Symbolic Deviation Tails, Finite-Prefix Limits and Eventual Equality
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batch-20261014-advance-20261030-r07-convergence-in-probability-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=1/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=1. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(1, 2, 1, 1/3)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 2, 1, 1/3).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
2. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=2/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=2. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(2, 3, 2, 2/5)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 3, 2, 2/5).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
3. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=3/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=3. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(3, 4, 3, 3/7)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 4, 3, 3/7).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
4. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=4/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=4. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(4, 5, 4, 4/9)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 5, 4, 4/9).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
5. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=5/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=5. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(5, 6, 5, 5/11)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 6, 5, 5/11).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
6. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=1/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=6. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(1, 7, 6, 1/13)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 7, 6, 1/13).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
7. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=2/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=7. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(2, 8, 7, 2/15)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 8, 7, 2/15).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
8. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=3/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=8. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(3, 2, 8, 3/10)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 2, 8, 3/10).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
9. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=4/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=9. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(4, 3, 9, 1/3)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 3, 9, 1/3).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
10. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=5/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=10. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(5, 4, 10, 5/14)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 4, 10, 5/14).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
11. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=1/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=11. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(1, 5, 11, 1/16)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 5, 11, 1/16).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
12. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=2/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=12. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(2, 6, 12, 1/9)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 6, 12, 1/9).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
13. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=3/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=13. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(3, 7, 13, 3/20)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 7, 13, 3/20).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
14. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=4/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=14. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(4, 8, 14, 2/11)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 8, 14, 2/11).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
15. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=5/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=15. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(5, 2, 15, 5/17)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 2, 15, 5/17).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
16. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=1/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=16. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(1, 3, 16, 1/19)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 3, 16, 1/19).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
17. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=2/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=17. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(2, 4, 17, 2/21)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 4, 17, 2/21).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
18. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=3/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=18. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(3, 5, 18, 3/23)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 5, 18, 3/23).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
19. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=4/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=19. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(4, 6, 19, 4/25)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 6, 19, 4/25).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
20. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=5/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=20. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(5, 7, 20, 5/27)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 7, 20, 5/27).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
21. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=1/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=21. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(1, 8, 21, 1/29)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (1, 8, 21, 1/29).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
22. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=2/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=22. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(2, 2, 22, 1/12)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (2, 2, 22, 1/12).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
23. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=3/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=23. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(3, 3, 23, 3/26)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 3, 23, 3/26).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
24. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=4/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=24. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(4, 4, 24, 1/7)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (4, 4, 24, 1/7).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
25. Evaluate the Deviation Profile Only at a Valid Probability Index
Use the declared rational deviation-probability tail p_n=5/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=25. Give (numerator,shift,n,deviation_probability).
View answer and explanation
(5, 5, 25, 1/6)
The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (5, 5, 25, 1/6).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-deviation-probability · candidate
26. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=1/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/10. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(1, 2, 1/10, 9, 1/11, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 2, 1/10, 9, 1/11, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
27. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=2/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/11. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(2, 3, 1/11, 20, 2/23, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 3, 1/11, 20, 2/23, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
28. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=3/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/12. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(3, 4, 1/12, 33, 3/37, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 4, 1/12, 33, 3/37, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
29. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=4/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/13. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(4, 5, 1/13, 48, 4/53, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 5, 1/13, 48, 4/53, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
30. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=5/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/14. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(5, 6, 1/14, 65, 5/71, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 6, 1/14, 65, 5/71, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
31. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=1/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/15. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(1, 7, 1/15, 9, 1/16, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 7, 1/15, 9, 1/16, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
32. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=2/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/16. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(2, 8, 1/16, 25, 2/33, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 8, 1/16, 25, 2/33, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
33. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=3/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/17. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(3, 2, 1/17, 50, 3/52, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 2, 1/17, 50, 3/52, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
34. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=4/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/18. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(4, 3, 1/18, 70, 4/73, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 3, 1/18, 70, 4/73, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
35. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=5/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/19. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(5, 4, 1/19, 92, 5/96, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 4, 1/19, 92, 5/96, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
36. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=1/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/20. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(1, 5, 1/20, 16, 1/21, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 5, 1/20, 16, 1/21, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
37. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=2/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/21. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(2, 6, 1/21, 37, 2/43, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 6, 1/21, 37, 2/43, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
38. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=3/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/22. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(3, 7, 1/22, 60, 3/67, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 7, 1/22, 60, 3/67, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
39. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=4/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/23. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(4, 8, 1/23, 85, 4/93, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 8, 1/23, 85, 4/93, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
40. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=5/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/24. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(5, 2, 1/24, 119, 5/121, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 2, 1/24, 119, 5/121, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
41. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=1/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/25. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(1, 3, 1/25, 23, 1/26, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 3, 1/25, 23, 1/26, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
42. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=2/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/26. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(2, 4, 1/26, 49, 2/53, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 4, 1/26, 49, 2/53, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
43. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=3/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/27. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(3, 5, 1/27, 77, 3/82, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 5, 1/27, 77, 3/82, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
44. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=4/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/28. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(4, 6, 1/28, 107, 4/113, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 6, 1/28, 107, 4/113, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
45. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=5/(n+7) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/29. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(5, 7, 1/29, 139, 5/146, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 7, 1/29, 139, 5/146, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
46. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=1/(n+8) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/30. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(1, 8, 1/30, 23, 1/31, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (1, 8, 1/30, 23, 1/31, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
47. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=2/(n+2) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/31. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(2, 2, 1/31, 61, 2/63, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (2, 2, 1/31, 61, 2/63, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
48. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=3/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/32. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(3, 3, 1/32, 94, 3/97, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (3, 3, 1/32, 94, 3/97, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
49. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=4/(n+4) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/33. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(4, 4, 1/33, 129, 4/133, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (4, 4, 1/33, 129, 4/133, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
50. A Strict Rational Tail Cutoff Is Different from a Finite Prefix
Use the declared rational deviation-probability tail p_n=5/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/34. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
View answer and explanation
(5, 5, 1/34, 166, 5/171, Yes, forall_n_ge_N_tail_bound)
Solve the strict inequality c/(N+s)<delta with the least nonnegative integer N. The returned N must also belong to the valid probability domain. Increasing positive denominators then give the bound for every later index, conditionally on the declared deviation profile; no sampled prefix or infinite-process construction is substituted. The exact result, retaining the requested order, is (5, 5, 1/34, 166, 5/171, Yes, forall_n_ge_N_tail_bound).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-epsilon-witness · candidate
51. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=1/(n+2) for n=0 through prefix_length-1, with prefix_length=1. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(1, 2, 1, [1/2], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 2, 1, [1/2], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
52. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=2/(n+3) for n=0 through prefix_length-1, with prefix_length=2. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(2, 3, 2, [2/3, 1/2], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 3, 2, [2/3, 1/2], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
53. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=3/(n+4) for n=0 through prefix_length-1, with prefix_length=3. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(3, 4, 3, [3/4, 3/5, 1/2], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 4, 3, [3/4, 3/5, 1/2], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
54. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=4/(n+5) for n=0 through prefix_length-1, with prefix_length=4. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(4, 5, 4, [4/5, 2/3, 4/7, 1/2], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 5, 4, [4/5, 2/3, 4/7, 1/2], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
55. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=5/(n+6) for n=0 through prefix_length-1, with prefix_length=5. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(5, 6, 5, [5/6, 5/7, 5/8, 5/9, 1/2], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 6, 5, [5/6, 5/7, 5/8, 5/9, 1/2], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
56. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=1/(n+7) for n=0 through prefix_length-1, with prefix_length=6. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(1, 7, 6, [1/7, 1/8, 1/9, 1/10, 1/11, 1/12], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 7, 6, [1/7, 1/8, 1/9, 1/10, 1/11, 1/12], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
57. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=2/(n+8) for n=0 through prefix_length-1, with prefix_length=7. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(2, 8, 7, [1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 8, 7, [1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
58. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=3/(n+3) for n=0 through prefix_length-1, with prefix_length=8. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(3, 3, 8, [1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 3, 8, [1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
59. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=4/(n+4) for n=0 through prefix_length-1, with prefix_length=9. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(4, 4, 9, [1, 4/5, 2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 4, 9, [1, 4/5, 2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
60. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=5/(n+5) for n=0 through prefix_length-1, with prefix_length=10. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(5, 5, 10, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 5, 10, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
61. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=1/(n+5) for n=0 through prefix_length-1, with prefix_length=11. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(1, 5, 11, [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 5, 11, [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
62. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=2/(n+6) for n=0 through prefix_length-1, with prefix_length=12. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(2, 6, 12, [1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 6, 12, [1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
63. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=3/(n+7) for n=0 through prefix_length-1, with prefix_length=13. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(3, 7, 13, [3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 7, 13, [3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
64. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=4/(n+8) for n=0 through prefix_length-1, with prefix_length=14. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(4, 8, 14, [1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 8, 14, [1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
65. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=5/(n+5) for n=0 through prefix_length-1, with prefix_length=15. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(5, 5, 15, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 5, 15, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
66. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=1/(n+3) for n=0 through prefix_length-1, with prefix_length=16. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(1, 3, 16, [1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 3, 16, [1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
67. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=2/(n+4) for n=0 through prefix_length-1, with prefix_length=17. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(2, 4, 17, [1/2, 2/5, 1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17, 1/9, 2/19, 1/10], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 4, 17, [1/2, 2/5, 1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17, 1/9, 2/19, 1/10], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
68. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=3/(n+5) for n=0 through prefix_length-1, with prefix_length=18. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(3, 5, 18, [3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19, 3/20, 1/7, 3/22], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 5, 18, [3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19, 3/20, 1/7, 3/22], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
69. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=4/(n+6) for n=0 through prefix_length-1, with prefix_length=19. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(4, 6, 19, [2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 6, 19, [2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
70. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=5/(n+7) for n=0 through prefix_length-1, with prefix_length=20. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(5, 7, 20, [5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19, 1/4, 5/21, 5/22, 5/23, 5/24, 1/5, 5/26], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 7, 20, [5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19, 1/4, 5/21, 5/22, 5/23, 5/24, 1/5, 5/26], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
71. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=1/(n+8) for n=0 through prefix_length-1, with prefix_length=21. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(1, 8, 21, [1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (1, 8, 21, [1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
72. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=2/(n+2) for n=0 through prefix_length-1, with prefix_length=22. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(2, 2, 22, [1, 2/3, 1/2, 2/5, 1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17, 1/9, 2/19, 1/10, 2/21, 1/11, 2/23], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (2, 2, 22, [1, 2/3, 1/2, 2/5, 1/3, 2/7, 1/4, 2/9, 1/5, 2/11, 1/6, 2/13, 1/7, 2/15, 1/8, 2/17, 1/9, 2/19, 1/10, 2/21, 1/11, 2/23], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
73. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=3/(n+3) for n=0 through prefix_length-1, with prefix_length=23. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(3, 3, 23, [1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19, 3/20, 1/7, 3/22, 3/23, 1/8, 3/25], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (3, 3, 23, [1, 3/4, 3/5, 1/2, 3/7, 3/8, 1/3, 3/10, 3/11, 1/4, 3/13, 3/14, 1/5, 3/16, 3/17, 1/6, 3/19, 3/20, 1/7, 3/22, 3/23, 1/8, 3/25], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
74. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=4/(n+4) for n=0 through prefix_length-1, with prefix_length=24. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(4, 4, 24, [1, 4/5, 2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6, 4/25, 2/13, 4/27], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (4, 4, 24, [1, 4/5, 2/3, 4/7, 1/2, 4/9, 2/5, 4/11, 1/3, 4/13, 2/7, 4/15, 1/4, 4/17, 2/9, 4/19, 1/5, 4/21, 2/11, 4/23, 1/6, 4/25, 2/13, 4/27], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
75. A Valid Zero-Indexed Probability Prefix Is Still Not a Convergence Certificate
Use only the explicitly supplied finite list p_n=5/(n+5) for n=0 through prefix_length-1, with prefix_length=25. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
View answer and explanation
(5, 5, 25, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19, 1/4, 5/21, 5/22, 5/23, 5/24, 1/5, 5/26, 5/27, 5/28, 5/29], No, finite_prefix_not_convergence_certificate)
List c/(n+s) for every integer0 through prefix_length-1 in order, without clipping or dropping an invalid leading value. The sufficiency flag is No: finitely many correct decreasing values alone do not establish the universal tail obligation. The exact result, retaining the requested order, is (5, 5, 25, [1, 5/6, 5/7, 5/8, 5/9, 1/2, 5/11, 5/12, 5/13, 5/14, 1/3, 5/16, 5/17, 5/18, 5/19, 1/4, 5/21, 5/22, 5/23, 5/24, 1/5, 5/26, 5/27, 5/28, 5/29], No, finite_prefix_not_convergence_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-finite-prefix-boundary · candidate
76. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=1; inspect n=0. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(1, 0, not_yet_zero, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (1, 0, not_yet_zero, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
77. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=2; inspect n=1. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(2, 1, not_yet_zero, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (2, 1, not_yet_zero, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
78. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=3; inspect n=2. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(3, 2, not_yet_zero, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (3, 2, not_yet_zero, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
79. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=4; inspect n=4. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(4, 4, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (4, 4, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
80. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=5; inspect n=5. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(5, 5, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (5, 5, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
81. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=6; inspect n=6. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(6, 6, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (6, 6, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
82. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=7; inspect n=8. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(7, 8, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (7, 8, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
83. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=8; inspect n=9. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(8, 9, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (8, 9, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
84. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=9; inspect n=10. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(9, 10, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (9, 10, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
85. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=10; inspect n=12. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(10, 12, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (10, 12, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
86. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=11; inspect n=13. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(11, 13, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (11, 13, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
87. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=12; inspect n=14. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(12, 14, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (12, 14, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
88. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=13; inspect n=16. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(13, 16, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (13, 16, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
89. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=14; inspect n=17. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(14, 17, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (14, 17, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
90. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=15; inspect n=18. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(15, 18, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (15, 18, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
91. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=16; inspect n=20. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(16, 20, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (16, 20, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
92. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=17; inspect n=21. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(17, 21, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (17, 21, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
93. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=18; inspect n=22. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(18, 22, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (18, 22, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
94. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=19; inspect n=24. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(19, 24, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (19, 24, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
95. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=20; inspect n=25. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(20, 25, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (20, 25, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
96. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=21; inspect n=26. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(21, 26, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (21, 26, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
97. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=22; inspect n=28. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(22, 28, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (22, 28, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
98. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=23; inspect n=29. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(23, 29, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (23, 29, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
99. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=24; inspect n=30. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(24, 30, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (24, 30, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
100. Eventual Equality Supplies a Conditional Universal-Tail Certificate
On the declared common probability space assume X_n=X almost surely for every n≥settling_index=25; inspect n=32. Give (settling_index,n,zero_probability_status,certificate?,eventual_equality_universal_tail_certificate).
View answer and explanation
(25, 32, 0, Yes, eventual_equality_universal_tail_certificate)
At or beyond the settling index the deviation probability is0 for every positive epsilon. Earlier not_yet_zero means this certificate has not yet supplied a zero value; it does not independently establish a positive deviation probability. The final Yes is the conditional eventual-equality tail certificate, not a proof from this single inspected index. The exact result, retaining the requested order, is (25, 32, 0, Yes, eventual_equality_universal_tail_certificate).
Reviewed step-by-step derivation
Convergence in probability · qf-mod30-convprob-eventual-equality · candidate
Canonical question JSONL · English JSONL · Validation report