Question 15371: Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[8, 8, 8, 8], [8, 8, 8, 8]]. 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 (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

• 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=[[8, 8, 8, 8], [8, 8, 8, 8]]. 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 (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time 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-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=[8, 8, 8, 8]

distinct current values first-encounter order=[8];constant on block=Yes

time=1;block label=0;indices=[0, 1];current values in original order=[8, 8]

distinct current values first-encounter order=[8];constant on block=Yes

time=1;block label=1;indices=[2, 3];current values in original order=[8, 8]

distinct current values first-encounter order=[8];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, 8)

martingale time0 block0 next weighted outcome0 = 2

martingale time0 block0 next weighted outcome1 := mul(1/4, 8)

martingale time0 block0 next weighted outcome1 = 2

martingale time0 block0 next weighted outcome2 := mul(1/4, 8)

martingale time0 block0 next weighted outcome2 = 2

martingale time0 block0 next weighted outcome3 := mul(1/4, 8)

martingale time0 block0 next weighted outcome3 = 2

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

martingale time0 block0 next weighted sum = 8

martingale time0 block0 next conditional mean := div(8, 1)

martingale time0 block0 next conditional mean = 8

time=0;block label=0;current distinct values=[8];next weighted sum=8;positive block mass=1

conditional mean=8;conditional step row=[0, 0, 8, [8], Yes]

martingale expectation time0 weighted outcome0 := mul(1/4, 8)

martingale expectation time0 weighted outcome0 = 2

martingale expectation time0 weighted outcome1 := mul(1/4, 8)

martingale expectation time0 weighted outcome1 = 2

martingale expectation time0 weighted outcome2 := mul(1/4, 8)

martingale expectation time0 weighted outcome2 = 2

martingale expectation time0 weighted outcome3 := mul(1/4, 8)

martingale expectation time0 weighted outcome3 = 2

martingale expectation time0 full sum := sum(2, 2, 2, 2)

martingale expectation time0 full sum = 8

martingale expectation time1 weighted outcome0 := mul(1/4, 8)

martingale expectation time1 weighted outcome0 = 2

martingale expectation time1 weighted outcome1 := mul(1/4, 8)

martingale expectation time1 weighted outcome1 = 2

martingale expectation time1 weighted outcome2 := mul(1/4, 8)

martingale expectation time1 weighted outcome2 = 2

martingale expectation time1 weighted outcome3 := mul(1/4, 8)

martingale expectation time1 weighted outcome3 = 2

martingale expectation time1 full sum := sum(2, 2, 2, 2)

martingale expectation time1 full sum = 8

unconditional exact expectations in original time order=[8, 8];full conditional rows=[[0, 0, 8, [8], Yes]]

expectation invariant=Yes;expectation_invariance_is_consequence_not_definition

Reverse checks

• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute martingale expectation time1 full sum with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute martingale expectation time1 weighted outcome3 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute martingale expectation time1 weighted outcome2 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute martingale expectation time1 weighted outcome1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute martingale expectation time1 weighted outcome0 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute martingale expectation time0 full sum with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute martingale expectation time0 weighted outcome3 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute martingale expectation time0 weighted outcome2 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute martingale expectation time0 weighted outcome1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute martingale expectation time0 weighted outcome0 with exact rational arithmetic, not rounded decimals.
• 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

([8, 8], Yes, expectation_invariance_is_consequence_not_definition)

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.