Step-by-step derivation
S001 · implies
Before
source operation=totalexp-zero-mass-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary]
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(1/13, 2/13, 10/13, 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(1/13, 3)
After
Total expectation original atom0 weighted X = 3/13
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, S002
S004 · calculation
Before
Total expectation original atom1 weighted X := mul(2/13, 4)
After
Total expectation original atom1 weighted X = 8/13
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, S003
S005 · calculation
Before
Total expectation original atom2 weighted X := mul(10/13, 6)
After
Total expectation original atom2 weighted X = 60/13
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, S004
S006 · calculation
Before
Total expectation original atom3 weighted X := mul(0, 102)
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(3/13, 8/13, 60/13, 0)
After
Total expectation full original mean = 71/13
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, S006
S008 · implies
Before
ORIGINAL masses=[1/13, 2/13, 10/13, 0],SIGNED values X=[3, 4, 6, 102];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=71/13;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(1/13, 2/13, 10/13)
After
Total expectation cell0 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, S008
S010 · calculation
Before
Total expectation cell0 numerator := sum(3/13, 8/13, 60/13)
After
Total expectation cell0 numerator = 71/13
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, S009
S011 · calculation
Before
Total expectation cell0 positive-mass conditional mean := div(71/13, 1)
After
Total expectation cell0 positive-mass conditional mean = 71/13
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, S010
S012 · implies
Before
ORIGINAL cell index=0,atom labels=[0, 1, 2];mass=1;weighted numerator=71/13
After
conditional quotient=71/13;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, S011
S013 · calculation
Before
Total expectation cell1 mass := sum(0)
After
Total expectation cell1 mass = 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, S012
S014 · calculation
Before
Total expectation cell1 numerator := sum(0)
After
Total expectation cell1 numerator = 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, S013
S015 · implies
Before
ORIGINAL cell index=1,atom labels=[3];mass=0;weighted numerator=0
After
conditional quotient=null;null cell:every nonnegative atom mass is0,numerator0;NO division performed
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, S014
S016 · calculation
Before
Total expectation positive cell0 weighted mean := mul(1, 71/13)
After
Total expectation positive cell0 weighted mean = 71/13
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, S015
S017 · calculation
Before
Total expectation sum positive-cell weighted means := sum(71/13)
After
Total expectation sum positive-cell weighted means = 71/13
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, S016
S018 · implies
Before
ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1, 2], 1, 71/13, 71/13], [1, [3], 0, null, 0]]
After
positive-cell weighted sum=71/13;ORIGINAL global mean=71/13;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, S017
S019 · implies
Before
ACTUAL null cells=[1];positive weighted sum=71/13;mean=71/13
After
null_cell_division_claim fixed No;undefined cell quotients remain null,while the weighted total-expectation identity still holds
Conditions: none
The final No rejects null-cell division,not the identity.A conditional-expectation version may choose a value on a null cell,but not as a uniquely computed0/0 quotient.
Step identifiers and dependencies
Rule: totalexp_null_division_nonclaim_not_tower_failure · Uses: G001, S018