# Question 25562: Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law [Return to catalogue page 128](https://ocme.evemisslab.com/en/questions/page/128/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations/#q-20261015-advance-20261115-r03-bias-variance-decomposition-0062) 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. R0050-25562 · q-20261015-advance-20261115-r03-bias-variance-decomposition-0062 ## Original question and givens Use complete laws=[[-1, [-1, 2, 5], [[1, 4], [1, 2], [1, 4]]], [0, [-1, 0, 1], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate). - G001 — Use complete laws=[[-1, [-1, 2, 5], [[1, 4], [1, 2], [1, 4]]], [0, [-1, 0, 1], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate). ## 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-bias-variance-decomposition](https://ocme.evemisslab.com/en/mko/mko-bias-variance-decomposition/) ## Step-by-step derivation S001 · impliesBefore source roles = [biasvar-multi-theta-table] After derived = [[theta_identity_rows, all_identities_hold, certificate]] Conditions: none List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Biasvar ORIGINAL law index0 ORIGINAL mass sum := sum(1/4, 1/2, 1/4) After Biasvar ORIGINAL law index0 ORIGINAL mass sum = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore source roles = [Biasvar ORIGINAL law index0, -1, [-1, 2, 5], [[1, 4], [1, 2], [1, 4]]] After derived = [[(-1, 1/4), (2, 1/2), (5, 1/4)], 1] Conditions: none Qualify this same complete P_theta law,with nonnegative original-order mass summing to1,and sort values numerically with their masses.Keep null rows.Unlike the earlier MSE node,this node explicitly proves the decomposition. Step identifiers and dependenciesRule: Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S002 S004 · calculationBefore Biasvar ORIGINAL law index0 mean product0 := mul(-1, 1/4) After Biasvar ORIGINAL law index0 mean product0 = -1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Biasvar ORIGINAL law index0 mean product1 := mul(2, 1/2) After Biasvar ORIGINAL law index0 mean product1 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Biasvar ORIGINAL law index0 mean product2 := mul(5, 1/4) After Biasvar ORIGINAL law index0 mean product2 = 5/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Biasvar ORIGINAL law index0 mean sum := sum(-1/4, 1, 5/4) After Biasvar ORIGINAL law index0 mean sum = 2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Biasvar ORIGINAL law index0 bias mean minus theta := sub(2, -1) After Biasvar ORIGINAL law index0 bias mean minus theta = 3 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Biasvar ORIGINAL law index0 centered deviation0 := sub(-1, 2) After Biasvar ORIGINAL law index0 centered deviation0 = -3 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Biasvar ORIGINAL law index0 centered square0 := mul(-3, -3) After Biasvar ORIGINAL law index0 centered square0 = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Biasvar ORIGINAL law index0 variance weighted term0 := mul(1/4, 9) After Biasvar ORIGINAL law index0 variance weighted term0 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Biasvar ORIGINAL law index0 centered deviation1 := sub(2, 2) After Biasvar ORIGINAL law index0 centered deviation1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Biasvar ORIGINAL law index0 centered square1 := mul(0, 0) After Biasvar ORIGINAL law index0 centered square1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Biasvar ORIGINAL law index0 variance weighted term1 := mul(1/2, 0) After Biasvar ORIGINAL law index0 variance weighted term1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Biasvar ORIGINAL law index0 centered deviation2 := sub(5, 2) After Biasvar ORIGINAL law index0 centered deviation2 = 3 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Biasvar ORIGINAL law index0 centered square2 := mul(3, 3) After Biasvar ORIGINAL law index0 centered square2 = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Biasvar ORIGINAL law index0 variance weighted term2 := mul(1/4, 9) After Biasvar ORIGINAL law index0 variance weighted term2 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Biasvar ORIGINAL law index0 variance sum := sum(9/4, 0, 9/4) After Biasvar ORIGINAL law index0 variance sum = 9/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Biasvar ORIGINAL law index0 theta error0 := sub(-1, -1) After Biasvar ORIGINAL law index0 theta error0 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Biasvar ORIGINAL law index0 theta square0 := mul(0, 0) After Biasvar ORIGINAL law index0 theta square0 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Biasvar ORIGINAL law index0 direct MSE term0 := mul(1/4, 0) After Biasvar ORIGINAL law index0 direct MSE term0 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Biasvar ORIGINAL law index0 theta error1 := sub(2, -1) After Biasvar ORIGINAL law index0 theta error1 = 3 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Biasvar ORIGINAL law index0 theta square1 := mul(3, 3) After Biasvar ORIGINAL law index0 theta square1 = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Biasvar ORIGINAL law index0 direct MSE term1 := mul(1/2, 9) After Biasvar ORIGINAL law index0 direct MSE term1 = 9/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Biasvar ORIGINAL law index0 theta error2 := sub(5, -1) After Biasvar ORIGINAL law index0 theta error2 = 6 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Biasvar ORIGINAL law index0 theta square2 := mul(6, 6) After Biasvar ORIGINAL law index0 theta square2 = 36 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Biasvar ORIGINAL law index0 direct MSE term2 := mul(1/4, 36) After Biasvar ORIGINAL law index0 direct MSE term2 = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Biasvar ORIGINAL law index0 direct MSE sum := sum(0, 9/2, 9) After Biasvar ORIGINAL law index0 direct MSE sum = 27/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Biasvar ORIGINAL law index0 centered mean product0 := mul(1/4, -3) After Biasvar ORIGINAL law index0 centered mean product0 = -3/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Biasvar ORIGINAL law index0 centered mean product1 := mul(1/2, 0) After Biasvar ORIGINAL law index0 centered mean product1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Biasvar ORIGINAL law index0 centered mean product2 := mul(1/4, 3) After Biasvar ORIGINAL law index0 centered mean product2 = 3/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Biasvar ORIGINAL law index0 centered mean sum := sum(-3/4, 0, 3/4) After Biasvar ORIGINAL law index0 centered mean sum = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Biasvar ORIGINAL law index0 bias times centered mean := mul(3, 0) After Biasvar ORIGINAL law index0 bias times centered mean = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Biasvar ORIGINAL law index0 cross twice inner := mul(2, 0) After Biasvar ORIGINAL law index0 cross twice inner = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Biasvar ORIGINAL law index0 squared bias := mul(3, 3) After Biasvar ORIGINAL law index0 squared bias = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Biasvar ORIGINAL law index0 variance plus squared bias := sum(9/2, 9) After Biasvar ORIGINAL law index0 variance plus squared bias = 27/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · impliesBefore source roles = [Biasvar ORIGINAL law index0, -1, 2, 3, [(-1, 1/4), (2, 1/2), (5, 1/4)]] After derived = [[[-1, 1/4, -3, 3, 0, 9, -18, 9, 0], [2, 1/2, 0, 3, 3, 0, 0, 9, 9], [5, 1/4, 3, 3, 6, 9, 18, 9, 36]], 0, 0, 9/2, 27/2, 27/2, Yes] Conditions: none For every value,T−theta=(T−mean)+bias.Square to centered²+2*bias*centered+bias² and weight under the same law.E[T−mean]=E[T]−mean*1=0 cancels the cross term,without requiring bias0 or every deviation0.Total mass1 leaves bias².Pointwise cross terms are extra exact teaching values,not new source-profile restrictions. Step identifiers and dependenciesRule: Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation · Uses: G001, S036 S038 · calculationBefore Biasvar ORIGINAL law index1 ORIGINAL mass sum := sum(1/4, 1/2, 1/4) After Biasvar ORIGINAL law index1 ORIGINAL mass sum = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore source roles = [Biasvar ORIGINAL law index1, 0, [-1, 0, 1], [[1, 4], [1, 2], [1, 4]]] After derived = [[(-1, 1/4), (0, 1/2), (1, 1/4)], 1] Conditions: none Qualify this same complete P_theta law,with nonnegative original-order mass summing to1,and sort values numerically with their masses.Keep null rows.Unlike the earlier MSE node,this node explicitly proves the decomposition. Step identifiers and dependenciesRule: Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S038 S040 · calculationBefore Biasvar ORIGINAL law index1 mean product0 := mul(-1, 1/4) After Biasvar ORIGINAL law index1 mean product0 = -1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Biasvar ORIGINAL law index1 mean product1 := mul(0, 1/2) After Biasvar ORIGINAL law index1 mean product1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Biasvar ORIGINAL law index1 mean product2 := mul(1, 1/4) After Biasvar ORIGINAL law index1 mean product2 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Biasvar ORIGINAL law index1 mean sum := sum(-1/4, 0, 1/4) After Biasvar ORIGINAL law index1 mean sum = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Biasvar ORIGINAL law index1 bias mean minus theta := sub(0, 0) After Biasvar ORIGINAL law index1 bias mean minus theta = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Biasvar ORIGINAL law index1 centered deviation0 := sub(-1, 0) After Biasvar ORIGINAL law index1 centered deviation0 = -1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Biasvar ORIGINAL law index1 centered square0 := mul(-1, -1) After Biasvar ORIGINAL law index1 centered square0 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Biasvar ORIGINAL law index1 variance weighted term0 := mul(1/4, 1) After Biasvar ORIGINAL law index1 variance weighted term0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Biasvar ORIGINAL law index1 centered deviation1 := sub(0, 0) After Biasvar ORIGINAL law index1 centered deviation1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore Biasvar ORIGINAL law index1 centered square1 := mul(0, 0) After Biasvar ORIGINAL law index1 centered square1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Biasvar ORIGINAL law index1 variance weighted term1 := mul(1/2, 0) After Biasvar ORIGINAL law index1 variance weighted term1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Biasvar ORIGINAL law index1 centered deviation2 := sub(1, 0) After Biasvar ORIGINAL law index1 centered deviation2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore Biasvar ORIGINAL law index1 centered square2 := mul(1, 1) After Biasvar ORIGINAL law index1 centered square2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore Biasvar ORIGINAL law index1 variance weighted term2 := mul(1/4, 1) After Biasvar ORIGINAL law index1 variance weighted term2 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore Biasvar ORIGINAL law index1 variance sum := sum(1/4, 0, 1/4) After Biasvar ORIGINAL law index1 variance sum = 1/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore Biasvar ORIGINAL law index1 theta error0 := sub(-1, 0) After Biasvar ORIGINAL law index1 theta error0 = -1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore Biasvar ORIGINAL law index1 theta square0 := mul(-1, -1) After Biasvar ORIGINAL law index1 theta square0 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Biasvar ORIGINAL law index1 direct MSE term0 := mul(1/4, 1) After Biasvar ORIGINAL law index1 direct MSE term0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore Biasvar ORIGINAL law index1 theta error1 := sub(0, 0) After Biasvar ORIGINAL law index1 theta error1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore Biasvar ORIGINAL law index1 theta square1 := mul(0, 0) After Biasvar ORIGINAL law index1 theta square1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore Biasvar ORIGINAL law index1 direct MSE term1 := mul(1/2, 0) After Biasvar ORIGINAL law index1 direct MSE term1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore Biasvar ORIGINAL law index1 theta error2 := sub(1, 0) After Biasvar ORIGINAL law index1 theta error2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore Biasvar ORIGINAL law index1 theta square2 := mul(1, 1) After Biasvar ORIGINAL law index1 theta square2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore Biasvar ORIGINAL law index1 direct MSE term2 := mul(1/4, 1) After Biasvar ORIGINAL law index1 direct MSE term2 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · calculationBefore Biasvar ORIGINAL law index1 direct MSE sum := sum(1/4, 0, 1/4) After Biasvar ORIGINAL law index1 direct MSE sum = 1/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore Biasvar ORIGINAL law index1 centered mean product0 := mul(1/4, -1) After Biasvar ORIGINAL law index1 centered mean product0 = -1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore Biasvar ORIGINAL law index1 centered mean product1 := mul(1/2, 0) After Biasvar ORIGINAL law index1 centered mean product1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore Biasvar ORIGINAL law index1 centered mean product2 := mul(1/4, 1) After Biasvar ORIGINAL law index1 centered mean product2 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore Biasvar ORIGINAL law index1 centered mean sum := sum(-1/4, 0, 1/4) After Biasvar ORIGINAL law index1 centered mean sum = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore Biasvar ORIGINAL law index1 bias times centered mean := mul(0, 0) After Biasvar ORIGINAL law index1 bias times centered mean = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore Biasvar ORIGINAL law index1 cross twice inner := mul(2, 0) After Biasvar ORIGINAL law index1 cross twice inner = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore Biasvar ORIGINAL law index1 squared bias := mul(0, 0) After Biasvar ORIGINAL law index1 squared bias = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore Biasvar ORIGINAL law index1 variance plus squared bias := sum(1/2, 0) After Biasvar ORIGINAL law index1 variance plus squared bias = 1/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · impliesBefore source roles = [Biasvar ORIGINAL law index1, 0, 0, 0, [(-1, 1/4), (0, 1/2), (1, 1/4)]] After derived = [[[-1, 1/4, -1, 0, -1, 1, 0, 0, 1], [0, 1/2, 0, 0, 0, 0, 0, 0, 0], [1, 1/4, 1, 0, 1, 1, 0, 0, 1]], 0, 0, 1/2, 1/2, 1/2, Yes] Conditions: none For every value,T−theta=(T−mean)+bias.Square to centered²+2*bias*centered+bias² and weight under the same law.E[T−mean]=E[T]−mean*1=0 cancels the cross term,without requiring bias0 or every deviation0.Total mass1 leaves bias².Pointwise cross terms are extra exact teaching values,not new source-profile restrictions. Step identifiers and dependenciesRule: Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation · Uses: G001, S072 S074 · impliesBefore source roles = [[[[-1, 1], [[[-1, 1], [1, 4]], [[2, 1], [1, 2]], [[5, 1], [1, 4]]]], [[0, 1], [[[-1, 1], [1, 4]], [[0, 1], [1, 2]], [[1, 1], [1, 4]]]]]] After derived = [[0, 1], [[-1, 2, 3, 9/2, 27/2, 27/2, Yes], [0, 0, 0, 1/2, 1/2, 1/2, Yes]]] Conditions: none Compute each theta under its own complete law,without prior pooling.Source order is English canonical normalized RAT[theta,rows],not numeric theta or display strings.This is a pointwise identity,not asymptotic consistency. Step identifiers and dependenciesRule: Biasvar_EACH_theta_RAW_RAT_English_law_order_not_prior_pool · Uses: G001, S073 ## Reverse checks - S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S072 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 variance plus squared bias with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 squared bias with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 cross twice inner with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 bias times centered mean with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered mean sum with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered mean product2 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered mean product1 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered mean product0 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 direct MSE sum with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 direct MSE term2 with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 theta square2 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 theta error2 with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 direct MSE term1 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 theta square1 with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 theta error1 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 direct MSE term0 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 theta square0 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 theta error0 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 variance sum with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 variance weighted term2 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered square2 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered deviation2 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 variance weighted term1 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered square1 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered deviation1 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 variance weighted term0 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered square0 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 centered deviation0 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 bias mean minus theta with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 mean sum with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 mean product2 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 mean product1 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 mean product0 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index1 ORIGINAL mass sum with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 variance plus squared bias with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 squared bias with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 cross twice inner with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 bias times centered mean with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered mean sum with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered mean product2 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered mean product1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered mean product0 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 direct MSE sum with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 direct MSE term2 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 theta square2 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 theta error2 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 direct MSE term1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 theta square1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 theta error1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 direct MSE term0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 theta square0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 theta error0 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 variance sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 variance weighted term2 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered square2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered deviation2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 variance weighted term1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered square1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered deviation1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 variance weighted term0 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered square0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 centered deviation0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 bias mean minus theta with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 mean sum with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 mean product2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 mean product1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Biasvar ORIGINAL law index0 mean product0 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 Biasvar ORIGINAL law index0 ORIGINAL mass sum 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 ([[-1, 2, 3, 9/2, 27/2, 27/2, Yes], [0, 0, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim) 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: e4148a43907a32420483c74886685e874553ad89b6799bb9ffbb1e951196942d - candidate record SHA-256: 678567e7d7f20f6083c28ba19be1d5fe86f8850c5e2f0947f43c53704725e55e - semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82 - question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r03-bias-variance-decomposition-0062.json)