [18, 18, 17]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(2, 0, [[[17, 18], 1/2], [[18, 17], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[17, 18], 1/2], [[18, 17], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0094","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd9508dc717087fa24f5f2e5f7e6f76f215dbc9348e9ff72918fcdf771678b17","family_id":"qf-mod23-process-two-time-joint","seed":202610236094,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[18,18,19],[18,19,19],[19,19,18]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[18, 18, 19], [18, 19, 19], [19, 19, 18]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[18, 18], 1/4], [[18, 19], 1/4], [[19, 19], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[18, 18], 1/4], [[18, 19], 1/4], [[19, 19], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0095","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bc567a729773b93dec18d76d8f997e562ab3af1da9015d7852cff12269ae265","family_id":"qf-mod23-process-two-time-joint","seed":202610236095,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[19,19,20],[19,20,20],[20,20,19]],\"time_a\":1,\"time_b\":2}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[19, 19, 20], [19, 20, 20], [20, 20, 19]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(1, 2, [[[19, 20], 1/4], [[20, 19], 1/2], [[20, 20], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[19, 20], 1/4], [[20, 19], 1/2], [[20, 20], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0096","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a31bc44ebd4d0615b97e1bfdc818d1e5695588d8cde02c8af9ff50d35a5c2c85","family_id":"qf-mod23-process-two-time-joint","seed":202610236096,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[20,20,21],[20,21,21],[21,21,20]],\"time_a\":2,\"time_b\":0}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[20, 20, 21], [20, 21, 21], [21, 21, 20]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(2, 0, [[[20, 21], 1/2], [[21, 20], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[20, 21], 1/2], [[21, 20], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0097","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"821f0d86dad05a6c3da18182c023bad464fabb18a47466427d70daec2c7d0f25","family_id":"qf-mod23-process-two-time-joint","seed":202610236097,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[21,21,22],[21,22,22],[22,22,21]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[21, 21, 22], [21, 22, 22], [22, 22, 21]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[21, 21], 1/4], [[21, 22], 1/4], [[22, 22], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[21, 21], 1/4], [[21, 22], 1/4], [[22, 22], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0098","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b47825c1ee40aaf602b517ccfe68da0a399d30bf6d94e26afa3ed2503d9fc71","family_id":"qf-mod23-process-two-time-joint","seed":202610236098,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[22,22,23],[22,23,23],[23,23,22]],\"time_a\":1,\"time_b\":2}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[22, 22, 23], [22, 23, 23], [23, 23, 22]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=1, time_b=2. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(1, 2, [[[22, 23], 1/4], [[23, 22], 1/2], [[23, 23], 1/4]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (1, 2, [[[22, 23], 1/4], [[23, 22], 1/2], [[23, 23], 1/4]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0099","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6924dd0db8fec4813b76222c5a926d828a3392aff2d88a611c54438551e671df","family_id":"qf-mod23-process-two-time-joint","seed":202610236099,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[23,23,24],[23,24,24],[24,24,23]],\"time_a\":2,\"time_b\":0}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[23, 23, 24], [23, 24, 24], [24, 24, 23]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=2, time_b=0. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(2, 0, [[[23, 24], 1/2], [[24, 23], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (2, 0, [[[23, 24], 1/2], [[24, 23], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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-20261023-r06-discrete-stochastic-process-0100","batch_id":"batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e32e21e023395a3b068e09bbf19a47b48ef363648362bbbc9c04af0bd4ece8d6","family_id":"qf-mod23-process-two-time-joint","seed":202610236100,"semantic_template":"qf-mod23-process-two-time-joint/{\"masses\":[[1,4],[1,4],[1,2]],\"paths\":[[24,24,25],[24,25,25],[25,25,24]],\"time_a\":0,\"time_b\":1}"},"fields":{"title":"A Two-Time Joint Distribution Keeps the Requested Time Order","stem":"Use one common finite probability space specified by the complete distinct equal-horizon path list=[[24, 24, 25], [24, 25, 25], [25, 25, 24]] and aligned exact masses=[1/4, 1/4, 1/2]. Preserve the original path-mass pairing. A qualified joint path law has nonnegative masses summing exactly to 1. Time starts at 0. Marginals do not determine the joint law or Markov property; a finite horizon gives no stationarity, ergodicity or infinite-time extension. Use exact reduced rational outputs with positive denominators, omitting /1 for integers. Require a qualified law. time_a=0, time_b=1. Give (time_a,time_b,every [[state at time_a,state at time_b],aggregated mass] row). Order distinct output pairs by the source’s en-locale comparison of their JSON strings, not numeric coordinate sorting. Equal requested times are allowed.","answer":{"kind":"expression","value":"(0, 1, [[[24, 24], 1/4], [[24, 25], 1/4], [[25, 25], 1/2]])","explanation":"Aggregate mass for each ordered state pair, preserving time roles. Marginal tables alone would not supply these dependence data. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (0, 1, [[[24, 24], 1/4], [[24, 25], 1/4], [[25, 25], 1/2]])."},"skills":["A Two-Time Joint Distribution Keeps the Requested Time Order"],"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."}}