Step-by-step derivation
S001 · implies
Before
source operation=totalvar-zero-mass-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[cell_rows, within, between, variance, equal, null_cell_division_claim, boundary]
Conditions: none
Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
Total variance complete probability mass := sum(1/33, 2/33, 10/11, 0)
After
Total variance complete probability mass = 1
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · implies
Before
ORIGINAL masses=[1/33, 2/33, 10/11, 0];SIGNED original X=[23, 24, 26, 122];ORIGINAL cell order=[[0, 1, 2], [3]]
After
ALL cells NONEMPTY and every atom occurs EXACTLY ONCE,including null atoms;normalized mass=1;no independence premise
Conditions: none
Unlike the earlier total-expectation convention,every cell here is nonempty.A null cell may still contain actual atoms.Use the same law and a complete disjoint partition;X may be signed and independence is not required.
Step identifiers and dependencies
Rule: totalvar_NONEMPTY_cells_complete_disjoint_same_law · Uses: G001, S002
S004 · calculation
Before
Total variance original atom0 weighted X := mul(1/33, 23)
After
Total variance original atom0 weighted X = 23/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Total variance original atom1 weighted X := mul(2/33, 24)
After
Total variance original atom1 weighted X = 16/11
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
Total variance original atom2 weighted X := mul(10/11, 26)
After
Total variance original atom2 weighted X = 260/11
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
Total variance original atom3 weighted X := mul(0, 122)
After
Total variance original atom3 weighted X = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
Total variance ORIGINAL global mean := sum(23/33, 16/11, 260/11, 0)
After
Total variance ORIGINAL global mean = 851/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
Total variance original atom0 centered X := sub(23, 851/33)
After
Total variance original atom0 centered X = -92/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Total variance original atom0 squared deviation := mul(-92/33, -92/33)
After
Total variance original atom0 squared deviation = 8464/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
Total variance original atom0 weighted squared deviation := mul(1/33, 8464/1089)
After
Total variance original atom0 weighted squared deviation = 8464/35937
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
Total variance original atom1 centered X := sub(24, 851/33)
After
Total variance original atom1 centered X = -59/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
Total variance original atom1 squared deviation := mul(-59/33, -59/33)
After
Total variance original atom1 squared deviation = 3481/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
Total variance original atom1 weighted squared deviation := mul(2/33, 3481/1089)
After
Total variance original atom1 weighted squared deviation = 6962/35937
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
Total variance original atom2 centered X := sub(26, 851/33)
After
Total variance original atom2 centered X = 7/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
Total variance original atom2 squared deviation := mul(7/33, 7/33)
After
Total variance original atom2 squared deviation = 49/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
Total variance original atom2 weighted squared deviation := mul(10/11, 49/1089)
After
Total variance original atom2 weighted squared deviation = 490/11979
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · calculation
Before
Total variance original atom3 centered X := sub(122, 851/33)
After
Total variance original atom3 centered X = 3175/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · calculation
Before
Total variance original atom3 squared deviation := mul(3175/33, 3175/33)
After
Total variance original atom3 squared deviation = 10080625/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · calculation
Before
Total variance original atom3 weighted squared deviation := mul(0, 10080625/1089)
After
Total variance original atom3 weighted squared deviation = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S019
S021 · calculation
Before
Total variance ORIGINAL global variance := sum(8464/35937, 6962/35937, 490/11979, 0)
After
Total variance ORIGINAL global variance = 512/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · calculation
Before
Total variance cell0 mass := sum(1/33, 2/33, 10/11)
After
Total variance cell0 mass = 1
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S021
S023 · calculation
Before
Total variance cell0 weighted numerator := sum(23/33, 16/11, 260/11)
After
Total variance cell0 weighted numerator = 851/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · calculation
Before
Total variance positive cell0 mean := div(851/33, 1)
After
Total variance positive cell0 mean = 851/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S023
S025 · calculation
Before
Total variance cell0 atom0 conditional centering := sub(23, 851/33)
After
Total variance cell0 atom0 conditional centering = -92/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · calculation
Before
Total variance cell0 atom0 conditional square := mul(-92/33, -92/33)
After
Total variance cell0 atom0 conditional square = 8464/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S025
S027 · calculation
Before
Total variance cell0 atom0 normalized conditional mass := div(1/33, 1)
After
Total variance cell0 atom0 normalized conditional mass = 1/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · calculation
Before
Total variance cell0 atom0 conditional variance term := mul(1/33, 8464/1089)
After
Total variance cell0 atom0 conditional variance term = 8464/35937
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S027
S029 · calculation
Before
Total variance cell0 atom0 weighted residual := mul(1/33, -92/33)
After
Total variance cell0 atom0 weighted residual = -92/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S028
S030 · calculation
Before
Total variance cell0 atom1 conditional centering := sub(24, 851/33)
After
Total variance cell0 atom1 conditional centering = -59/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S029
S031 · calculation
Before
Total variance cell0 atom1 conditional square := mul(-59/33, -59/33)
After
Total variance cell0 atom1 conditional square = 3481/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S030
S032 · calculation
Before
Total variance cell0 atom1 normalized conditional mass := div(2/33, 1)
After
Total variance cell0 atom1 normalized conditional mass = 2/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S031
S033 · calculation
Before
Total variance cell0 atom1 conditional variance term := mul(2/33, 3481/1089)
After
Total variance cell0 atom1 conditional variance term = 6962/35937
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · calculation
Before
Total variance cell0 atom1 weighted residual := mul(2/33, -59/33)
After
Total variance cell0 atom1 weighted residual = -118/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S033
S035 · calculation
Before
Total variance cell0 atom2 conditional centering := sub(26, 851/33)
After
Total variance cell0 atom2 conditional centering = 7/33
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S034
S036 · calculation
Before
Total variance cell0 atom2 conditional square := mul(7/33, 7/33)
After
Total variance cell0 atom2 conditional square = 49/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S035
S037 · calculation
Before
Total variance cell0 atom2 normalized conditional mass := div(10/11, 1)
After
Total variance cell0 atom2 normalized conditional mass = 10/11
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S036
S038 · calculation
Before
Total variance cell0 atom2 conditional variance term := mul(10/11, 49/1089)
After
Total variance cell0 atom2 conditional variance term = 490/11979
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S037
S039 · calculation
Before
Total variance cell0 atom2 weighted residual := mul(10/11, 7/33)
After
Total variance cell0 atom2 weighted residual = 70/363
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · calculation
Before
Total variance positive cell0 conditional variance := sum(8464/35937, 6962/35937, 490/11979)
After
Total variance positive cell0 conditional variance = 512/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S039
S041 · calculation
Before
Total variance positive cell0 residual mean zero := sum(-92/1089, -118/1089, 70/363)
After
Total variance positive cell0 residual mean zero = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S040
S042 · implies
Before
ORIGINAL cell index=0;actual atoms=[0, 1, 2];mass=1;numerator=851/33
After
conditional mean=851/33;conditional variance=512/1089;positive-cell weighted residual sum0,so cross term vanishes
Conditions: none
A positive cell uses p_i/cellmass for its conditional variance,and its weighted residual sum is0.A nonempty null cell is never divided:both mean and variance remain null,while both outer contributions are0.Undefined quotients are not computed zeros.
Step identifiers and dependencies
Rule: totalvar_conditional_mean_variance_null_and_cross_term · Uses: G001, S041
S043 · calculation
Before
Total variance cell1 mass := sum(0)
After
Total variance cell1 mass = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S042
S044 · calculation
Before
Total variance cell1 weighted numerator := sum(0)
After
Total variance cell1 weighted numerator = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S043
S045 · implies
Before
ORIGINAL cell index=1;actual atoms=[3];mass=0;numerator=0
After
conditional mean=null;conditional variance=null;NONEMPTY null cell:all atom masses0,no conditional division;BOTH quotients null
Conditions: none
A positive cell uses p_i/cellmass for its conditional variance,and its weighted residual sum is0.A nonempty null cell is never divided:both mean and variance remain null,while both outer contributions are0.Undefined quotients are not computed zeros.
Step identifiers and dependencies
Rule: totalvar_conditional_mean_variance_null_and_cross_term · Uses: G001, S044
S046 · calculation
Before
Total variance cell0 weighted WITHIN term := mul(1, 512/1089)
After
Total variance cell0 weighted WITHIN term = 512/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S045
S047 · calculation
Before
Total variance cell0 mean minus global mean := sub(851/33, 851/33)
After
Total variance cell0 mean minus global mean = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S046
S048 · calculation
Before
Total variance cell0 BETWEEN squared deviation := mul(0, 0)
After
Total variance cell0 BETWEEN squared deviation = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S047
S049 · calculation
Before
Total variance cell0 weighted BETWEEN term := mul(1, 0)
After
Total variance cell0 weighted BETWEEN term = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S048
S050 · calculation
Before
Total variance sum WITHIN terms := sum(512/1089)
After
Total variance sum WITHIN terms = 512/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S049
S051 · calculation
Before
Total variance sum BETWEEN terms := sum(0)
After
Total variance sum BETWEEN terms = 0
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S050
S052 · calculation
Before
Total variance WITHIN plus BETWEEN := sum(512/1089, 0)
After
Total variance WITHIN plus BETWEEN = 512/1089
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S051
S053 · implies
Before
ORIGINAL SAME-law mean=851/33;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1, 2], 1, 851/33, 512/1089], [1, [3], 0, null, null]]
After
Var(X)=512/1089;within=512/1089;between=0;sum=512/1089;X-mu=(X-c_j)+(c_j-mu),and each positive-cell cross term2(c_j-mu)*sum_i p_i(X_i-c_j)=0
Conditions: none
Expand the square cellwise.The first term gives mass times conditional variance,the second the weighted squared cell-mean deviation,and the cross term vanishes by the checked residual identity.Null atoms contribute0.No independence assumption or omitted term replaces this decomposition.
Step identifiers and dependencies
Rule: totalvar_general_centered_square_decomposition_same_partition · Uses: G001, S052
S054 · implies
Before
ACTUAL NONEMPTY null cell indices=[1];within=512/1089,between=0,Var=512/1089
After
null_cell_division_claim fixed No;both conditional quotients remain null;total-variance identity remains TRUE
Conditions: none
The final No rejects null-cell division,not the proven identity.A genuine zero mean or variance in a positive cell remains numeric0,not null.
Step identifiers and dependencies
Rule: totalvar_null_division_No_not_identity_failure · Uses: G001, S053