# Positive-Atom Tails, Null-Atom Exceptions and Finite-Prefix Evidence Limits

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261101-r04-almost-sure-convergence-0100/)

Almost-sure convergence · qf-mod32-asconv-finite-prefix-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations/validation-report.json)