Step-by-step derivation
S001 · implies
Before
source operation=sampling-family-readback;original snapshot and mathematics retained
After
explicit ordered source output fields=[statistic, parameter_laws, model_qualified, certificate]
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
sampling parameter0 full law mass := sum(1/8, 1/4, 3/8, 1/4)
After
sampling 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
sampling:original parameter SLOT=0;same outcome carrier Fin(4);ALL aligned law masses=[1/8, 1/4, 3/8, 1/4]
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
sampling parameter1 full law mass := sum(1/6, 1/4, 1/3, 1/4)
After
sampling 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
sampling:original parameter SLOT=1;same outcome carrier Fin(4);ALL aligned law masses=[1/6, 1/4, 1/3, 1/4]
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
sampling:parameter SLOT carrier Fin(2);common outcome carrier Fin(4);ONE total original statistic array=[0, 0, 1, 2]
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 · calculation
Before
sampling parameter0 statistic value0 full fibre mass := sum(1/8, 1/4)
After
sampling parameter0 statistic value0 full fibre mass = 3/8
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, S006
S008 · implies
Before
FIXED original parameter SLOT=0;statistic value=0;FULL original fibre outcome indices=[0, 1];aligned law masses=[1/8, 1/4]
After
pushforward mass at value0=3/8;retain this row even if mass0
Conditions: none
Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter.
Step identifiers and dependencies
Rule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S007
S009 · calculation
Before
sampling parameter0 statistic value1 full fibre mass := sum(3/8)
After
sampling parameter0 statistic value1 full fibre mass = 3/8
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 · implies
Before
FIXED original parameter SLOT=0;statistic value=1;FULL original fibre outcome indices=[2];aligned law masses=[3/8]
After
pushforward mass at value1=3/8;retain this row even if mass0
Conditions: none
Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter.
Step identifiers and dependencies
Rule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S009
S011 · calculation
Before
sampling parameter0 statistic value2 full fibre mass := sum(1/4)
After
sampling parameter0 statistic value2 full fibre mass = 1/4
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 · implies
Before
FIXED original parameter SLOT=0;statistic value=2;FULL original fibre outcome indices=[3];aligned law masses=[1/4]
After
pushforward mass at value2=1/4;retain this row even if mass0
Conditions: none
Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter.
Step identifiers and dependencies
Rule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S011
S013 · calculation
Before
sampling parameter0 complete pushforward mass := sum(3/8, 3/8, 1/4)
After
sampling parameter0 complete pushforward 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, S012
S014 · implies
Before
fixed parameter SLOT=0;ALL increasingly ordered statistic values=[0, 1, 2];disjoint complete fibres=[[0, 1], [2], [3]]
After
pushforward law=[[0, 3/8], [1, 3/8], [2, 1/4]];total mass=1;ANY set of statistic values has preimage equal to the union of its fibres
Conditions: none
The disjoint fibres cover the entire common sample carrier,so the preimage probability of any set of statistic values is the sum of its rows.This gives the complete discrete sampling law,not observed frequencies or a distribution over parameters.
Step identifiers and dependencies
Rule: sampling_pushforward_full_partition_probability_law · Uses: G001, S013
S015 · calculation
Before
sampling parameter1 statistic value0 full fibre mass := sum(1/6, 1/4)
After
sampling parameter1 statistic value0 full fibre mass = 5/12
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, S014
S016 · implies
Before
FIXED original parameter SLOT=1;statistic value=0;FULL original fibre outcome indices=[0, 1];aligned law masses=[1/6, 1/4]
After
pushforward mass at value0=5/12;retain this row even if mass0
Conditions: none
Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter.
Step identifiers and dependencies
Rule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S015
S017 · calculation
Before
sampling parameter1 statistic value1 full fibre mass := sum(1/3)
After
sampling parameter1 statistic value1 full fibre mass = 1/3
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, S016
S018 · implies
Before
FIXED original parameter SLOT=1;statistic value=1;FULL original fibre outcome indices=[2];aligned law masses=[1/3]
After
pushforward mass at value1=1/3;retain this row even if mass0
Conditions: none
Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter.
Step identifiers and dependencies
Rule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S017
S019 · calculation
Before
sampling parameter1 statistic value2 full fibre mass := sum(1/4)
After
sampling parameter1 statistic value2 full fibre mass = 1/4
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, S018
S020 · implies
Before
FIXED original parameter SLOT=1;statistic value=2;FULL original fibre outcome indices=[3];aligned law masses=[1/4]
After
pushforward mass at value2=1/4;retain this row even if mass0
Conditions: none
Add masses from every outcome with the same statistic value,not one representative.Retain every value present in the original statistic even when its fibre has mass0 under this parameter.
Step identifiers and dependencies
Rule: sampling_complete_fibre_not_one_representative_or_null_deletion · Uses: G001, S019
S021 · calculation
Before
sampling parameter1 complete pushforward mass := sum(5/12, 1/3, 1/4)
After
sampling parameter1 complete pushforward 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, S020
S022 · implies
Before
fixed parameter SLOT=1;ALL increasingly ordered statistic values=[0, 1, 2];disjoint complete fibres=[[0, 1], [2], [3]]
After
pushforward law=[[0, 5/12], [1, 1/3], [2, 1/4]];total mass=1;ANY set of statistic values has preimage equal to the union of its fibres
Conditions: none
The disjoint fibres cover the entire common sample carrier,so the preimage probability of any set of statistic values is the sum of its rows.This gives the complete discrete sampling law,not observed frequencies or a distribution over parameters.
Step identifiers and dependencies
Rule: sampling_pushforward_full_partition_probability_law · Uses: G001, S021
S023 · implies
Before
ONE original statistic=[0, 0, 1, 2];ALL original parameter slots0..1
After
ordered full parameter-indexed family=[[0, [[0, 3/8], [1, 3/8], [2, 1/4]]], [1, [[0, 5/12], [1, 1/3], [2, 1/4]]]];NO parameter averaging or unstated prior
Conditions: none
Retain each parameter's own law;the same statistic may have different sampling distributions.This is an indexed family,not an equal-weight or otherwise invented mixture over parameters.
Step identifiers and dependencies
Rule: sampling_one_statistic_many_laws_not_parameter_prior · Uses: G001, S022