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(1/18, 1/9, 5/6, 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(1/18, 8)
After
Total expectation original atom0 weighted X = 4/9
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(1/9, 9)
After
Total expectation original atom1 weighted X = 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, S003
S005 · calculation
Before
Total expectation original atom2 weighted X := mul(5/6, 11)
After
Total expectation original atom2 weighted X = 55/6
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, 107)
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(4/9, 1, 55/6, 0)
After
Total expectation full original mean = 191/18
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, S006
S008 · implies
Before
ORIGINAL masses=[1/18, 1/9, 5/6, 0],SIGNED values X=[8, 9, 11, 107];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=191/18;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(1/18, 1/9)
After
Total expectation cell0 mass = 1/6
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(4/9, 1)
After
Total expectation cell0 numerator = 13/9
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(13/9, 1/6)
After
Total expectation cell0 positive-mass conditional mean = 26/3
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=1/6;weighted numerator=13/9
After
conditional quotient=26/3;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(5/6, 0)
After
Total expectation cell1 mass = 5/6
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(55/6, 0)
After
Total expectation cell1 numerator = 55/6
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(55/6, 5/6)
After
Total expectation cell1 positive-mass conditional mean = 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, S014
S016 · implies
Before
ORIGINAL cell index=1,atom labels=[2, 3];mass=5/6;weighted numerator=55/6
After
conditional quotient=11;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(1/6, 26/3)
After
Total expectation positive cell0 weighted mean = 13/9
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(5/6, 11)
After
Total expectation positive cell1 weighted mean = 55/6
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(13/9, 55/6)
After
Total expectation sum positive-cell weighted means = 191/18
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, S018
S020 · implies
Before
ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1], 1/6, 26/3, 13/9], [1, [2, 3], 5/6, 11, 55/6]]
After
positive-cell weighted sum=191/18;ORIGINAL global mean=191/18;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