Exact Finite-Atomic p-Moments, Moment Tolerances and Symbolic Tails
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batch-20261014-advance-20261104-r07-lp-convergence-foundations · 2026-10-14 · 100 questions
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1. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[1, -2], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=1. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 1, 1/2, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 1, 1/2, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
2. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[2, -3], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=2. Give (p, n, p_moment, certificate).
View answer and explanation
(2, 2, 13/32, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 2, 13/32, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
3. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[3, -4], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=3. Give (p, n, p_moment, certificate).
View answer and explanation
(3, 3, 91/250, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 3, 91/250, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
4. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[4, -5], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=4. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 4, 3/4, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 4, 3/4, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
5. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[5, -6], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=5. Give (p, n, p_moment, certificate).
View answer and explanation
(2, 5, 61/98, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 5, 61/98, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
6. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[6, -7], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=6. Give (p, n, p_moment, certificate).
View answer and explanation
(3, 6, 559/1024, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 6, 559/1024, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
7. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=7. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 7, 5/6, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 7, 5/6, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
8. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[8, -9], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=8. Give (p, n, p_moment, certificate).
View answer and explanation
(2, 8, 29/40, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 8, 29/40, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
9. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[9, -10], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=9. Give (p, n, p_moment, certificate).
View answer and explanation
(3, 9, 1729/2662, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 9, 1729/2662, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
10. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=10. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 10, 7/8, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 10, 7/8, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
11. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[11, -12], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=11. Give (p, n, p_moment, certificate).
View answer and explanation
(2, 11, 265/338, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 11, 265/338, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
12. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[12, -13], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=12. Give (p, n, p_moment, certificate).
View answer and explanation
(3, 12, 3925/5488, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 12, 3925/5488, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
13. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[13, -14], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=13. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 13, 9/10, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 13, 9/10, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
14. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=14. Give (p, n, p_moment, certificate).
View answer and explanation
(2, 14, 421/512, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 14, 421/512, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
15. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[15, -16], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=15. Give (p, n, p_moment, certificate).
View answer and explanation
(3, 15, 7471/9826, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 15, 7471/9826, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
16. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[16, -17], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=16. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 16, 11/12, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 16, 11/12, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
17. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[17, -18], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=17. Give (p, n, p_moment, certificate).
View answer and explanation
(2, 17, 613/722, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 17, 613/722, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
18. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=18. Give (p, n, p_moment, certificate).
View answer and explanation
(3, 18, 12691/16000, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 18, 12691/16000, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
19. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[19, -20], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=19. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 19, 13/14, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 19, 13/14, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
20. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[20, -21], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=20. Give (p, n, p_moment, certificate).
View answer and explanation
(2, 20, 841/968, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 20, 841/968, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
21. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[21, -22], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=21. Give (p, n, p_moment, certificate).
View answer and explanation
(3, 21, 19909/24334, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 21, 19909/24334, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
22. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[22, -23], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=22. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 22, 15/16, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 22, 15/16, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
23. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[23, -24], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=23. Give (p, n, p_moment, certificate).
View answer and explanation
(2, 23, 221/250, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (2, 23, 221/250, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
24. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[24, -25], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=24. Give (p, n, p_moment, certificate).
View answer and explanation
(3, 24, 29449/35152, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (3, 24, 29449/35152, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
25. The p-Moment Is a Weighted Absolute-Power Error
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[25, -26], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. n=25. Give (p, n, p_moment, certificate).
View answer and explanation
(1, 25, 17/18, exact_finite_atomic_moment)
Compute each absolute amplitude raised to p, weight by that atom mass and add exactly. This output is E|error|^p, not its p-th-root norm or an unweighted atom average. The exact result, retaining the requested order, is (1, 25, 17/18, exact_finite_atomic_moment).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-moment-at-n · candidate
26. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[1, -2], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/10 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/10, 14, 3/32, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/10, 14, 3/32, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
27. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[2, -3], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/11 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(2, 1/11, 7, 13/162, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/11, 7, 13/162, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
28. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[3, -4], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/12 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(3, 1/12, 7, 91/1458, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/12, 7, 91/1458, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
29. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[4, -5], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/13 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/13, 57, 9/118, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/13, 57, 9/118, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
30. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[5, -6], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/14 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(2, 1/14, 19, 61/882, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/14, 19, 61/882, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
31. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[6, -7], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/15 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(3, 1/15, 15, 559/9826, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/15, 15, 559/9826, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
32. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/16 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/16, 119, 15/242, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/16, 119, 15/242, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
33. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[8, -9], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/17 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(2, 1/17, 34, 145/2592, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/17, 34, 145/2592, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
34. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[9, -10], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/18 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(3, 1/18, 23, 1729/31250, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/18, 23, 1729/31250, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
35. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/19 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/19, 198, 21/400, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/19, 198, 21/400, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
36. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[11, -12], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/20 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(2, 1/20, 50, 265/5408, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/20, 50, 265/5408, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
37. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[12, -13], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/21 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(3, 1/21, 33, 157/3430, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/21, 33, 157/3430, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
38. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[13, -14], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/22 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/22, 296, 27/596, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/22, 296, 27/596, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
39. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/23 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(2, 1/23, 68, 421/9800, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/23, 68, 421/9800, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
40. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[15, -16], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/24 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(3, 1/24, 43, 7471/182250, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/24, 43, 7471/182250, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
41. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[16, -17], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/25 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/25, 411, 33/826, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/25, 411, 33/826, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
42. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[17, -18], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/26 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(2, 1/26, 88, 613/16200, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/26, 88, 613/16200, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
43. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/27 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(3, 1/27, 54, 37/1024, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/27, 54, 37/1024, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
44. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[19, -20], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/28 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/28, 545, 39/1094, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/28, 545, 39/1094, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
45. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[20, -21], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/29 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(2, 1/29, 109, 841/24642, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/29, 109, 841/24642, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
46. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[21, -22], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/30 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(3, 1/30, 65, 19909/601526, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/30, 65, 19909/601526, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
47. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[22, -23], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/31 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/31, 696, 45/1396, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/31, 696, 45/1396, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
48. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[23, -24], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/32 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(2, 1/32, 131, 1105/35378, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (2, 1/32, 131, 1105/35378, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
49. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[24, -25], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/33 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(3, 1/33, 77, 29449/986078, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (3, 1/33, 77, 29449/986078, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
50. Epsilon Bounds the Moment, Not the p-th-Root Norm by the Same Number
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[25, -26], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Positive input epsilon=1/34 is a moment_tolerance. Give (p, moment_tolerance, N, p_moment_at_N, strict_tail, certificate).
Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance.
View answer and explanation
(1, 1/34, 866, 51/1736, Yes, forall_n_ge_N_symbolic_monotone_tail)
Choose the least nonnegative N with E|error_N|^p<epsilon. Each error moment decreases with n, so every n≥N satisfies that strict moment bound. A norm bound with the same epsilon would require comparison with epsilon^p instead; it is not the criterion computed here. The exact result, retaining the requested order, is (1, 1/34, 866, 51/1736, Yes, forall_n_ge_N_symbolic_monotone_tail).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-epsilon-witness · candidate
51. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[1, -2], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [1, -2], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [1, -2], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
52. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[2, -3], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(2, [1/2, 1/2], [2, -3], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [2, -3], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
53. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[3, -4], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(3, [1/2, 1/2], [3, -4], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [3, -4], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
54. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[4, -5], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [4, -5], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [4, -5], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
55. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[5, -6], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(2, [1/2, 1/2], [5, -6], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [5, -6], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
56. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[6, -7], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(3, [1/2, 1/2], [6, -7], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [6, -7], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
57. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [7, -8], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [7, -8], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
58. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[8, -9], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(2, [1/2, 1/2], [8, -9], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [8, -9], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
59. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[9, -10], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(3, [1/2, 1/2], [9, -10], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [9, -10], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
60. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [10, -11], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [10, -11], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
61. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[11, -12], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(2, [1/2, 1/2], [11, -12], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [11, -12], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
62. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[12, -13], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(3, [1/2, 1/2], [12, -13], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [12, -13], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
63. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[13, -14], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [13, -14], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [13, -14], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
64. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(2, [1/2, 1/2], [14, -15], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [14, -15], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
65. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[15, -16], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(3, [1/2, 1/2], [15, -16], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [15, -16], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
66. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[16, -17], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [16, -17], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [16, -17], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
67. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[17, -18], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(2, [1/2, 1/2], [17, -18], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [17, -18], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
68. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(3, [1/2, 1/2], [18, -19], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [18, -19], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
69. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[19, -20], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [19, -20], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [19, -20], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
70. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[20, -21], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(2, [1/2, 1/2], [20, -21], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [20, -21], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
71. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[21, -22], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(3, [1/2, 1/2], [21, -22], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [21, -22], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
72. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[22, -23], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [22, -23], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [22, -23], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
73. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[23, -24], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(2, [1/2, 1/2], [23, -24], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (2, [1/2, 1/2], [23, -24], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
74. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[24, -25], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(3, [1/2, 1/2], [24, -25], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (3, [1/2, 1/2], [24, -25], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
75. Keep the Fixed p, Common Law and Whole Symbolic Tail Together
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[25, -26], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Give (p, masses, coefficients, shift, qualified, certificate).
View answer and explanation
(1, [1/2, 1/2], [25, -26], 2, Yes, common_probability_space_and_finite_integer_p_moments)
The finite weighted absolute-power error has a common positive denominator (n+shift)^p and tends to zero under this explicit tail model. This supplies the stated finite-model moment certificate, not a new Lean proof or unrestricted cross-mode theorem. The exact result, retaining the requested order, is (1, [1/2, 1/2], [25, -26], 2, Yes, common_probability_space_and_finite_integer_p_moments).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-qualification · candidate
76. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[1, -2], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=1. Give (prefix_length, sufficient, boundary).
View answer and explanation
(1, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (1, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
77. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[2, -3], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=2. Give (prefix_length, sufficient, boundary).
View answer and explanation
(2, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (2, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
78. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[3, -4], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=3. Give (prefix_length, sufficient, boundary).
View answer and explanation
(3, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (3, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
79. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[4, -5], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=4. Give (prefix_length, sufficient, boundary).
View answer and explanation
(4, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (4, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
80. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[5, -6], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=5. Give (prefix_length, sufficient, boundary).
View answer and explanation
(5, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (5, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
81. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[6, -7], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=6. Give (prefix_length, sufficient, boundary).
View answer and explanation
(6, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (6, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
82. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=7. Give (prefix_length, sufficient, boundary).
View answer and explanation
(7, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (7, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
83. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[8, -9], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=8. Give (prefix_length, sufficient, boundary).
View answer and explanation
(8, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (8, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
84. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[9, -10], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=9. Give (prefix_length, sufficient, boundary).
View answer and explanation
(9, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (9, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
85. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[10, -11], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=10. Give (prefix_length, sufficient, boundary).
View answer and explanation
(10, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (10, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
86. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[11, -12], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=11. Give (prefix_length, sufficient, boundary).
View answer and explanation
(11, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (11, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
87. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[12, -13], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=12. Give (prefix_length, sufficient, boundary).
View answer and explanation
(12, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (12, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
88. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[13, -14], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=13. Give (prefix_length, sufficient, boundary).
View answer and explanation
(13, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (13, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
89. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[14, -15], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=14. Give (prefix_length, sufficient, boundary).
View answer and explanation
(14, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (14, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
90. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[15, -16], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=15. Give (prefix_length, sufficient, boundary).
View answer and explanation
(15, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (15, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
91. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[16, -17], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=16. Give (prefix_length, sufficient, boundary).
View answer and explanation
(16, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (16, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
92. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[17, -18], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=17. Give (prefix_length, sufficient, boundary).
View answer and explanation
(17, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (17, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
93. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[18, -19], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=18. Give (prefix_length, sufficient, boundary).
View answer and explanation
(18, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (18, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
94. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[19, -20], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=19. Give (prefix_length, sufficient, boundary).
View answer and explanation
(19, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (19, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
95. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[20, -21], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=20. Give (prefix_length, sufficient, boundary).
View answer and explanation
(20, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (20, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
96. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[21, -22], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=21. Give (prefix_length, sufficient, boundary).
View answer and explanation
(21, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (21, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
97. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[22, -23], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=22. Give (prefix_length, sufficient, boundary).
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(22, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (22, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
98. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[23, -24], positive shift=2, and fixed integer p=2 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=23. Give (prefix_length, sufficient, boundary).
View answer and explanation
(23, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (23, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
99. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[24, -25], positive shift=2, and fixed integer p=3 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=24. Give (prefix_length, sufficient, boundary).
View answer and explanation
(24, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (24, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
100. A Finite Prefix Does Not Establish Lp Convergence or Mode Implications
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[25, -26], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=25. Give (prefix_length, sufficient, boundary).
View answer and explanation
(25, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication)
The No concerns inference from a fixed finite prefix alone. It does not deny the separately declared rational-tail certificate or additional consequences available under explicit finite-space assumptions. General almost-sure, probability and other-Lq relations require their own stated hypotheses. The exact result, retaining the requested order, is (25, No, finite_prefix_not_Lp_convergence_certificate_or_mode_implication).
Reviewed step-by-step derivation
Lp convergence · qf-mod35-lp-finite-prefix-boundary · candidate
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