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(1/18, 1/9, 5/6, 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=[1/18, 1/9, 5/6, 0];SIGNED original X=[8, 9, 11, 107];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(1/18, 8)
After
Total variance original atom0 weighted X = 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, S003
S005 · calculation
Before
Total variance original atom1 weighted X := mul(1/9, 9)
After
Total variance original atom1 weighted X = 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, S004
S006 · calculation
Before
Total variance original atom2 weighted X := mul(5/6, 11)
After
Total variance original atom2 weighted X = 55/6
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, 107)
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(4/9, 1, 55/6, 0)
After
Total variance ORIGINAL global mean = 191/18
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(8, 191/18)
After
Total variance original atom0 centered X = -47/18
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(-47/18, -47/18)
After
Total variance original atom0 squared deviation = 2209/324
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(1/18, 2209/324)
After
Total variance original atom0 weighted squared deviation = 2209/5832
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(9, 191/18)
After
Total variance original atom1 centered X = -29/18
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(-29/18, -29/18)
After
Total variance original atom1 squared deviation = 841/324
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(1/9, 841/324)
After
Total variance original atom1 weighted squared deviation = 841/2916
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(11, 191/18)
After
Total variance original atom2 centered X = 7/18
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(7/18, 7/18)
After
Total variance original atom2 squared deviation = 49/324
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(5/6, 49/324)
After
Total variance original atom2 weighted squared deviation = 245/1944
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(107, 191/18)
After
Total variance original atom3 centered X = 1735/18
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(1735/18, 1735/18)
After
Total variance original atom3 squared deviation = 3010225/324
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, 3010225/324)
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(2209/5832, 841/2916, 245/1944, 0)
After
Total variance ORIGINAL global variance = 257/324
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(1/18, 1/9, 5/6)
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(4/9, 1, 55/6)
After
Total variance cell0 weighted numerator = 191/18
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(191/18, 1)
After
Total variance positive cell0 mean = 191/18
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(8, 191/18)
After
Total variance cell0 atom0 conditional centering = -47/18
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(-47/18, -47/18)
After
Total variance cell0 atom0 conditional square = 2209/324
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(1/18, 1)
After
Total variance cell0 atom0 normalized conditional mass = 1/18
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(1/18, 2209/324)
After
Total variance cell0 atom0 conditional variance term = 2209/5832
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(1/18, -47/18)
After
Total variance cell0 atom0 weighted residual = -47/324
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(9, 191/18)
After
Total variance cell0 atom1 conditional centering = -29/18
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(-29/18, -29/18)
After
Total variance cell0 atom1 conditional square = 841/324
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(1/9, 1)
After
Total variance cell0 atom1 normalized conditional mass = 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, S031
S033 · calculation
Before
Total variance cell0 atom1 conditional variance term := mul(1/9, 841/324)
After
Total variance cell0 atom1 conditional variance term = 841/2916
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(1/9, -29/18)
After
Total variance cell0 atom1 weighted residual = -29/162
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(11, 191/18)
After
Total variance cell0 atom2 conditional centering = 7/18
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(7/18, 7/18)
After
Total variance cell0 atom2 conditional square = 49/324
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(5/6, 1)
After
Total variance cell0 atom2 normalized conditional mass = 5/6
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(5/6, 49/324)
After
Total variance cell0 atom2 conditional variance term = 245/1944
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(5/6, 7/18)
After
Total variance cell0 atom2 weighted residual = 35/108
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(2209/5832, 841/2916, 245/1944)
After
Total variance positive cell0 conditional variance = 257/324
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(-47/324, -29/162, 35/108)
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=191/18
After
conditional mean=191/18;conditional variance=257/324;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, 257/324)
After
Total variance cell0 weighted WITHIN term = 257/324
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(191/18, 191/18)
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(257/324)
After
Total variance sum WITHIN terms = 257/324
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(257/324, 0)
After
Total variance WITHIN plus BETWEEN = 257/324
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=191/18;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1, 2], 1, 191/18, 257/324], [1, [3], 0, null, null]]
After
Var(X)=257/324;within=257/324;between=0;sum=257/324;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=257/324,between=0,Var=257/324
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