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R0027-14179 · q-20261014-advance-20261029-r04-conditional-expectation-0079
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 3/49, 24/49, 4/49], values X=[-9, 0, 5, 9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
G001 — Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 3/49, 24/49, 4/49], values X=[-9, 0, 5, 9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
conditional model whole mass sum := sum(18/49, 3/49, 24/49, 4/49)
After
conditional model whole mass sum = 1
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001
Before
conditional model:raw aligned masses=[[18, 49], [3, 49], [24, 49], [4, 49]]
After
normalized masses=[18/49, 3/49, 24/49, 4/49];nonnegative=Yes;total=1
Conditions: none
First qualify nonnegativity and exact unit total on one complete finite probability model.Retain zero-mass atoms;do not assume uniformity or substitute another model.
Rule: whole_finite_probability_law_not_uniform_assumption · Uses: G001, S001
Before
same outcome indices=[0, 1, 2, 3];X values=[-9, 0, 5, 9];block labels=[0, 0, 1, 1]
After
distinct conditioning labels in increasing order=[0, 1]
Conditions: none
Each label fiber is a conditioning block.Labels need not be consecutive.Sort summaries by label but retain original outcome indices within blocks;labels are not array positions.
Rule: conditional_partition_same_space_original_indices · Uses: G001, S002
Before
block0 mass := sum(18/49, 3/49)
After
block0 mass = 3/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
block0 atom0 weighted X := mul(18/49, -9)
After
block0 atom0 weighted X = -162/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
block0 atom1 weighted X := mul(3/49, 0)
After
block0 atom1 weighted X = 0
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
block0 original weighted integral := sum(-162/49, 0)
After
block0 original weighted integral = -162/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
block0 positive-mass conditional mean := div(-162/49, 3/7)
After
block0 positive-mass conditional mean = -54/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
block label=0;indices=[0, 1];positive mass=3/7;weighted integral=-162/49
After
block-constant conditional value=-54/7
Conditions: none
Divide the weighted integral by mass only when mass is positive.A zero mean remains the determined numeric value0,not a null-block marker.Copy the same value to every atom in the block.
Rule: positive_block_mean_including_zero_mean · Uses: G001, S008
Before
block1 mass := sum(24/49, 4/49)
After
block1 mass = 4/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
block1 atom2 weighted X := mul(24/49, 5)
After
block1 atom2 weighted X = 120/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
block1 atom3 weighted X := mul(4/49, 9)
After
block1 atom3 weighted X = 36/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
block1 original weighted integral := sum(120/49, 36/49)
After
block1 original weighted integral = 156/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
block1 positive-mass conditional mean := div(156/49, 4/7)
After
block1 positive-mass conditional mean = 39/7
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
block label=1;indices=[2, 3];positive mass=4/7;weighted integral=156/49
After
block-constant conditional value=39/7
Conditions: none
Divide the weighted integral by mass only when mass is positive.A zero mean remains the determined numeric value0,not a null-block marker.Copy the same value to every atom in the block.
Rule: positive_block_mean_including_zero_mean · Uses: G001, S014
Before
total original expectation := sum(-162/49, 156/49)
After
total original expectation = -6/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
total expectation block0 conditional integral := mul(3/7, -54/7)
After
total expectation block0 conditional integral = -162/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
total expectation block1 conditional integral := mul(4/7, 39/7)
After
total expectation block1 conditional integral = 156/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
total conditional expectation := sum(-162/49, 156/49)
After
total conditional expectation = -6/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
all conditioning blocks partition the same carrier;E_X=-6/49;E_conditional=-6/49
After
equal=Yes;tower_to_trivial_sigma_algebra
Conditions: none
Sum the blockwise integral identities to obtain total expectation equality.This is the tower consequence to the trivial sigma-algebra,not a reverse substitution for measurability and every block obligation.
Rule: total_expectation_consequence_not_whole_definition · Uses: G001, S019
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Recompute total conditional expectation with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute total expectation block1 conditional integral with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute total expectation block0 conditional integral with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute total original expectation with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Recompute block1 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute block1 original weighted integral with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute block1 atom3 weighted X with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute block1 atom2 weighted X with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute block1 mass with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute block0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute block0 original weighted integral with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute block0 atom1 weighted X with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute block0 atom0 weighted X with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute block0 mass with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Recompute conditional model whole mass sum with exact rational arithmetic, not rounded decimals.(-6/49, -6/49, Yes, tower_to_trivial_sigma_algebra)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
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