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(1/25, 2/25, 22/25, 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(1/25, 15)
After
Total expectation original atom0 weighted X = 3/5
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, S002
S004 · calculation
Before
Total expectation original atom1 weighted X := mul(2/25, 16)
After
Total expectation original atom1 weighted X = 32/25
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(22/25, 18)
After
Total expectation original atom2 weighted X = 396/25
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, 114)
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(3/5, 32/25, 396/25, 0)
After
Total expectation full original mean = 443/25
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=[1/25, 2/25, 22/25, 0],SIGNED values X=[15, 16, 18, 114];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=443/25;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(1/25, 2/25)
After
Total expectation cell0 mass = 3/25
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(3/5, 32/25)
After
Total expectation cell0 numerator = 47/25
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(47/25, 3/25)
After
Total expectation cell0 positive-mass conditional mean = 47/3
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=3/25;weighted numerator=47/25
After
conditional quotient=47/3;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(22/25, 0)
After
Total expectation cell1 mass = 22/25
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(396/25, 0)
After
Total expectation cell1 numerator = 396/25
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(396/25, 22/25)
After
Total expectation cell1 positive-mass conditional mean = 18
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=22/25;weighted numerator=396/25
After
conditional quotient=18;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(3/25, 47/3)
After
Total expectation positive cell0 weighted mean = 47/25
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(22/25, 18)
After
Total expectation positive cell1 weighted mean = 396/25
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(47/25, 396/25)
After
Total expectation sum positive-cell weighted means = 443/25
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], 3/25, 47/3, 47/25], [1, [2, 3], 22/25, 18, 396/25]]
After
positive-cell weighted sum=443/25;ORIGINAL global mean=443/25;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(1/25, 47/3)
After
Total expectation tower atom0 weighted conditional version = 47/75
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(2/25, 47/3)
After
Total expectation tower atom1 weighted conditional version = 94/75
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(22/25, 18)
After
Total expectation tower atom2 weighted conditional version = 396/25
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, 18)
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(47/75, 94/75, 396/25, 0)
After
Total expectation full tower expectation = 443/25
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=[47/3, 47/3, 18, 18];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))=443/25=E(X)=443/25
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