Step-by-step derivation
S001 · implies
Before
source operation=totalexp-positive-cells-formula;original snapshot and mathematics retained
After
explicit ordered source output fields=[cell_rows, weighted_sum, mean, equal]
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(1/11, 2/11, 8/11, 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(1/11, 1)
After
Total expectation original atom0 weighted X = 1/11
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(2/11, 2)
After
Total expectation original atom1 weighted X = 4/11
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(8/11, 4)
After
Total expectation original atom2 weighted X = 32/11
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, 100)
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(1/11, 4/11, 32/11, 0)
After
Total expectation full original mean = 37/11
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=[1/11, 2/11, 8/11, 0],SIGNED values X=[1, 2, 4, 100];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=37/11;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(1/11, 2/11)
After
Total expectation cell0 mass = 3/11
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(1/11, 4/11)
After
Total expectation cell0 numerator = 5/11
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(5/11, 3/11)
After
Total expectation cell0 positive-mass conditional mean = 5/3
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=3/11;weighted numerator=5/11
After
conditional quotient=5/3;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(8/11, 0)
After
Total expectation cell1 mass = 8/11
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(32/11, 0)
After
Total expectation cell1 numerator = 32/11
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(32/11, 8/11)
After
Total expectation cell1 positive-mass conditional mean = 4
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=8/11;weighted numerator=32/11
After
conditional quotient=4;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(3/11, 5/3)
After
Total expectation positive cell0 weighted mean = 5/11
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(8/11, 4)
After
Total expectation positive cell1 weighted mean = 32/11
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(5/11, 32/11)
After
Total expectation sum positive-cell weighted means = 37/11
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], 3/11, 5/3, 5/11], [1, [2, 3], 8/11, 4, 32/11]]
After
positive-cell weighted sum=37/11;ORIGINAL global mean=37/11;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