Step-by-step derivation
S001 · implies
Before
source operation=totalvar-within-between-components;original snapshot and mathematics retained
After
explicit ordered source output fields=[cell_rows, within, between, certificate]
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(1/23, 2/23, 20/23, 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=[1/23, 2/23, 20/23, 0];SIGNED original X=[13, 14, 16, 112];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(1/23, 13)
After
Total variance original atom0 weighted X = 13/23
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(2/23, 14)
After
Total variance original atom1 weighted X = 28/23
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(20/23, 16)
After
Total variance original atom2 weighted X = 320/23
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, 112)
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(13/23, 28/23, 320/23, 0)
After
Total variance ORIGINAL global mean = 361/23
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(13, 361/23)
After
Total variance original atom0 centered X = -62/23
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(-62/23, -62/23)
After
Total variance original atom0 squared deviation = 3844/529
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(1/23, 3844/529)
After
Total variance original atom0 weighted squared deviation = 3844/12167
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(14, 361/23)
After
Total variance original atom1 centered X = -39/23
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(-39/23, -39/23)
After
Total variance original atom1 squared deviation = 1521/529
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(2/23, 1521/529)
After
Total variance original atom1 weighted squared deviation = 3042/12167
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(16, 361/23)
After
Total variance original atom2 centered X = 7/23
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(7/23, 7/23)
After
Total variance original atom2 squared deviation = 49/529
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(20/23, 49/529)
After
Total variance original atom2 weighted squared deviation = 980/12167
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(112, 361/23)
After
Total variance original atom3 centered X = 2215/23
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(2215/23, 2215/23)
After
Total variance original atom3 squared deviation = 4906225/529
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, 4906225/529)
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(3844/12167, 3042/12167, 980/12167, 0)
After
Total variance ORIGINAL global variance = 342/529
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(1/23, 2/23)
After
Total variance cell0 mass = 3/23
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(13/23, 28/23)
After
Total variance cell0 weighted numerator = 41/23
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(41/23, 3/23)
After
Total variance positive cell0 mean = 41/3
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(13, 41/3)
After
Total variance cell0 atom0 conditional centering = -2/3
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(-2/3, -2/3)
After
Total variance cell0 atom0 conditional square = 4/9
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(1/23, 3/23)
After
Total variance cell0 atom0 normalized conditional mass = 1/3
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(1/3, 4/9)
After
Total variance cell0 atom0 conditional variance term = 4/27
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(1/23, -2/3)
After
Total variance cell0 atom0 weighted residual = -2/69
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(14, 41/3)
After
Total variance cell0 atom1 conditional centering = 1/3
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(1/3, 1/3)
After
Total variance cell0 atom1 conditional square = 1/9
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(2/23, 3/23)
After
Total variance cell0 atom1 normalized conditional mass = 2/3
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(2/3, 1/9)
After
Total variance cell0 atom1 conditional variance term = 2/27
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(2/23, 1/3)
After
Total variance cell0 atom1 weighted residual = 2/69
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 positive cell0 conditional variance := sum(4/27, 2/27)
After
Total variance positive cell0 conditional variance = 2/9
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 positive cell0 residual mean zero := sum(-2/69, 2/69)
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, S035
S037 · implies
Before
ORIGINAL cell index=0;actual atoms=[0, 1];mass=3/23;numerator=41/23
After
conditional mean=41/3;conditional variance=2/9;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, S036
S038 · calculation
Before
Total variance cell1 mass := sum(20/23, 0)
After
Total variance cell1 mass = 20/23
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 cell1 weighted numerator := sum(320/23, 0)
After
Total variance cell1 weighted numerator = 320/23
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 cell1 mean := div(320/23, 20/23)
After
Total variance positive cell1 mean = 16
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 cell1 atom2 conditional centering := sub(16, 16)
After
Total variance cell1 atom2 conditional centering = 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 · calculation
Before
Total variance cell1 atom2 conditional square := mul(0, 0)
After
Total variance cell1 atom2 conditional square = 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, S041
S043 · calculation
Before
Total variance cell1 atom2 normalized conditional mass := div(20/23, 20/23)
After
Total variance cell1 atom2 normalized conditional 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, S042
S044 · calculation
Before
Total variance cell1 atom2 conditional variance term := mul(1, 0)
After
Total variance cell1 atom2 conditional variance 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, S043
S045 · calculation
Before
Total variance cell1 atom2 weighted residual := mul(20/23, 0)
After
Total variance cell1 atom2 weighted residual = 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, S044
S046 · calculation
Before
Total variance cell1 atom3 conditional centering := sub(112, 16)
After
Total variance cell1 atom3 conditional centering = 96
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 cell1 atom3 conditional square := mul(96, 96)
After
Total variance cell1 atom3 conditional square = 9216
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 cell1 atom3 normalized conditional mass := div(0, 20/23)
After
Total variance cell1 atom3 normalized conditional 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, S047
S049 · calculation
Before
Total variance cell1 atom3 conditional variance term := mul(0, 9216)
After
Total variance cell1 atom3 conditional variance 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 cell1 atom3 weighted residual := mul(0, 96)
After
Total variance cell1 atom3 weighted residual = 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, S049
S051 · calculation
Before
Total variance positive cell1 conditional variance := sum(0, 0)
After
Total variance positive cell1 conditional variance = 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 positive cell1 residual mean zero := sum(0, 0)
After
Total variance positive cell1 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, S051
S053 · implies
Before
ORIGINAL cell index=1;actual atoms=[2, 3];mass=20/23;numerator=320/23
After
conditional mean=16;conditional variance=0;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, S052
S054 · calculation
Before
Total variance cell0 weighted WITHIN term := mul(3/23, 2/9)
After
Total variance cell0 weighted WITHIN term = 2/69
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, S053
S055 · calculation
Before
Total variance cell0 mean minus global mean := sub(41/3, 361/23)
After
Total variance cell0 mean minus global mean = -140/69
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, S054
S056 · calculation
Before
Total variance cell0 BETWEEN squared deviation := mul(-140/69, -140/69)
After
Total variance cell0 BETWEEN squared deviation = 19600/4761
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, S055
S057 · calculation
Before
Total variance cell0 weighted BETWEEN term := mul(3/23, 19600/4761)
After
Total variance cell0 weighted BETWEEN term = 19600/36501
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, S056
S058 · calculation
Before
Total variance cell1 weighted WITHIN term := mul(20/23, 0)
After
Total variance cell1 weighted WITHIN 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, S057
S059 · calculation
Before
Total variance cell1 mean minus global mean := sub(16, 361/23)
After
Total variance cell1 mean minus global mean = 7/23
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, S058
S060 · calculation
Before
Total variance cell1 BETWEEN squared deviation := mul(7/23, 7/23)
After
Total variance cell1 BETWEEN squared deviation = 49/529
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, S059
S061 · calculation
Before
Total variance cell1 weighted BETWEEN term := mul(20/23, 49/529)
After
Total variance cell1 weighted BETWEEN term = 980/12167
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, S060
S062 · calculation
Before
Total variance sum WITHIN terms := sum(2/69, 0)
After
Total variance sum WITHIN terms = 2/69
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, S061
S063 · calculation
Before
Total variance sum BETWEEN terms := sum(19600/36501, 980/12167)
After
Total variance sum BETWEEN terms = 980/1587
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, S062
S064 · calculation
Before
Total variance WITHIN plus BETWEEN := sum(2/69, 980/1587)
After
Total variance WITHIN plus BETWEEN = 342/529
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, S063
S065 · implies
Before
ORIGINAL SAME-law mean=361/23;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1], 3/23, 41/3, 2/9], [1, [2, 3], 20/23, 16, 0]]
After
Var(X)=342/529;within=2/69;between=980/1587;sum=342/529;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, S064