Question 15323: Adaptedness Alone Is Not Martingale Qualification

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=[[10, 10, 10, 10], [10, 10, 10, 10]]. 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,adapted?). Failure rows use time order and first-seen block-label order; adaptedness failures retain [t,label,indices,distinct_values].

• 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=[[10, 10, 10, 10], [10, 10, 10, 10]]. 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,adapted?). Failure rows use time order and first-seen block-label order; adaptedness failures retain [t,label,indices,distinct_values].

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=[10, 10, 10, 10]

distinct current values first-encounter order=[10];constant on block=Yes

time=1;block label=0;indices=[0, 1];current values in original order=[10, 10]

distinct current values first-encounter order=[10];constant on block=Yes

time=1;block label=1;indices=[2, 3];current values in original order=[10, 10]

distinct current values first-encounter order=[10];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, 10)

martingale time0 block0 next weighted outcome0 = 5/2

martingale time0 block0 next weighted outcome1 := mul(1/4, 10)

martingale time0 block0 next weighted outcome1 = 5/2

martingale time0 block0 next weighted outcome2 := mul(1/4, 10)

martingale time0 block0 next weighted outcome2 = 5/2

martingale time0 block0 next weighted outcome3 := mul(1/4, 10)

martingale time0 block0 next weighted outcome3 = 5/2

martingale time0 block0 next weighted sum := sum(5/2, 5/2, 5/2, 5/2)

martingale time0 block0 next weighted sum = 10

martingale time0 block0 next conditional mean := div(10, 1)

martingale time0 block0 next conditional mean = 10

time=0;block label=0;current distinct values=[10];next weighted sum=10;positive block mass=1

conditional mean=10;conditional step row=[0, 0, 10, [10], Yes]

Reverse checks

• 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

([], [], 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.