# Question 15390: Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0029-15390 · q-20261014-advance-20261030-r03-martingale-0015



## 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=[[2, 2, 2, 2], [2, 2, 2, 2]]. 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=[[2, 2, 2, 2], [2, 2, 2, 2]]. 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](https://ocme.evemisslab.com/en/mko/mko-martingale/)





## Step-by-step derivation

S001 · calculationBefore

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

After

martingale full probability mass = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · impliesBefore

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]]

After

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

Conditions: none

Every later block must lie inside one earlier block.Check all outcome pairs;more block labels alone do not prove refinement.Nonnegative masses totaling1 are the source English probability-law premise;loose raw-oracle guards do not license negative probability.

Step identifiers and dependenciesRule: martingale_filtration_refinement_all_outcome_pairs · Uses: G001, S001





S003 · impliesBefore

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

After

distinct current values first-encounter order=[2];constant on block=Yes

Conditions: none

Measurability for this finite partition requires the current value to be constant on each block.Keep distinct values in first-encounter order.Null outcomes are not removed from this specified version's adaptedness audit.

Step identifiers and dependenciesRule: martingale_adaptedness_current_time_including_null_atoms · Uses: G001, S002





S004 · impliesBefore

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

After

distinct current values first-encounter order=[2];constant on block=Yes

Conditions: none

Measurability for this finite partition requires the current value to be constant on each block.Keep distinct values in first-encounter order.Null outcomes are not removed from this specified version's adaptedness audit.

Step identifiers and dependenciesRule: martingale_adaptedness_current_time_including_null_atoms · Uses: G001, S003





S005 · impliesBefore

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

After

distinct current values first-encounter order=[2];constant on block=Yes

Conditions: none

Measurability for this finite partition requires the current value to be constant on each block.Keep distinct values in first-encounter order.Null outcomes are not removed from this specified version's adaptedness audit.

Step identifiers and dependenciesRule: martingale_adaptedness_current_time_including_null_atoms · Uses: G001, S004





S006 · calculationBefore

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

After

martingale time0 block0 mass = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

martingale time0 block0 next weighted outcome0 := mul(1/4, 2)

After

martingale time0 block0 next weighted outcome0 = 1/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

martingale time0 block0 next weighted outcome1 := mul(1/4, 2)

After

martingale time0 block0 next weighted outcome1 = 1/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

martingale time0 block0 next weighted outcome2 := mul(1/4, 2)

After

martingale time0 block0 next weighted outcome2 = 1/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

martingale time0 block0 next weighted outcome3 := mul(1/4, 2)

After

martingale time0 block0 next weighted outcome3 = 1/2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

martingale time0 block0 next weighted sum := sum(1/2, 1/2, 1/2, 1/2)

After

martingale time0 block0 next weighted sum = 2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

martingale time0 block0 next conditional mean := div(2, 1)

After

martingale time0 block0 next conditional mean = 2

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · impliesBefore

time=0;block label=0;current distinct values=[2];next weighted sum=2;positive block mass=1

After

conditional mean=2;conditional step row=[0, 0, 2, [2], Yes]

Conditions: none

Weight next-time values,divide by this current block's probability,and compare with its current constant value.Do not substitute a next-time value or a global expectation.A nonconstant current block does not pass this step either.

Step identifiers and dependenciesRule: martingale_next_time_conditional_mean_equals_current_not_next_value · Uses: G001, S012





S014 · impliesBefore

nesting failures=[];adaptedness failures=[];conditional rows=[[0, 0, 2, [2], Yes]]

After

finite integrable process;all three qualifications=Yes

Conditions: none

Finite values on a finite probability carrier give integrability.Still require filtration refinement,current measurability and each next-step conditional expectation equaling the current value.A global mean or a single passing row is insufficient.

Step identifiers and dependenciesRule: martingale_qualification_three_separate_obligations · Uses: G001, S013









## 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, 2, [2], 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 608d228a0f05c1ef0c294642148fadf1505c8d07fc7c19d6c499e41ee3cedf13
- candidate record SHA-256: 5c9b993cbddc49e54b5c8968d8ca8e7708d971be9d9f2ae64e29d29b1098d1b0
- semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725
- question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117
- Lean: NOT_RUN · human review: NOT_PERFORMED

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