Positive-Atom Tails, Null-Atom Exceptions and Finite-Prefix Evidence Limits
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batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[1, 2, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
2. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[2, 3, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
3. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
4. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[4, 5, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
5. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
6. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[6, 7, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
7. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[7, 8, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
8. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[8, 9, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
9. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[9, 10, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
10. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[10, 11, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
11. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[11, 12, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
12. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[12, 13, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
13. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
14. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[14, 15, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
15. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[15, 16, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
16. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[16, 17, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
17. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[17, 18, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
18. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[18, 19, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
19. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[19, 20, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
20. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[20, 21, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
21. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[21, 22, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
22. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[22, 23, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
23. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[23, 24, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
24. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[24, 25, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
25. Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[25, 26, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.
View answer and explanation
([[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)
Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-qualification · candidate
26. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[1, 2, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes]], 2, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes]], 2, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
27. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[2, 3, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes]], 3, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes]], 3, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
28. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes]], 4, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes]], 4, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
29. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[4, 5, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes]], 5, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes]], 5, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
30. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes]], 6, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes]], 6, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
31. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[6, 7, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes]], 7, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes]], 7, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
32. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[7, 8, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes]], 8, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes]], 8, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
33. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[8, 9, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes]], 9, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes]], 9, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
34. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[9, 10, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes]], 10, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes]], 10, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
35. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[10, 11, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes]], 11, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes]], 11, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
36. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[11, 12, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes]], 12, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes]], 12, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
37. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[12, 13, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes]], 13, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes]], 13, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
38. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes]], 14, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes]], 14, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
39. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[14, 15, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes]], 15, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes]], 15, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
40. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[15, 16, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes]], 16, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes]], 16, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
41. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[16, 17, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes]], 17, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes]], 17, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
42. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[17, 18, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes]], 18, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes]], 18, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
43. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[18, 19, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes]], 19, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes]], 19, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
44. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[19, 20, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes]], 20, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes]], 20, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
45. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[20, 21, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes]], 21, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes]], 21, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
46. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[21, 22, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes]], 22, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes]], 22, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
47. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[22, 23, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes]], 23, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes]], 23, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
48. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[23, 24, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes]], 24, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes]], 24, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
49. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[24, 25, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes]], 25, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes]], 25, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
50. A Finite Number of Positive Atoms Gives a Maximum Settling Index
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[25, 26, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.
View answer and explanation
([[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes]], 26, Yes, uniform_finite_atom_tail_after_maximum)
Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes]], 26, Yes, uniform_finite_atom_tail_after_maximum).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-settling-index · candidate
51. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[1, 2, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
52. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[2, 3, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
53. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
54. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[4, 5, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
55. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
56. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[6, 7, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
57. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[7, 8, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
58. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[8, 9, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
59. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[9, 10, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
60. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[10, 11, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
61. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[11, 12, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
62. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[12, 13, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
63. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
64. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[14, 15, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
65. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[15, 16, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
66. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[16, 17, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
67. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[17, 18, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
68. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[18, 19, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
69. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[19, 20, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
70. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[20, 21, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
71. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[21, 22, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
72. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[22, 23, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
73. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[23, 24, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
74. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[24, 25, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
75. A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[25, 26, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).
View answer and explanation
(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)
Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-zero-mass-boundary · candidate
76. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[1, 2, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=1. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(1, [[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (1, [[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
77. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[2, 3, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=2. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(2, [[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (2, [[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
78. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=3. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(3, [[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (3, [[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
79. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[4, 5, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=4. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(4, [[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (4, [[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
80. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=5. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(5, [[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (5, [[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
81. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[6, 7, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=6. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(6, [[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (6, [[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
82. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[7, 8, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=7. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(7, [[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (7, [[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
83. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[8, 9, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=8. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(8, [[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (8, [[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
84. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[9, 10, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=9. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(9, [[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (9, [[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
85. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[10, 11, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=10. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(10, [[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (10, [[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
86. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[11, 12, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=11. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(11, [[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (11, [[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
87. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[12, 13, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=12. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(12, [[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (12, [[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
88. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=13. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(13, [[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (13, [[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
89. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[14, 15, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=14. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(14, [[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (14, [[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
90. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[15, 16, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=15. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(15, [[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (15, [[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
91. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[16, 17, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=16. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(16, [[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (16, [[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
92. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[17, 18, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=17. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(17, [[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (17, [[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
93. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[18, 19, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=18. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(18, [[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (18, [[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
94. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[19, 20, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=19. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(19, [[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (19, [[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
95. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[20, 21, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=20. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(20, [[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (20, [[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
96. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[21, 22, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=21. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(21, [[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (21, [[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
97. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[22, 23, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=22. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(22, [[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (22, [[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
98. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[23, 24, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=23. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(23, [[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (23, [[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
99. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[24, 25, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=24. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(24, [[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (24, [[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
100. A Finite Prefix Alone Does Not Establish Every Atomwise Tail
Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[25, 26, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=25. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).
View answer and explanation
(25, [[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)
The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (25, [[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate).
Reviewed step-by-step derivation
Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate
Canonical question JSONL · English JSONL · Validation report