Step-by-step derivation
S001 · implies
Before
source operation=totalvar-partition-qualification;original snapshot and mathematics retained
After
explicit ordered source output fields=[partition, cell_mass_rows, complete_disjoint_certificate, mean, variance]
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(1/20, 1/10, 17/20, 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=[1/20, 1/10, 17/20, 0];SIGNED original X=[10, 11, 13, 109];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(1/20, 10)
After
Total variance original atom0 weighted X = 1/2
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(1/10, 11)
After
Total variance original atom1 weighted X = 11/10
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(17/20, 13)
After
Total variance original atom2 weighted X = 221/20
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, 109)
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(1/2, 11/10, 221/20, 0)
After
Total variance ORIGINAL global mean = 253/20
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(10, 253/20)
After
Total variance original atom0 centered X = -53/20
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(-53/20, -53/20)
After
Total variance original atom0 squared deviation = 2809/400
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(1/20, 2809/400)
After
Total variance original atom0 weighted squared deviation = 2809/8000
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(11, 253/20)
After
Total variance original atom1 centered X = -33/20
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(-33/20, -33/20)
After
Total variance original atom1 squared deviation = 1089/400
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(1/10, 1089/400)
After
Total variance original atom1 weighted squared deviation = 1089/4000
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(13, 253/20)
After
Total variance original atom2 centered X = 7/20
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(7/20, 7/20)
After
Total variance original atom2 squared deviation = 49/400
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(17/20, 49/400)
After
Total variance original atom2 weighted squared deviation = 833/8000
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(109, 253/20)
After
Total variance original atom3 centered X = 1927/20
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(1927/20, 1927/20)
After
Total variance original atom3 squared deviation = 3713329/400
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, 3713329/400)
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(2809/8000, 1089/4000, 833/8000, 0)
After
Total variance ORIGINAL global variance = 291/400
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(1/20, 1/10)
After
Total variance cell0 mass = 3/20
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(1/2, 11/10)
After
Total variance cell0 weighted numerator = 8/5
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(8/5, 3/20)
After
Total variance positive cell0 mean = 32/3
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(10, 32/3)
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(1/20, 3/20)
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(1/20, -2/3)
After
Total variance cell0 atom0 weighted residual = -1/30
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(11, 32/3)
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(1/10, 3/20)
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(1/10, 1/3)
After
Total variance cell0 atom1 weighted residual = 1/30
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(-1/30, 1/30)
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=3/20;numerator=8/5
After
conditional mean=32/3;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(17/20, 0)
After
Total variance cell1 mass = 17/20
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(221/20, 0)
After
Total variance cell1 weighted numerator = 221/20
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(221/20, 17/20)
After
Total variance positive cell1 mean = 13
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(13, 13)
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(17/20, 17/20)
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(17/20, 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(109, 13)
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, 17/20)
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=17/20;numerator=221/20
After
conditional mean=13;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(3/20, 2/9)
After
Total variance cell0 weighted WITHIN term = 1/30
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(32/3, 253/20)
After
Total variance cell0 mean minus global mean = -119/60
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(-119/60, -119/60)
After
Total variance cell0 BETWEEN squared deviation = 14161/3600
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(3/20, 14161/3600)
After
Total variance cell0 weighted BETWEEN term = 14161/24000
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(17/20, 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(13, 253/20)
After
Total variance cell1 mean minus global mean = 7/20
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(7/20, 7/20)
After
Total variance cell1 BETWEEN squared deviation = 49/400
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(17/20, 49/400)
After
Total variance cell1 weighted BETWEEN term = 833/8000
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(1/30, 0)
After
Total variance sum WITHIN terms = 1/30
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(14161/24000, 833/8000)
After
Total variance sum BETWEEN terms = 833/1200
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(1/30, 833/1200)
After
Total variance WITHIN plus BETWEEN = 291/400
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=253/20;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1], 3/20, 32/3, 2/9], [1, [2, 3], 17/20, 13, 0]]
After
Var(X)=291/400;within=1/30;between=833/1200;sum=291/400;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