# Separate Nesting, Adaptedness, Conditional Steps and Expectation Consequences

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261030-r03-martingale-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[-12, -12, -12, -12], [-11, -12, -12, -12]]. 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.

View answer and explanation([], [[1, 0, [0, 1], [-11, -12]]], [[0, 0, -47/4, [-12], No]], No)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [[1, 0, [0, 1], [-11, -12]]], [[0, 0, -47/4, [-12], No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0001/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 2. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[-11, -11, -11, -11], [-11, -11, -11, -11]]. 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.

View answer and explanation([], [], [[0, 0, -11, [-11], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -11, [-11], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0002/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 3. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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,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.

View answer and explanation([], [], [[0, 0, -10, [-10], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -10, [-10], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0003/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 4. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[-9, -9, -9, -9], [-9, -9, -9, -9]]. 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.

View answer and explanation([], [], [[0, 0, -9, [-9], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -9, [-9], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0004/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 5. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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 (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.

View answer and explanation([], [], [[0, 0, -8, [-8], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -8, [-8], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0005/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 6. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[-7, -7, -7, -7], [-6, -7, -7, -7]]. 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.

View answer and explanation([], [[1, 0, [0, 1], [-6, -7]]], [[0, 0, -27/4, [-7], No]], No)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [[1, 0, [0, 1], [-6, -7]]], [[0, 0, -27/4, [-7], No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0006/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 7. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[-6, -6, -6, -6], [-6, -6, -6, -6]]. 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.

View answer and explanation([], [], [[0, 0, -6, [-6], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -6, [-6], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0007/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 8. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[-5, -5, -5, -5], [-5, -5, -5, -5]]. 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.

View answer and explanation([], [], [[0, 0, -5, [-5], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -5, [-5], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0008/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 9. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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.

View answer and explanation([], [], [[0, 0, -4, [-4], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -4, [-4], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0009/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 10. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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. 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.

View answer and explanation([], [], [[0, 0, -3, [-3], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -3, [-3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0010/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 11. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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], [-1, -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.

View answer and explanation([], [[1, 0, [0, 1], [-1, -2]]], [[0, 0, -7/4, [-2], No]], No)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [[1, 0, [0, 1], [-1, -2]]], [[0, 0, -7/4, [-2], No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0011/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 12. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[-1, -1, -1, -1], [-1, -1, -1, -1]]. 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.

View answer and explanation([], [], [[0, 0, -1, [-1], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, -1, [-1], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0012/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 13. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[0, 0, 0, 0], [0, 0, 0, 0]]. 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.

View answer and explanation([], [], [[0, 0, 0, [0], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 0, [0], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0013/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 14. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[1, 1, 1, 1], [1, 1, 1, 1]]. 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.

View answer and explanation([], [], [[0, 0, 1, [1], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 1, [1], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0014/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 15. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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.

View answer and explanation([], [], [[0, 0, 2, [2], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 2, [2], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0015/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 16. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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], [4, 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. 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.

View answer and explanation([], [[1, 0, [0, 1], [4, 3]]], [[0, 0, 13/4, [3], No]], No)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [[1, 0, [0, 1], [4, 3]]], [[0, 0, 13/4, [3], No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0016/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 17. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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.

View answer and explanation([], [], [[0, 0, 4, [4], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 4, [4], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0017/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 18. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[5, 5, 5, 5], [5, 5, 5, 5]]. 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.

View answer and explanation([], [], [[0, 0, 5, [5], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 5, [5], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0018/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 19. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[6, 6, 6, 6], [6, 6, 6, 6]]. 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.

View answer and explanation([], [], [[0, 0, 6, [6], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 6, [6], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0019/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 20. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[7, 7, 7, 7], [7, 7, 7, 7]]. 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.

View answer and explanation([], [], [[0, 0, 7, [7], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 7, [7], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0020/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 21. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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], [9, 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 (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.

View answer and explanation([], [[1, 0, [0, 1], [9, 8]]], [[0, 0, 33/4, [8], No]], No)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [[1, 0, [0, 1], [9, 8]]], [[0, 0, 33/4, [8], No]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0021/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 22. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[9, 9, 9, 9], [9, 9, 9, 9]]. 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.

View answer and explanation([], [], [[0, 0, 9, [9], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 9, [9], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0022/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 23. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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,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.

View answer and explanation([], [], [[0, 0, 10, [10], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 10, [10], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0023/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 24. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[11, 11, 11, 11], [11, 11, 11, 11]]. 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.

View answer and explanation([], [], [[0, 0, 11, [11], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 11, [11], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0024/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 25. Conjoin Nesting, Adaptedness and Every Conditional-Step Equation

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=[[12, 12, 12, 12], [12, 12, 12, 12]]. 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.

View answer and explanation([], [], [[0, 0, 12, [12], Yes]], Yes)

First check filtration refinement and process measurability at every time. For each time block, average the next process value under the same conditional law and compare with the current block value. Conjoin all obligations; constant total expectations alone are insufficient. The exact result, retaining the requested order, is ([], [], [[0, 0, 12, [12], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0025/)

Finite martingale · qf-mod30-martingale-qualification · candidate





## 26. Adaptedness Alone Is Not Martingale Qualification

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=[[-12, -12, -12, -12], [-12, -12, -12, -12]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0026/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 27. Adaptedness Alone Is Not Martingale Qualification

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=[[-11, -11, -11, -11], [-11, -11, -11, -11]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0027/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 28. Adaptedness Alone Is Not Martingale Qualification

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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0028/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 29. Adaptedness Alone Is Not Martingale Qualification

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=[[-9, -9, -9, -9], [-9, -9, -9, -9]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0029/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 30. Adaptedness Alone Is Not Martingale Qualification

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

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0030/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 31. Adaptedness Alone Is Not Martingale Qualification

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=[[-7, -7, -7, -7], [-7, -7, -7, -7]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0031/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 32. Adaptedness Alone Is Not Martingale Qualification

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=[[-6, -6, -6, -6], [-6, -6, -6, -6]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0032/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 33. Adaptedness Alone Is Not Martingale Qualification

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=[[-5, -5, -5, -5], [-5, -5, -5, -5]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0033/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 34. Adaptedness Alone Is Not Martingale Qualification

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

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0034/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 35. Adaptedness Alone Is Not Martingale Qualification

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. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0035/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 36. Adaptedness Alone Is Not Martingale Qualification

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

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0036/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 37. Adaptedness Alone Is Not Martingale Qualification

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=[[-1, -1, -1, -1], [-1, -1, -1, -1]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0037/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 38. Adaptedness Alone Is Not Martingale Qualification

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=[[0, 0, 0, 0], [0, 0, 0, 0]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0038/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 39. Adaptedness Alone Is Not Martingale Qualification

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=[[1, 1, 1, 1], [1, 1, 1, 1]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0039/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 40. Adaptedness Alone Is Not Martingale Qualification

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

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0040/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 41. Adaptedness Alone Is Not Martingale Qualification

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. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0041/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 42. Adaptedness Alone Is Not Martingale Qualification

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

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0042/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 43. Adaptedness Alone Is Not Martingale Qualification

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=[[5, 5, 5, 5], [5, 5, 5, 5]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0043/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 44. Adaptedness Alone Is Not Martingale Qualification

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=[[6, 6, 6, 6], [6, 6, 6, 6]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0044/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 45. Adaptedness Alone Is Not Martingale Qualification

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=[[7, 7, 7, 7], [7, 7, 7, 7]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0045/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 46. Adaptedness Alone Is Not Martingale Qualification

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

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0046/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 47. Adaptedness Alone Is Not Martingale Qualification

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=[[9, 9, 9, 9], [9, 9, 9, 9]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0047/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 48. Adaptedness Alone Is Not Martingale Qualification

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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0048/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 49. Adaptedness Alone Is Not Martingale Qualification

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=[[11, 11, 11, 11], [11, 11, 11, 11]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0049/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 50. Adaptedness Alone Is Not Martingale Qualification

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=[[12, 12, 12, 12], [12, 12, 12, 12]]. 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].

View answer and explanation([], [], Yes)

A current process value must be constant on each current information block. The final adapted flag here records only absence of these measurability failures. Retain nesting failures separately; conditional-step equality still needs its own check. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0050/)

Finite martingale · qf-mod30-martingale-adaptedness · candidate





## 51. Check One Conditional Step and Retain the Null-Block Variant

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=[[-12, -12, -12, -12], [-12, -12, -12, -12]]. 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.

View answer and explanation(0, 0, -12, [-12], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -12, [-12], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0051/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 52. Check One Conditional Step and Retain the Null-Block Variant

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=[[-11, -11, -11, -11], [-11, -11, -11, -11]]. 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.

View answer and explanation(0, 0, -11, [-11], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -11, [-11], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0052/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 53. Check One Conditional Step and Retain the Null-Block Variant

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. 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.

View answer and explanation(0, 0, -10, [-10], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -10, [-10], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0053/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 54. Check One Conditional Step and Retain the Null-Block Variant

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=[[-9, -9, -9, -9], [-9, -9, -9, -9]]. 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.

View answer and explanation(0, 0, -9, [-9], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -9, [-9], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0054/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 55. Check One Conditional Step and Retain the Null-Block Variant

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. 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.

View answer and explanation(0, 0, -8, [-8], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -8, [-8], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0055/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 56. Check One Conditional Step and Retain the Null-Block Variant

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=[[-7, -7, -7, -7], [-7, -7, -7, -7]]. 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.

View answer and explanation(0, 0, -7, [-7], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -7, [-7], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0056/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 57. Check One Conditional Step and Retain the Null-Block Variant

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=[[-6, -6, -6, -6], [-6, -6, -6, -6]]. 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.

View answer and explanation(0, 0, -6, [-6], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -6, [-6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0057/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 58. Check One Conditional Step and Retain the Null-Block Variant

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=[[-5, -5, -5, -5], [-5, -5, -5, -5]]. 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.

View answer and explanation(0, 0, -5, [-5], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -5, [-5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0058/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 59. Check One Conditional Step and Retain the Null-Block Variant

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. 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.

View answer and explanation(0, 0, -4, [-4], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -4, [-4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0059/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 60. Check One Conditional Step and Retain the Null-Block Variant

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.

View answer and explanation(0, 0, -3, [-3], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -3, [-3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0060/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 61. Check One Conditional Step and Retain the Null-Block Variant

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. 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.

View answer and explanation(0, 0, -2, [-2], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -2, [-2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0061/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 62. Check One Conditional Step and Retain the Null-Block Variant

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=[[-1, -1, -1, -1], [-1, -1, -1, -1]]. 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.

View answer and explanation(0, 0, -1, [-1], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, -1, [-1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0062/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 63. Check One Conditional Step and Retain the Null-Block Variant

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=[[0, 0, 0, 0], [0, 0, 0, 0]]. 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.

View answer and explanation(0, 0, 0, [0], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 0, [0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0063/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 64. Check One Conditional Step and Retain the Null-Block Variant

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=[[1, 1, 1, 1], [1, 1, 1, 1]]. 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.

View answer and explanation(0, 0, 1, [1], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 1, [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0064/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 65. Check One Conditional Step and Retain the Null-Block Variant

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. 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.

View answer and explanation(0, 0, 2, [2], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 2, [2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0065/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 66. Check One Conditional Step and Retain the Null-Block Variant

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.

View answer and explanation(0, 0, 3, [3], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 3, [3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0066/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 67. Check One Conditional Step and Retain the Null-Block Variant

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. 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.

View answer and explanation(0, 0, 4, [4], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 4, [4], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0067/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 68. Check One Conditional Step and Retain the Null-Block Variant

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=[[5, 5, 5, 5], [5, 5, 5, 5]]. 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.

View answer and explanation(0, 0, 5, [5], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 5, [5], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0068/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 69. Check One Conditional Step and Retain the Null-Block Variant

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=[[6, 6, 6, 6], [6, 6, 6, 6]]. 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.

View answer and explanation(0, 0, 6, [6], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 6, [6], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0069/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 70. Check One Conditional Step and Retain the Null-Block Variant

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=[[7, 7, 7, 7], [7, 7, 7, 7]]. 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.

View answer and explanation(0, 0, 7, [7], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 7, [7], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0070/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 71. Check One Conditional Step and Retain the Null-Block Variant

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. 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.

View answer and explanation(0, 0, 8, [8], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 8, [8], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0071/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 72. Check One Conditional Step and Retain the Null-Block Variant

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=[[9, 9, 9, 9], [9, 9, 9, 9]]. 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.

View answer and explanation(0, 0, 9, [9], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 9, [9], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0072/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 73. Check One Conditional Step and Retain the Null-Block Variant

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. 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.

View answer and explanation(0, 0, 10, [10], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 10, [10], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0073/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 74. Check One Conditional Step and Retain the Null-Block Variant

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=[[11, 11, 11, 11], [11, 11, 11, 11]]. 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.

View answer and explanation(0, 0, 11, [11], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 11, [11], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0074/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 75. Check One Conditional Step and Retain the Null-Block Variant

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=[[12, 12, 12, 12], [12, 12, 12, 12]]. 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.

View answer and explanation(0, 0, 12, [12], Yes)

Compute the next-time weighted mean over this current information block. For positive mass it must equal the unique current value on that block. A null-block equation is only an almost-sure obligation and uses the explicit shorter tuple; one passing block does not certify the whole process. The exact result, retaining the requested order, is (0, 0, 12, [12], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0075/)

Finite martingale · qf-mod30-martingale-conditional-step · candidate





## 76. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[-12, -12, -12, -12], [-12, -12, -12, -12]]. 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.

View answer and explanation([-12, -12], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-12, -12], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0076/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 77. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[-11, -11, -11, -11], [-11, -11, -11, -11]]. 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.

View answer and explanation([-11, -11], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-11, -11], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0077/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 78. Equal Expectations Are a Consequence, Not a Martingale Definition

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 (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

View answer and explanation([-10, -10], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-10, -10], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0078/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 79. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[-9, -9, -9, -9], [-9, -9, -9, -9]]. 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.

View answer and explanation([-9, -9], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-9, -9], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0079/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 80. Equal Expectations Are a Consequence, Not a Martingale Definition

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.

View answer and explanation([-8, -8], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-8, -8], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0080/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 81. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[-7, -7, -7, -7], [-7, -7, -7, -7]]. 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.

View answer and explanation([-7, -7], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-7, -7], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0081/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 82. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[-6, -6, -6, -6], [-6, -6, -6, -6]]. 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.

View answer and explanation([-6, -6], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-6, -6], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0082/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 83. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[-5, -5, -5, -5], [-5, -5, -5, -5]]. 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.

View answer and explanation([-5, -5], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-5, -5], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0083/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 84. Equal Expectations Are a Consequence, Not a Martingale Definition

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 (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

View answer and explanation([-4, -4], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-4, -4], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0084/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 85. Equal Expectations Are a Consequence, Not a Martingale Definition

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. Give (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

View answer and explanation([-3, -3], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-3, -3], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0085/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 86. Equal Expectations Are a Consequence, Not a Martingale Definition

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 (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

View answer and explanation([-2, -2], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-2, -2], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0086/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 87. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[-1, -1, -1, -1], [-1, -1, -1, -1]]. 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.

View answer and explanation([-1, -1], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([-1, -1], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0087/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 88. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[0, 0, 0, 0], [0, 0, 0, 0]]. 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.

View answer and explanation([0, 0], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([0, 0], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0088/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 89. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[1, 1, 1, 1], [1, 1, 1, 1]]. 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.

View answer and explanation([1, 1], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([1, 1], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0089/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 90. Equal Expectations Are a Consequence, Not a Martingale Definition

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 (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

View answer and explanation([2, 2], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([2, 2], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0090/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 91. Equal Expectations Are a Consequence, Not a Martingale Definition

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. Give (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

View answer and explanation([3, 3], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([3, 3], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0091/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 92. Equal Expectations Are a Consequence, Not a Martingale Definition

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 (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

View answer and explanation([4, 4], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([4, 4], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0092/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 93. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[5, 5, 5, 5], [5, 5, 5, 5]]. 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.

View answer and explanation([5, 5], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([5, 5], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0093/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 94. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[6, 6, 6, 6], [6, 6, 6, 6]]. 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.

View answer and explanation([6, 6], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([6, 6], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0094/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 95. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[7, 7, 7, 7], [7, 7, 7, 7]]. 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.

View answer and explanation([7, 7], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([7, 7], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0095/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 96. Equal Expectations Are a Consequence, Not a Martingale Definition

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.

View answer and explanation([8, 8], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([8, 8], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0096/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 97. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[9, 9, 9, 9], [9, 9, 9, 9]]. 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.

View answer and explanation([9, 9], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([9, 9], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0097/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 98. Equal Expectations Are a Consequence, Not a Martingale Definition

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 (expectations,invariant?,expectation_invariance_is_consequence_not_definition), preserving time order.

View answer and explanation([10, 10], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([10, 10], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0098/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 99. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[11, 11, 11, 11], [11, 11, 11, 11]]. 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.

View answer and explanation([11, 11], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([11, 11], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0099/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate





## 100. Equal Expectations Are a Consequence, Not a Martingale Definition

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=[[12, 12, 12, 12], [12, 12, 12, 12]]. 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.

View answer and explanation([12, 12], Yes, expectation_invariance_is_consequence_not_definition)

Calculate each total expectation with the same aligned masses and compare the exact values. A process can keep the same total expectation while failing conditional means or adaptedness, so a Yes here is not martingale qualification. The exact result, retaining the requested order, is ([12, 12], Yes, expectation_invariance_is_consequence_not_definition).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r03-martingale-0100/)

Finite martingale · qf-mod30-martingale-expectation-invariant · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261030-r03-martingale-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261030-r03-martingale-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261030-r03-martingale-foundations/validation-report.json)