Step-by-step derivation
S001 · implies
Before
source roles = [biasvar-biased-unbiased-boundary]
After
derived = [[theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary]]
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 dependencies
Rule: source_ordered_fields_without_JS · Uses: G001
S002 · calculation
Before
Biasvar ORIGINAL single theta ORIGINAL mass sum := sum(1/4, 1/2, 1/4)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · implies
Before
source roles = [Biasvar ORIGINAL single theta, 1, [1, 3, 5], [[1, 4], [1, 2], [1, 4]]]
After
derived = [[(1, 1/4), (3, 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 dependencies
Rule: Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S002
S004 · calculation
Before
Biasvar ORIGINAL single theta mean product0 := mul(1, 1/4)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Biasvar ORIGINAL single theta mean product1 := mul(3, 1/2)
After
Biasvar ORIGINAL single theta mean product1 = 3/2
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
Biasvar ORIGINAL single theta mean product2 := mul(5, 1/4)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
Biasvar ORIGINAL single theta mean sum := sum(1/4, 3/2, 5/4)
After
Biasvar ORIGINAL single theta mean sum = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
Biasvar ORIGINAL single theta bias mean minus theta := sub(3, 1)
After
Biasvar ORIGINAL single theta bias mean minus theta = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
Biasvar ORIGINAL single theta centered deviation0 := sub(1, 3)
After
Biasvar ORIGINAL single theta centered deviation0 = -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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Biasvar ORIGINAL single theta centered square0 := mul(-2, -2)
After
Biasvar ORIGINAL single theta centered square0 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
Biasvar ORIGINAL single theta variance weighted term0 := mul(1/4, 4)
After
Biasvar ORIGINAL single theta variance weighted term0 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
Biasvar ORIGINAL single theta centered deviation1 := sub(3, 3)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
Biasvar ORIGINAL single theta centered square1 := mul(0, 0)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
Biasvar ORIGINAL single theta variance weighted term1 := mul(1/2, 0)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
Biasvar ORIGINAL single theta centered deviation2 := sub(5, 3)
After
Biasvar ORIGINAL single theta centered deviation2 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
Biasvar ORIGINAL single theta centered square2 := mul(2, 2)
After
Biasvar ORIGINAL single theta centered square2 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
Biasvar ORIGINAL single theta variance weighted term2 := mul(1/4, 4)
After
Biasvar ORIGINAL single theta variance weighted term2 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · calculation
Before
Biasvar ORIGINAL single theta variance sum := sum(1, 0, 1)
After
Biasvar ORIGINAL single theta variance 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · calculation
Before
Biasvar ORIGINAL single theta theta error0 := sub(1, 1)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · calculation
Before
Biasvar ORIGINAL single theta theta square0 := mul(0, 0)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S019
S021 · calculation
Before
Biasvar ORIGINAL single theta direct MSE term0 := mul(1/4, 0)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · calculation
Before
Biasvar ORIGINAL single theta theta error1 := sub(3, 1)
After
Biasvar ORIGINAL single theta theta error1 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S021
S023 · calculation
Before
Biasvar ORIGINAL single theta theta square1 := mul(2, 2)
After
Biasvar ORIGINAL single theta theta square1 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · calculation
Before
Biasvar ORIGINAL single theta direct MSE term1 := mul(1/2, 4)
After
Biasvar ORIGINAL single theta direct MSE term1 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S023
S025 · calculation
Before
Biasvar ORIGINAL single theta theta error2 := sub(5, 1)
After
Biasvar ORIGINAL single theta theta error2 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · calculation
Before
Biasvar ORIGINAL single theta theta square2 := mul(4, 4)
After
Biasvar ORIGINAL single theta theta square2 = 16
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S025
S027 · calculation
Before
Biasvar ORIGINAL single theta direct MSE term2 := mul(1/4, 16)
After
Biasvar ORIGINAL single theta direct MSE term2 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · calculation
Before
Biasvar ORIGINAL single theta direct MSE sum := sum(0, 2, 4)
After
Biasvar ORIGINAL single theta direct MSE sum = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S027
S029 · calculation
Before
Biasvar ORIGINAL single theta centered mean product0 := mul(1/4, -2)
After
Biasvar ORIGINAL single theta centered mean product0 = -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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S028
S030 · calculation
Before
Biasvar ORIGINAL single theta centered mean product1 := mul(1/2, 0)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S029
S031 · calculation
Before
Biasvar ORIGINAL single theta centered mean product2 := mul(1/4, 2)
After
Biasvar ORIGINAL single theta centered mean product2 = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S030
S032 · calculation
Before
Biasvar ORIGINAL single theta centered mean sum := sum(-1/2, 0, 1/2)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S031
S033 · calculation
Before
Biasvar ORIGINAL single theta bias times centered mean := mul(2, 0)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · calculation
Before
Biasvar ORIGINAL single theta cross twice inner := mul(2, 0)
After
Biasvar ORIGINAL single theta 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S033
S035 · calculation
Before
Biasvar ORIGINAL single theta squared bias := mul(2, 2)
After
Biasvar ORIGINAL single theta squared bias = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S034
S036 · calculation
Before
Biasvar ORIGINAL single theta variance plus squared bias := sum(2, 4)
After
Biasvar ORIGINAL single theta variance plus squared bias = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S035
S037 · implies
Before
source roles = [Biasvar ORIGINAL single theta, 1, 3, 2, [(1, 1/4), (3, 1/2), (5, 1/4)]]
After
derived = [[[1, 1/4, -2, 2, 0, 4, -8, 4, 0], [3, 1/2, 0, 2, 2, 0, 0, 4, 4], [5, 1/4, 2, 2, 4, 4, 8, 4, 16]], 0, 0, 2, 6, 6, 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 dependencies
Rule: Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation · Uses: G001, S036
S038 · calculation
Before
Biasvar ORIGINAL observed minus theta := sub(1, 1)
After
Biasvar ORIGINAL observed 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S037
S039 · calculation
Before
Biasvar ORIGINAL observed square := mul(0, 0)
After
Biasvar ORIGINAL observed square = 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · implies
Before
source roles = [1, 0, 2, 2, 6]
After
derived = [No, Yes, No, No]
Conditions: none
Unbiasedness only removes bias²,giving MSE=variance,not variance0 or consistency.When biased,MSE>=variance.One observed loss is computed separately and may coincide with MSE without defining it.
Step identifiers and dependencies
Rule: Biasvar_observed_error_unbiased_variance_risk_consistency_separate · Uses: G001, S039