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(1/29, 2/29, 26/29, 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=[1/29, 2/29, 26/29, 0];SIGNED original X=[19, 20, 22, 118];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(1/29, 19)
After
Total variance original atom0 weighted X = 19/29
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(2/29, 20)
After
Total variance original atom1 weighted X = 40/29
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(26/29, 22)
After
Total variance original atom2 weighted X = 572/29
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, 118)
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(19/29, 40/29, 572/29, 0)
After
Total variance ORIGINAL global mean = 631/29
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(19, 631/29)
After
Total variance original atom0 centered X = -80/29
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(-80/29, -80/29)
After
Total variance original atom0 squared deviation = 6400/841
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(1/29, 6400/841)
After
Total variance original atom0 weighted squared deviation = 6400/24389
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(20, 631/29)
After
Total variance original atom1 centered X = -51/29
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(-51/29, -51/29)
After
Total variance original atom1 squared deviation = 2601/841
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(2/29, 2601/841)
After
Total variance original atom1 weighted squared deviation = 5202/24389
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(22, 631/29)
After
Total variance original atom2 centered X = 7/29
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(7/29, 7/29)
After
Total variance original atom2 squared deviation = 49/841
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(26/29, 49/841)
After
Total variance original atom2 weighted squared deviation = 1274/24389
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(118, 631/29)
After
Total variance original atom3 centered X = 2791/29
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(2791/29, 2791/29)
After
Total variance original atom3 squared deviation = 7789681/841
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, 7789681/841)
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(6400/24389, 5202/24389, 1274/24389, 0)
After
Total variance ORIGINAL global variance = 444/841
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(1/29, 2/29, 26/29)
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(19/29, 40/29, 572/29)
After
Total variance cell0 weighted numerator = 631/29
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(631/29, 1)
After
Total variance positive cell0 mean = 631/29
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(19, 631/29)
After
Total variance cell0 atom0 conditional centering = -80/29
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(-80/29, -80/29)
After
Total variance cell0 atom0 conditional square = 6400/841
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(1/29, 1)
After
Total variance cell0 atom0 normalized conditional mass = 1/29
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(1/29, 6400/841)
After
Total variance cell0 atom0 conditional variance term = 6400/24389
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(1/29, -80/29)
After
Total variance cell0 atom0 weighted residual = -80/841
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(20, 631/29)
After
Total variance cell0 atom1 conditional centering = -51/29
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(-51/29, -51/29)
After
Total variance cell0 atom1 conditional square = 2601/841
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(2/29, 1)
After
Total variance cell0 atom1 normalized conditional mass = 2/29
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(2/29, 2601/841)
After
Total variance cell0 atom1 conditional variance term = 5202/24389
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(2/29, -51/29)
After
Total variance cell0 atom1 weighted residual = -102/841
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(22, 631/29)
After
Total variance cell0 atom2 conditional centering = 7/29
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(7/29, 7/29)
After
Total variance cell0 atom2 conditional square = 49/841
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(26/29, 1)
After
Total variance cell0 atom2 normalized conditional mass = 26/29
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(26/29, 49/841)
After
Total variance cell0 atom2 conditional variance term = 1274/24389
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(26/29, 7/29)
After
Total variance cell0 atom2 weighted residual = 182/841
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(6400/24389, 5202/24389, 1274/24389)
After
Total variance positive cell0 conditional variance = 444/841
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(-80/841, -102/841, 182/841)
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=631/29
After
conditional mean=631/29;conditional variance=444/841;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, 444/841)
After
Total variance cell0 weighted WITHIN term = 444/841
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(631/29, 631/29)
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(444/841)
After
Total variance sum WITHIN terms = 444/841
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(444/841, 0)
After
Total variance WITHIN plus BETWEEN = 444/841
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=631/29;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1, 2], 1, 631/29, 444/841], [1, [3], 0, null, null]]
After
Var(X)=444/841;within=444/841;between=0;sum=444/841;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=444/841,between=0,Var=444/841
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