# Question 22686: Undefined Null-Cell Mean and Variance Have Zero Outer Contribution [Return to catalogue page 114](https://ocme.evemisslab.com/en/questions/page/114/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261110-r04-law-of-total-variance-foundations/#q-20261014-advance-20261110-r04-law-of-total-variance-0086) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0044-22686 · q-20261014-advance-20261110-r04-law-of-total-variance-0086 ## Original question and givens Use the same complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1, 2], [3]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary). - G001 — Use the same complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1, 2], [3]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-law-of-total-variance](https://ocme.evemisslab.com/en/mko/mko-law-of-total-variance/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore 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 dependenciesRule: totalvar_NONEMPTY_cells_complete_disjoint_same_law · Uses: G001, S002 S004 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Total variance cell0 mass := sum(1/20, 1/10, 17/20) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Total variance cell0 weighted numerator := sum(1/2, 11/10, 221/20) After Total variance cell0 weighted numerator = 253/20 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Total variance positive cell0 mean := div(253/20, 1) After Total variance positive cell0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Total variance cell0 atom0 conditional centering := sub(10, 253/20) After Total variance cell0 atom0 conditional centering = -53/20 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Total variance cell0 atom0 conditional square := mul(-53/20, -53/20) After Total variance cell0 atom0 conditional square = 2809/400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Total variance cell0 atom0 normalized conditional mass := div(1/20, 1) After Total variance cell0 atom0 normalized conditional mass = 1/20 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Total variance cell0 atom0 conditional variance term := mul(1/20, 2809/400) After Total variance cell0 atom0 conditional variance term = 2809/8000 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Total variance cell0 atom0 weighted residual := mul(1/20, -53/20) After Total variance cell0 atom0 weighted residual = -53/400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Total variance cell0 atom1 conditional centering := sub(11, 253/20) After Total variance cell0 atom1 conditional centering = -33/20 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Total variance cell0 atom1 conditional square := mul(-33/20, -33/20) After Total variance cell0 atom1 conditional square = 1089/400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Total variance cell0 atom1 normalized conditional mass := div(1/10, 1) After Total variance cell0 atom1 normalized conditional mass = 1/10 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Total variance cell0 atom1 conditional variance term := mul(1/10, 1089/400) After Total variance cell0 atom1 conditional variance term = 1089/4000 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Total variance cell0 atom1 weighted residual := mul(1/10, -33/20) After Total variance cell0 atom1 weighted residual = -33/200 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Total variance cell0 atom2 conditional centering := sub(13, 253/20) After Total variance cell0 atom2 conditional centering = 7/20 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Total variance cell0 atom2 conditional square := mul(7/20, 7/20) After Total variance cell0 atom2 conditional square = 49/400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Total variance cell0 atom2 normalized conditional mass := div(17/20, 1) After Total variance cell0 atom2 normalized conditional 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Total variance cell0 atom2 conditional variance term := mul(17/20, 49/400) After Total variance cell0 atom2 conditional variance 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Total variance cell0 atom2 weighted residual := mul(17/20, 7/20) After Total variance cell0 atom2 weighted residual = 119/400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Total variance positive cell0 conditional variance := sum(2809/8000, 1089/4000, 833/8000) After Total variance positive cell0 conditional 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Total variance positive cell0 residual mean zero := sum(-53/400, -33/200, 119/400) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · impliesBefore ORIGINAL cell index=0;actual atoms=[0, 1, 2];mass=1;numerator=253/20 After conditional mean=253/20;conditional variance=291/400;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 dependenciesRule: totalvar_conditional_mean_variance_null_and_cross_term · Uses: G001, S041 S043 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · impliesBefore 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 dependenciesRule: totalvar_conditional_mean_variance_null_and_cross_term · Uses: G001, S044 S046 · calculationBefore Total variance cell0 weighted WITHIN term := mul(1, 291/400) After Total variance cell0 weighted WITHIN term = 291/400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Total variance cell0 mean minus global mean := sub(253/20, 253/20) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Total variance sum WITHIN terms := sum(291/400) After Total variance sum WITHIN terms = 291/400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore Total variance WITHIN plus BETWEEN := sum(291/400, 0) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · impliesBefore ORIGINAL SAME-law mean=253/20;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1, 2], 1, 253/20, 291/400], [1, [3], 0, null, null]] After Var(X)=291/400;within=291/400;between=0;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 dependenciesRule: totalvar_general_centered_square_decomposition_same_partition · Uses: G001, S052 S054 · impliesBefore ACTUAL NONEMPTY null cell indices=[1];within=291/400,between=0,Var=291/400 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 dependenciesRule: totalvar_null_division_No_not_identity_failure · Uses: G001, S053 ## Reverse checks - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Recompute Total variance WITHIN plus BETWEEN with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Total variance sum BETWEEN terms with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Total variance sum WITHIN terms with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Total variance cell0 weighted BETWEEN term with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Total variance cell0 BETWEEN squared deviation with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Total variance cell0 mean minus global mean with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Total variance cell0 weighted WITHIN term with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Recompute Total variance cell1 weighted numerator with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Total variance cell1 mass with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Recompute Total variance positive cell0 residual mean zero with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Total variance positive cell0 conditional variance with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Total variance cell0 atom2 weighted residual with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Total variance cell0 atom2 conditional variance term with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Total variance cell0 atom2 normalized conditional mass with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Total variance cell0 atom2 conditional square with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Total variance cell0 atom2 conditional centering with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Total variance cell0 atom1 weighted residual with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Total variance cell0 atom1 conditional variance term with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Total variance cell0 atom1 normalized conditional mass with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Total variance cell0 atom1 conditional square with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Total variance cell0 atom1 conditional centering with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Total variance cell0 atom0 weighted residual with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Total variance cell0 atom0 conditional variance term with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Total variance cell0 atom0 normalized conditional mass with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Total variance cell0 atom0 conditional square with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Total variance cell0 atom0 conditional centering with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Total variance positive cell0 mean with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Total variance cell0 weighted numerator with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Total variance cell0 mass with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Total variance ORIGINAL global variance with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Total variance original atom3 weighted squared deviation with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Total variance original atom3 squared deviation with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Total variance original atom3 centered X with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Total variance original atom2 weighted squared deviation with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Total variance original atom2 squared deviation with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Total variance original atom2 centered X with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Total variance original atom1 weighted squared deviation with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Total variance original atom1 squared deviation with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Total variance original atom1 centered X with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Total variance original atom0 weighted squared deviation with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Total variance original atom0 squared deviation with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Total variance original atom0 centered X with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Total variance ORIGINAL global mean with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Total variance original atom3 weighted X with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Total variance original atom2 weighted X with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Total variance original atom1 weighted X with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Total variance original atom0 weighted X with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute Total variance complete probability mass with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([[0, [0, 1, 2], 1, 253/20, 291/400], [1, [3], 0, null, null]], 291/400, 0, 291/400, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: ef42b57de3edbfa505bbd13f676aa7e4eb4217381489b19e14a8ab13084c5b90 - candidate record SHA-256: 90c266833a9dcafdf1cede019d6bb9a192e1c1ffc317d939150b5919c434f23c - semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81 - question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r04-law-of-total-variance-0086.json)