Question 15335: Check One Conditional Step and Retain the Null-Block Variant

Original question and givens

Use one complete finite probability space with aligned exact masses=[1/4, 1/4, 1/4, 1/4], time-ordered partitions=[[0, 0, 0, 0], [0, 0, 1, 1]], and time-by-outcome process_values=[[-3, -3, -3, -3], [-3, -3, -3, -3]]. Masses are nonnegative and sum to 1; all process values are finite, hence integrable. Information nesting, blockwise adaptedness and the conditional-step equation are separate obligations. Process and partition rows use the same outcomes. Null-block conditional values are only almost-surely specified, with no division by zero. No independent-increment, Markov, stopping or convergence theorem follows. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Selected time=0, block_label=0 must identify a block before the final time. On positive mass give (time,block_label,conditional_mean,current_values,pass?). On zero mass give (time,block_label,null_block_a.s._arbitrary,Yes), omitting the point-value field.

• G001 — Use one complete finite probability space with aligned exact masses=[1/4, 1/4, 1/4, 1/4], time-ordered partitions=[[0, 0, 0, 0], [0, 0, 1, 1]], and time-by-outcome process_values=[[-3, -3, -3, -3], [-3, -3, -3, -3]]. Masses are nonnegative and sum to 1; all process values are finite, hence integrable. Information nesting, blockwise adaptedness and the conditional-step equation are separate obligations. Process and partition rows use the same outcomes. Null-block conditional values are only almost-surely specified, with no division by zero. No independent-increment, Markov, stopping or convergence theorem follows. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Selected time=0, block_label=0 must identify a block before the final time. On positive mass give (time,block_label,conditional_mean,current_values,pass?). On zero mass give (time,block_label,null_block_a.s._arbitrary,Yes), omitting the point-value field.

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-martingale

Step-by-step derivation

martingale full probability mass := sum(1/4, 1/4, 1/4, 1/4)

martingale full probability mass = 1

same finite probability carrier size=4;exact masses=[1/4, 1/4, 1/4, 1/4];partitions in time order=[[0, 0, 0, 0], [0, 0, 1, 1]]

total mass=1;all later-equal pairs with earlier-different labels=[]

time=0;block label=0;indices=[0, 1, 2, 3];current values in original order=[-3, -3, -3, -3]

distinct current values first-encounter order=[-3];constant on block=Yes

time=1;block label=0;indices=[0, 1];current values in original order=[-3, -3]

distinct current values first-encounter order=[-3];constant on block=Yes

time=1;block label=1;indices=[2, 3];current values in original order=[-3, -3]

distinct current values first-encounter order=[-3];constant on block=Yes

martingale time0 block0 mass := sum(1/4, 1/4, 1/4, 1/4)

martingale time0 block0 mass = 1

martingale time0 block0 next weighted outcome0 := mul(1/4, -3)

martingale time0 block0 next weighted outcome0 = -3/4

martingale time0 block0 next weighted outcome1 := mul(1/4, -3)

martingale time0 block0 next weighted outcome1 = -3/4

martingale time0 block0 next weighted outcome2 := mul(1/4, -3)

martingale time0 block0 next weighted outcome2 = -3/4

martingale time0 block0 next weighted outcome3 := mul(1/4, -3)

martingale time0 block0 next weighted outcome3 = -3/4

martingale time0 block0 next weighted sum := sum(-3/4, -3/4, -3/4, -3/4)

martingale time0 block0 next weighted sum = -3

martingale time0 block0 next conditional mean := div(-3, 1)

martingale time0 block0 next conditional mean = -3

time=0;block label=0;current distinct values=[-3];next weighted sum=-3;positive block mass=1

conditional mean=-3;conditional step row=[0, 0, -3, [-3], Yes]

requested current time=0;original block label=0;all source-bound conditional rows=[[0, 0, -3, [-3], Yes]]

selected conditional row=[0, 0, -3, [-3], Yes]

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute martingale time0 block0 next conditional mean with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute martingale time0 block0 next weighted sum with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute martingale time0 block0 next weighted outcome3 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute martingale time0 block0 next weighted outcome2 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute martingale time0 block0 next weighted outcome1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute martingale time0 block0 next weighted outcome0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute martingale time0 block0 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 martingale full probability mass with exact rational arithmetic, not rounded decimals.

Result

(0, 0, -3, [-3], 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.