Question 16049: Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model

Original question and givens

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.

• G001 — 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.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-almost-sure-convergence

Step-by-step derivation

almost-sure full probability mass := sum(1/2, 1/2, 0)

almost-sure full probability mass = 1

same complete finite atomic space;raw masses=[[1, 2], [1, 2], [0, 1]];raw settling declarations=[24, 25, null]

normalized nonnegative masses=[1/2, 1/2, 0];total=1;integer N means equality to common limit at EVERY later index;null means model-declared nonconvergence

original atom=0;exact mass=1/2;full-tail declaration=24

atom row=[0, 1/2, 24, Yes, Yes]

original atom=1;exact mass=1/2;full-tail declaration=25

atom row=[1, 1/2, 25, Yes, Yes]

original atom=2;exact mass=0;full-tail declaration=null

atom row=[2, 0, null, No, Yes]

almost-sure declared nonconvergent exceptional mass := sum(0)

almost-sure declared nonconvergent exceptional mass = 0

all model-declared nonconvergent atom indices=[2];their total mass=0

positive-mass failures=[];almost-sure qualification=Yes

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute almost-sure declared nonconvergent exceptional mass with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute almost-sure full probability mass with exact rational arithmetic, not rounded decimals.

Result

([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.