Separate Nesting, Adaptedness, Conditional Steps and Expectation Consequences
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Finite martingale · qf-mod30-martingale-expectation-invariant · candidate
Canonical question JSONL · English JSONL · Validation report