Question 15392: Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[4, 4, 4, 4], [4, 4, 4, 4]]. 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. Give (nesting_failures,adaptedness_failures,conditional_steps,martingale?). Nesting rows are [t,a,b], t increasing and a<b increasing. Adaptedness rows are [t,block_label,indices,distinct_values], traversing first-seen block labels within each row. Positive-mass step rows are [t,label,next_conditional_mean,current_distinct_values,pass?]; null steps are [t,label,null_block_a.s._arbitrary,Yes] with no forced current-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=[[4, 4, 4, 4], [4, 4, 4, 4]]. 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. Give (nesting_failures,adaptedness_failures,conditional_steps,martingale?). Nesting rows are [t,a,b], t increasing and a<b increasing. Adaptedness rows are [t,block_label,indices,distinct_values], traversing first-seen block labels within each row. Positive-mass step rows are [t,label,next_conditional_mean,current_distinct_values,pass?]; null steps are [t,label,null_block_a.s._arbitrary,Yes] with no forced current-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=[4, 4, 4, 4]

distinct current values first-encounter order=[4];constant on block=Yes

time=1;block label=0;indices=[0, 1];current values in original order=[4, 4]

distinct current values first-encounter order=[4];constant on block=Yes

time=1;block label=1;indices=[2, 3];current values in original order=[4, 4]

distinct current values first-encounter order=[4];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, 4)

martingale time0 block0 next weighted outcome0 = 1

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

martingale time0 block0 next weighted outcome1 = 1

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

martingale time0 block0 next weighted outcome2 = 1

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

martingale time0 block0 next weighted outcome3 = 1

martingale time0 block0 next weighted sum := sum(1, 1, 1, 1)

martingale time0 block0 next weighted sum = 4

martingale time0 block0 next conditional mean := div(4, 1)

martingale time0 block0 next conditional mean = 4

time=0;block label=0;current distinct values=[4];next weighted sum=4;positive block mass=1

conditional mean=4;conditional step row=[0, 0, 4, [4], Yes]

nesting failures=[];adaptedness failures=[];conditional rows=[[0, 0, 4, [4], Yes]]

finite integrable process;all three qualifications=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, 4, [4], Yes]], 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.