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R0050-25554 · q-20261015-advance-20261115-r03-bias-variance-decomposition-0054
Use complete laws=[[-9, [-9, -8, -7], [[1, 4], [1, 2], [1, 4]]], [-8, [-9, -8, -7], [[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=[[-9, [-9, -8, -7], [[1, 4], [1, 2], [1, 4]]], [-8, [-9, -8, -7], [[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).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.
Before
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [Biasvar ORIGINAL law index0, -9, [-9, -8, -7], [[1, 4], [1, 2], [1, 4]]]
After
derived = [[(-9, 1/4), (-8, 1/2), (-7, 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.
Rule: Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S002
Before
Biasvar ORIGINAL law index0 mean product0 := mul(-9, 1/4)
After
Biasvar ORIGINAL law index0 mean product0 = -9/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Biasvar ORIGINAL law index0 mean product1 := mul(-8, 1/2)
After
Biasvar ORIGINAL law index0 mean product1 = -4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Biasvar ORIGINAL law index0 mean product2 := mul(-7, 1/4)
After
Biasvar ORIGINAL law index0 mean product2 = -7/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Biasvar ORIGINAL law index0 mean sum := sum(-9/4, -4, -7/4)
After
Biasvar ORIGINAL law index0 mean sum = -8
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Biasvar ORIGINAL law index0 bias mean minus theta := sub(-8, -9)
After
Biasvar ORIGINAL law index0 bias mean minus theta = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Biasvar ORIGINAL law index0 centered deviation0 := sub(-9, -8)
After
Biasvar ORIGINAL law index0 centered deviation0 = -1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Biasvar ORIGINAL law index0 centered square0 := mul(-1, -1)
After
Biasvar ORIGINAL law index0 centered square0 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Biasvar ORIGINAL law index0 variance weighted term0 := mul(1/4, 1)
After
Biasvar ORIGINAL law index0 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.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
Biasvar ORIGINAL law index0 centered deviation1 := sub(-8, -8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
Biasvar ORIGINAL law index0 centered deviation2 := sub(-7, -8)
After
Biasvar ORIGINAL law index0 centered deviation2 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
Biasvar ORIGINAL law index0 centered square2 := mul(1, 1)
After
Biasvar ORIGINAL law index0 centered square2 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
Biasvar ORIGINAL law index0 variance weighted term2 := mul(1/4, 1)
After
Biasvar ORIGINAL law index0 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.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
Biasvar ORIGINAL law index0 variance sum := sum(1/4, 0, 1/4)
After
Biasvar ORIGINAL law index0 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.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
Biasvar ORIGINAL law index0 theta error0 := sub(-9, -9)
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.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
Biasvar ORIGINAL law index0 theta error1 := sub(-8, -9)
After
Biasvar ORIGINAL law index0 theta error1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
Biasvar ORIGINAL law index0 theta square1 := mul(1, 1)
After
Biasvar ORIGINAL law index0 theta square1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
Biasvar ORIGINAL law index0 direct MSE term1 := mul(1/2, 1)
After
Biasvar ORIGINAL law index0 direct MSE term1 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
Biasvar ORIGINAL law index0 theta error2 := sub(-7, -9)
After
Biasvar ORIGINAL law index0 theta error2 = 2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
Biasvar ORIGINAL law index0 theta square2 := mul(2, 2)
After
Biasvar ORIGINAL law index0 theta square2 = 4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
Biasvar ORIGINAL law index0 direct MSE term2 := mul(1/4, 4)
After
Biasvar ORIGINAL law index0 direct MSE term2 = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
Biasvar ORIGINAL law index0 direct MSE sum := sum(0, 1/2, 1)
After
Biasvar ORIGINAL law index0 direct MSE sum = 3/2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
Biasvar ORIGINAL law index0 centered mean product0 := mul(1/4, -1)
After
Biasvar ORIGINAL law index0 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.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
Biasvar ORIGINAL law index0 centered mean product2 := mul(1/4, 1)
After
Biasvar ORIGINAL law index0 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.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
Biasvar ORIGINAL law index0 centered mean sum := sum(-1/4, 0, 1/4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
Biasvar ORIGINAL law index0 bias times centered mean := mul(1, 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.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
Biasvar ORIGINAL law index0 squared bias := mul(1, 1)
After
Biasvar ORIGINAL law index0 squared bias = 1
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
Biasvar ORIGINAL law index0 variance plus squared bias := sum(1/2, 1)
After
Biasvar ORIGINAL law index0 variance plus squared bias = 3/2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
source roles = [Biasvar ORIGINAL law index0, -9, -8, 1, [(-9, 1/4), (-8, 1/2), (-7, 1/4)]]
After
derived = [[[-9, 1/4, -1, 1, 0, 1, -2, 1, 0], [-8, 1/2, 0, 1, 1, 0, 0, 1, 1], [-7, 1/4, 1, 1, 2, 1, 2, 1, 4]], 0, 0, 1/2, 3/2, 3/2, 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.
Rule: Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation · Uses: G001, S036
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
source roles = [Biasvar ORIGINAL law index1, -8, [-9, -8, -7], [[1, 4], [1, 2], [1, 4]]]
After
derived = [[(-9, 1/4), (-8, 1/2), (-7, 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.
Rule: Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S038
Before
Biasvar ORIGINAL law index1 mean product0 := mul(-9, 1/4)
After
Biasvar ORIGINAL law index1 mean product0 = -9/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
Biasvar ORIGINAL law index1 mean product1 := mul(-8, 1/2)
After
Biasvar ORIGINAL law index1 mean product1 = -4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
Biasvar ORIGINAL law index1 mean product2 := mul(-7, 1/4)
After
Biasvar ORIGINAL law index1 mean product2 = -7/4
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
Biasvar ORIGINAL law index1 mean sum := sum(-9/4, -4, -7/4)
After
Biasvar ORIGINAL law index1 mean sum = -8
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
Biasvar ORIGINAL law index1 bias mean minus theta := sub(-8, -8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
Biasvar ORIGINAL law index1 centered deviation0 := sub(-9, -8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
Biasvar ORIGINAL law index1 centered deviation1 := sub(-8, -8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
Biasvar ORIGINAL law index1 centered deviation2 := sub(-7, -8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
Biasvar ORIGINAL law index1 theta error0 := sub(-9, -8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
Biasvar ORIGINAL law index1 theta error1 := sub(-8, -8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
Biasvar ORIGINAL law index1 theta error2 := sub(-7, -8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S068
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
source roles = [Biasvar ORIGINAL law index1, -8, -8, 0, [(-9, 1/4), (-8, 1/2), (-7, 1/4)]]
After
derived = [[[-9, 1/4, -1, 0, -1, 1, 0, 0, 1], [-8, 1/2, 0, 0, 0, 0, 0, 0, 0], [-7, 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.
Rule: Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation · Uses: G001, S072
Before
source roles = [[[[-9, 1], [[[-9, 1], [1, 4]], [[-8, 1], [1, 2]], [[-7, 1], [1, 4]]]], [[-8, 1], [[[-9, 1], [1, 4]], [[-8, 1], [1, 2]], [[-7, 1], [1, 4]]]]]]
After
derived = [[1, 0], [[-8, -8, 0, 1/2, 1/2, 1/2, Yes], [-9, -8, 1, 1/2, 3/2, 3/2, 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.
Rule: Biasvar_EACH_theta_RAW_RAT_English_law_order_not_prior_pool · Uses: G001, S073
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.([[-8, -8, 0, 1/2, 1/2, 1/2, Yes], [-9, -8, 1, 1/2, 3/2, 3/2, 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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f00ec7c78e5fc5704aabec16696b0687ed2f472df838b4710f9a6ad42feeb358candidate record SHA-256: c3f31db8acdff4bd00990578740d07c8498ddfc71cb5f55867875ce340aafc7bsemantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95fLean: NOT_RUN · human review: NOT_PERFORMED