Original question and givens
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
G001 — Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
Step-by-step derivation
S001 · implies
Before
source operation=bias-observed-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
bias parameter0 full law mass := sum(1/30, 1/15, 1/10, 4/5)
After
bias parameter0 full law mass = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · implies
Before
bias:original parameter SLOT=0;same outcome carrier Fin(4);ALL aligned law masses=[1/30, 1/15, 1/10, 4/5]
After
full sum=1;all masses nonnegative=Yes;qualified probability law;zero-mass outcomes retained
Conditions: none
Each complete parameter law must separately be nonnegative and total1.Keep zero-mass outcomes in the original table;do not normalize across parameter rows to invent a prior or posterior.
Step identifiers and dependencies
Rule: finite_statistical_full_law_not_partial_or_prior · Uses: G001, S002
S004 · calculation
Before
bias parameter1 full law mass := sum(1/17, 3/34, 2/17, 25/34)
After
bias parameter1 full law mass = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
bias:original parameter SLOT=1;same outcome carrier Fin(4);ALL aligned law masses=[1/17, 3/34, 2/17, 25/34]
After
full sum=1;all masses nonnegative=Yes;qualified probability law;zero-mass outcomes retained
Conditions: none
Each complete parameter law must separately be nonnegative and total1.Keep zero-mass outcomes in the original table;do not normalize across parameter rows to invent a prior or posterior.
Step identifiers and dependencies
Rule: finite_statistical_full_law_not_partial_or_prior · Uses: G001, S004
S006 · implies
Before
bias:parameter SLOT carrier Fin(2);common outcome carrier Fin(4);ONE total original statistic array=[0, 1, 22, 23]
After
T takes only the observation index,not the unknown parameter;SAME function at all parameter slots;its sampling law may still vary with the slot
Conditions: none
Functional parameter-independence is different from a parameter-free distribution of T(X).Use the same original statistic value at every outcome rather than changing T between laws.
Step identifiers and dependencies
Rule: finite_statistic_definition_not_distribution_free · Uses: G001, S005
S007 · implies
Before
original parameter SLOT order0..1;aligned exact theta_values=[1/2, 21/23]
After
declared target g(theta)=theta;indices are not targets;claims range over THIS supplied finite parameter model only
Conditions: none
Keep target values separate from parameter indices;theta values need not be distinct.No unprovided continuous parameter space,prior or sequence of statistics is supplied,so a universal claim within this finite model is not extended beyond it.
Step identifiers and dependencies
Rule: bias_declared_target_and_finite_parameter_scope · Uses: G001, S006
S008 · calculation
Before
bias parameter0 original outcome0 weighted statistic := mul(1/30, 0)
After
bias parameter0 original outcome0 weighted statistic = 0
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
bias parameter0 original outcome1 weighted statistic := mul(1/15, 1)
After
bias parameter0 original outcome1 weighted statistic = 1/15
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
bias parameter0 original outcome2 weighted statistic := mul(1/10, 22)
After
bias parameter0 original outcome2 weighted statistic = 11/5
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
bias parameter0 original outcome3 weighted statistic := mul(4/5, 23)
After
bias parameter0 original outcome3 weighted statistic = 92/5
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
bias parameter0 same-law expectation := sum(0, 1/15, 11/5, 92/5)
After
bias parameter0 same-law expectation = 62/3
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
bias parameter0 expectation minus actual target := sub(62/3, 1/2)
After
bias parameter0 expectation minus actual target = 121/6
Conditions: none
Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
fixed original parameter SLOT=0;actual aligned theta VALUE=1/2;SAME complete law=[1/30, 1/15, 1/10, 4/5];same statistic=[0, 1, 22, 23];weighted terms=[0, 1/15, 11/5, 92/5]
After
E_at_slot[T]=62/3;Bias=E_at_slot[T]-ACTUAL theta value=121/6
Conditions: none
The target g(theta)=theta is now declared.Subtract the aligned theta_values entry,not its slot index or one observed estimate,and use no other parameter's law.
Step identifiers and dependencies
Rule: bias_same_law_expectation_minus_target_not_index · Uses: G001, S013
S015 · implies
Before
original parameter SLOT=0;original outcome=0;observed estimate=0;SEPARATE same-law bias=121/6
After
single-observation property sufficiency=No;NOT a comparison of whether estimate numerically equals bias;no zero-variance/consistency/sufficiency/minimum-MSE claim
Conditions: none
The No withholds property claims from one observation;it does not assert numerical inequality of estimate and bias.Unbiasedness itself does not imply zero variance,consistency,sufficiency or minimum MSE;these require their own target and sampling framework.
Step identifiers and dependencies
Rule: bias_observed_No_nonclaim_not_numerical_inequality · Uses: G001, S014