Step-by-step derivation
S001 · implies
Before
source operation=totalexp-tower-identity;original snapshot and mathematics retained
After
explicit ordered source output fields=[conditional_values, tower_expectation, mean, equal, certificate]
Conditions: none
Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
Total expectation complete original mass := sum(1/20, 1/10, 17/20, 0)
After
Total expectation complete original mass = 1
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Total expectation original atom0 weighted X := mul(1/20, 10)
After
Total expectation original atom0 weighted X = 1/2
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
Total expectation original atom1 weighted X := mul(1/10, 11)
After
Total expectation original atom1 weighted X = 11/10
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Total expectation original atom2 weighted X := mul(17/20, 13)
After
Total expectation original atom2 weighted X = 221/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
Total expectation original atom3 weighted X := mul(0, 109)
After
Total expectation original atom3 weighted X = 0
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
Total expectation full original mean := sum(1/2, 11/10, 221/20, 0)
After
Total expectation full original mean = 253/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL masses=[1/20, 1/10, 17/20, 0],SIGNED values X=[10, 11, 13, 109];ORIGINAL cell order=[[0, 1], [2, 3]]
After
flattened original atom labels=[0, 1, 2, 3] cover EVERY atom exactly ONCE including null atoms;total mass=1;integrable finite mean=253/20;NO independence premise
Conditions: none
The partition covers all original atoms,including null atoms,exactly once.Keep original cell order and source-normalized internal labels.X may be signed and is integrable in this finite model;independence is not required.
Step identifiers and dependencies
Rule: totalexp_same_law_complete_disjoint_partition_all_atoms · Uses: G001, S007
S009 · calculation
Before
Total expectation cell0 mass := sum(1/20, 1/10)
After
Total expectation cell0 mass = 3/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Total expectation cell0 numerator := sum(1/2, 11/10)
After
Total expectation cell0 numerator = 8/5
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
Total expectation cell0 positive-mass conditional mean := div(8/5, 3/20)
After
Total expectation cell0 positive-mass conditional mean = 32/3
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
ORIGINAL cell index=0,atom labels=[0, 1];mass=3/20;weighted numerator=8/5
After
conditional quotient=32/3;positive denominator qualifies division
Conditions: none
Divide only for positive cell mass.In a null cell,all nonnegative atom masses are0,so the numerator is0 but0/0 remains undefined.Keep null rather than an asserted quotient0,including empty cells.
Step identifiers and dependencies
Rule: totalexp_cell_quotient_positive_mass_only_null_preserved · Uses: G001, S011
S013 · calculation
Before
Total expectation cell1 mass := sum(17/20, 0)
After
Total expectation cell1 mass = 17/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
Total expectation cell1 numerator := sum(221/20, 0)
After
Total expectation cell1 numerator = 221/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
Total expectation cell1 positive-mass conditional mean := div(221/20, 17/20)
After
Total expectation cell1 positive-mass conditional mean = 13
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · implies
Before
ORIGINAL cell index=1,atom labels=[2, 3];mass=17/20;weighted numerator=221/20
After
conditional quotient=13;positive denominator qualifies division
Conditions: none
Divide only for positive cell mass.In a null cell,all nonnegative atom masses are0,so the numerator is0 but0/0 remains undefined.Keep null rather than an asserted quotient0,including empty cells.
Step identifiers and dependencies
Rule: totalexp_cell_quotient_positive_mass_only_null_preserved · Uses: G001, S015
S017 · calculation
Before
Total expectation positive cell0 weighted mean := mul(3/20, 32/3)
After
Total expectation positive cell0 weighted mean = 8/5
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · calculation
Before
Total expectation positive cell1 weighted mean := mul(17/20, 13)
After
Total expectation positive cell1 weighted mean = 221/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · calculation
Before
Total expectation sum positive-cell weighted means := sum(8/5, 221/20)
After
Total expectation sum positive-cell weighted means = 253/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1], 3/20, 32/3, 8/5], [1, [2, 3], 17/20, 13, 221/20]]
After
positive-cell weighted sum=253/20;ORIGINAL global mean=253/20;equal=Yes;disjoint complete cells regroup EVERY original p_i*X_i exactly ONCE,null contributions0
Conditions: none
Each positive mass times its conditional mean recovers its numerator;null cells contribute0.Disjoint complete cells then regroup every original weighted term exactly once.This proves the finite identity,not an unweighted average or an accidental numerical equality.
Step identifiers and dependencies
Rule: totalexp_regroup_complete_partition_general_identity · Uses: G001, S019
S021 · calculation
Before
Total expectation tower atom0 weighted conditional version := mul(1/20, 32/3)
After
Total expectation tower atom0 weighted conditional version = 8/15
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · calculation
Before
Total expectation tower atom1 weighted conditional version := mul(1/10, 32/3)
After
Total expectation tower atom1 weighted conditional version = 16/15
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S021
S023 · calculation
Before
Total expectation tower atom2 weighted conditional version := mul(17/20, 13)
After
Total expectation tower atom2 weighted conditional version = 221/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · calculation
Before
Total expectation tower atom3 weighted conditional version := mul(0, 13)
After
Total expectation tower atom3 weighted conditional version = 0
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S023
S025 · calculation
Before
Total expectation full tower expectation := sum(8/15, 16/15, 221/20, 0)
After
Total expectation full tower expectation = 253/20
Conditions: none
Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · implies
Before
ORIGINAL atom-order conditional version=[32/3, 32/3, 13, 13];ORIGINAL cells=[[0, 1], [2, 3]]
After
version constant on each cell,so measurable for sigma(partition);each cell integral equals original X cell integral;null-cell version CHOSEN0,not quotient;E(E(X|G))=253/20=E(X)=253/20
Conditions: none
Cell-constant values are measurable for the sigma-algebra generated by the partition.Cellwise integral identities extend to its finite cell unions,establishing a conditional-expectation version.On null cells0 is chosen,not a unique quotient.Weight by the original atom masses to obtain the tower identity.
Step identifiers and dependencies
Rule: totalexp_conditional_version_cell_measurability_and_tower · Uses: G001, S025