Complete Finite-Horizon Joint Path Laws and Marginals
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261023-r06-discrete-stochastic-process-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[0, 0, 1], [0, 1, 1], [1, 1, 0]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
2. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[1, 1, 2], [1, 2, 2], [2, 2, 1]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
3. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[2, 2, 3], [2, 3, 3], [3, 3, 2]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
4. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[3, 3, 4], [3, 4, 4], [4, 4, 3]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
5. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[4, 4, 5], [4, 5, 5], [5, 5, 4]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
6. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[5, 5, 6], [5, 6, 6], [6, 6, 5]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
7. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[6, 6, 7], [6, 7, 7], [7, 7, 6]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
8. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[7, 7, 8], [7, 8, 8], [8, 8, 7]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
9. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[8, 8, 9], [8, 9, 9], [9, 9, 8]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
10. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[9, 9, 10], [9, 10, 10], [10, 10, 9]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
11. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[10, 10, 11], [10, 11, 11], [11, 11, 10]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
12. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[11, 11, 12], [11, 12, 12], [12, 12, 11]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
13. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[12, 12, 13], [12, 13, 13], [13, 13, 12]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
14. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[13, 13, 14], [13, 14, 14], [14, 14, 13]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
15. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[14, 14, 15], [14, 15, 15], [15, 15, 14]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
16. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[15, 15, 16], [15, 16, 16], [16, 16, 15]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
17. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[16, 16, 17], [16, 17, 17], [17, 17, 16]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
18. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[17, 17, 18], [17, 18, 18], [18, 18, 17]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
19. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[18, 18, 19], [18, 19, 19], [19, 19, 18]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
20. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[19, 19, 20], [19, 20, 20], [20, 20, 19]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
21. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[20, 20, 21], [20, 21, 21], [21, 21, 20]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
22. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[21, 21, 22], [21, 22, 22], [22, 22, 21]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
23. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[22, 22, 23], [22, 23, 23], [23, 23, 22]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
24. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[23, 23, 24], [23, 24, 24], [24, 24, 23]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
25. Qualify a Complete Joint Path Law before Taking Marginals
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?).
View answer and explanation
([[24, 24, 25], [24, 25, 25], [25, 25, 24]], [1/4, 1/4, 1/2], 3, Yes, 1, Yes)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-path-law-qualification · candidate
26. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[0, 1/2], [1, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
27. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(1, [[1, 1/4], [2, 3/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
28. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(2, [[2, 1/2], [3, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
29. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[3, 1/2], [4, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
30. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(1, [[4, 1/4], [5, 3/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
31. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(2, [[5, 1/2], [6, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
32. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[6, 1/2], [7, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
33. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(1, [[7, 1/4], [8, 3/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
34. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(2, [[8, 1/2], [9, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
35. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[9, 1/2], [10, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
36. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(1, [[10, 1/4], [11, 3/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
37. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(2, [[11, 1/2], [12, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
38. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[12, 1/2], [13, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
39. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(1, [[13, 1/4], [14, 3/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
40. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(2, [[14, 1/2], [15, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
41. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[15, 1/2], [16, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
42. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(1, [[16, 1/4], [17, 3/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
43. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(2, [[17, 1/2], [18, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
44. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[18, 1/2], [19, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
45. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(1, [[19, 1/4], [20, 3/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
46. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(2, [[20, 1/2], [21, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
47. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[21, 1/2], [22, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
48. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(1, [[22, 1/4], [23, 3/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
49. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(2, [[23, 1/2], [24, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
50. Aggregate a Marginal from the Complete Joint Path Law
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.
View answer and explanation
(0, [[24, 1/2], [25, 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-marginal · candidate
51. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [1], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
52. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(1, [2], 3/4)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
53. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(2, [3], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
54. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [4], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
55. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(1, [5], 3/4)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
56. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(2, [6], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
57. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [7], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
58. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(1, [8], 3/4)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
59. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(2, [9], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
60. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [10], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
61. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(1, [11], 3/4)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
62. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(2, [12], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
63. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [13], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
64. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(1, [14], 3/4)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
65. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(2, [15], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
66. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [16], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
67. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(1, [17], 3/4)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
68. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(2, [18], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
69. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [19], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
70. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(1, [20], 3/4)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
71. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(2, [21], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
72. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [22], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
73. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(1, [23], 3/4)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
74. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(2, [24], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
75. One Time-Event Probability with Duplicate-Free Event Semantics
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.
View answer and explanation
(0, [25], 1/2)
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).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-time-event-probability · candidate
76. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[0, 0], 1/4], [[0, 1], 1/4], [[1, 1], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
77. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(1, 2, [[[1, 2], 1/4], [[2, 1], 1/2], [[2, 2], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
78. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(2, 0, [[[2, 3], 1/2], [[3, 2], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
79. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[3, 3], 1/4], [[3, 4], 1/4], [[4, 4], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
80. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(1, 2, [[[4, 5], 1/4], [[5, 4], 1/2], [[5, 5], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
81. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(2, 0, [[[5, 6], 1/2], [[6, 5], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
82. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[6, 6], 1/4], [[6, 7], 1/4], [[7, 7], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
83. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(1, 2, [[[7, 8], 1/4], [[8, 7], 1/2], [[8, 8], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
84. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(2, 0, [[[8, 9], 1/2], [[9, 8], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
85. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[10, 10], 1/2], [[9, 10], 1/4], [[9, 9], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
86. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(1, 2, [[[10, 11], 1/4], [[11, 10], 1/2], [[11, 11], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
87. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(2, 0, [[[11, 12], 1/2], [[12, 11], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
88. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[12, 12], 1/4], [[12, 13], 1/4], [[13, 13], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
89. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(1, 2, [[[13, 14], 1/4], [[14, 13], 1/2], [[14, 14], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
90. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(2, 0, [[[14, 15], 1/2], [[15, 14], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
91. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[15, 15], 1/4], [[15, 16], 1/4], [[16, 16], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
92. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(1, 2, [[[16, 17], 1/4], [[17, 16], 1/2], [[17, 17], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
93. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(2, 0, [[[17, 18], 1/2], [[18, 17], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
94. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[18, 18], 1/4], [[18, 19], 1/4], [[19, 19], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
95. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(1, 2, [[[19, 20], 1/4], [[20, 19], 1/2], [[20, 20], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
96. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(2, 0, [[[20, 21], 1/2], [[21, 20], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
97. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[21, 21], 1/4], [[21, 22], 1/4], [[22, 22], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
98. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(1, 2, [[[22, 23], 1/4], [[23, 22], 1/2], [[23, 23], 1/4]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
99. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(2, 0, [[[23, 24], 1/2], [[24, 23], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
100. A Two-Time Joint Distribution Keeps the Requested Time Order
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.
View answer and explanation
(0, 1, [[[24, 24], 1/4], [[24, 25], 1/4], [[25, 25], 1/2]])
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]]).
Reviewed step-by-step derivation
Discrete-time stochastic process · qf-mod23-process-two-time-joint · candidate
Canonical question JSONL · English JSONL · Validation report