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R0027-14178 · q-20261014-advance-20261029-r04-conditional-expectation-0078
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 4/49, 18/49, 3/49], values X=[-10, -1, 4, 10], and conditioning-block labels=[0, 1, 1, 2]. 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=[24/49, 4/49, 18/49, 3/49], values X=[-10, -1, 4, 10], and conditioning-block labels=[0, 1, 1, 2]. 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(24/49, 4/49, 18/49, 3/49)
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=[[24, 49], [4, 49], [18, 49], [3, 49]]
After
normalized masses=[24/49, 4/49, 18/49, 3/49];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=[-10, -1, 4, 10];block labels=[0, 1, 1, 2]
After
distinct conditioning labels in increasing order=[0, 1, 2]
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(24/49)
After
block0 mass = 24/49
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(24/49, -10)
After
block0 atom0 weighted X = -240/49
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 original weighted integral := sum(-240/49)
After
block0 original weighted integral = -240/49
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 positive-mass conditional mean := div(-240/49, 24/49)
After
block0 positive-mass conditional mean = -10
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
block label=0;indices=[0];positive mass=24/49;weighted integral=-240/49
After
block-constant conditional value=-10
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, S007
Before
block1 mass := sum(4/49, 18/49)
After
block1 mass = 22/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
block1 atom1 weighted X := mul(4/49, -1)
After
block1 atom1 weighted X = -4/49
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(18/49, 4)
After
block1 atom2 weighted X = 72/49
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 original weighted integral := sum(-4/49, 72/49)
After
block1 original weighted integral = 68/49
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 positive-mass conditional mean := div(68/49, 22/49)
After
block1 positive-mass conditional mean = 34/11
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
block label=1;indices=[1, 2];positive mass=22/49;weighted integral=68/49
After
block-constant conditional value=34/11
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, S013
Before
block2 mass := sum(3/49)
After
block2 mass = 3/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
block2 atom3 weighted X := mul(3/49, 10)
After
block2 atom3 weighted X = 30/49
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
block2 original weighted integral := sum(30/49)
After
block2 original weighted integral = 30/49
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
block2 positive-mass conditional mean := div(30/49, 3/49)
After
block2 positive-mass conditional mean = 10
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
block label=2;indices=[3];positive mass=3/49;weighted integral=30/49
After
block-constant conditional value=10
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, S018
Before
total original expectation := sum(-240/49, 68/49, 30/49)
After
total original expectation = -142/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
total expectation block0 conditional integral := mul(24/49, -10)
After
total expectation block0 conditional integral = -240/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
total expectation block1 conditional integral := mul(22/49, 34/11)
After
total expectation block1 conditional integral = 68/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
total expectation block2 conditional integral := mul(3/49, 10)
After
total expectation block2 conditional integral = 30/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
total conditional expectation := sum(-240/49, 68/49, 30/49)
After
total conditional expectation = -142/49
Conditions: none
Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
all conditioning blocks partition the same carrier;E_X=-142/49;E_conditional=-142/49
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, S024
S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Recompute total conditional expectation with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute total expectation block2 conditional integral with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute total expectation block1 conditional integral with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute total expectation block0 conditional integral with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute total original expectation with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Recompute block2 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute block2 original weighted integral with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute block2 atom3 weighted X with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute block2 mass with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Recompute block1 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute block1 original weighted integral 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 atom1 weighted X with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute block1 mass with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute block0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute block0 original weighted integral 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.(-142/49, -142/49, 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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