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P250-0128 · q-20261015-advance-20261116-r05-finite-sufficient-statistic-0028
Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).
G001 — Use complete theta labels=[a2, b2], samples=[20, 21, 22, 23], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary).The statistic is total on the full sample carrier. Normalize each parameter law and condition only on positive-mass fibers.
Before
Ω=[20, 21, 22, 23]; T=[0, 1, 0, 1]
After
fibers=[[0, [20, 22]], [1, [21, 23]]]
Conditions: none
Group the original sample labels by statistic value; distinguish a sample's position from its label.
Rule: statistic_fibers · Uses: G001
Before
[[0, [20, 22]], [1, [21, 23]]]
After
disjoint union of fibers = [20, 21, 22, 23]
Conditions: none
Each sample lies in exactly one fiber, and their union covers all of Omega.
Rule: partition_audit · Uses: G001, S001
Before
P_a2(Ω) := sum(2/5, 1/10, 2/5, 1/10)
After
P_a2(Ω) = 1
Conditions: none
Verify normalization of the probability row for this parameter.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
P_a2(T=0) := sum(2/5, 2/5)
After
P_a2(T=0) = 4/5
Conditions: none
Sum only the masses in fiber [20, 22].
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
P_a2(X=20|T=0) := div(2/5, 4/5)
After
P_a2(X=20|T=0) = 1/2
Conditions: none
The numerator is the atom mass 2/5 under the same law and the denominator is the fiber mass 4/5.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
reconstruct P_a2(X=20) := mul(1/2, 4/5)
After
reconstruct P_a2(X=20) = 2/5
Conditions: none
Multiply by the fiber mass to recover that atom's original probability.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
P_a2(X=22|T=0) := div(2/5, 4/5)
After
P_a2(X=22|T=0) = 1/2
Conditions: none
The numerator is the atom mass 2/5 under the same law and the denominator is the fiber mass 4/5.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
reconstruct P_a2(X=22) := mul(1/2, 4/5)
After
reconstruct P_a2(X=22) = 2/5
Conditions: none
Multiply by the fiber mass to recover that atom's original probability.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
conditional total (a2,0) := sum(1/2, 1/2)
After
conditional total (a2,0) = 1
Conditions: none
The full conditional law on a positive-mass fiber must sum to 1.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
P_a2(T=1) := sum(1/10, 1/10)
After
P_a2(T=1) = 1/5
Conditions: none
Sum only the masses in fiber [21, 23].
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
P_a2(X=21|T=1) := div(1/10, 1/5)
After
P_a2(X=21|T=1) = 1/2
Conditions: none
The numerator is the atom mass 1/10 under the same law and the denominator is the fiber mass 1/5.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
reconstruct P_a2(X=21) := mul(1/2, 1/5)
After
reconstruct P_a2(X=21) = 1/10
Conditions: none
Multiply by the fiber mass to recover that atom's original probability.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
P_a2(X=23|T=1) := div(1/10, 1/5)
After
P_a2(X=23|T=1) = 1/2
Conditions: none
The numerator is the atom mass 1/10 under the same law and the denominator is the fiber mass 1/5.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
reconstruct P_a2(X=23) := mul(1/2, 1/5)
After
reconstruct P_a2(X=23) = 1/10
Conditions: none
Multiply by the fiber mass to recover that atom's original probability.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
conditional total (a2,1) := sum(1/2, 1/2)
After
conditional total (a2,1) = 1
Conditions: none
The full conditional law on a positive-mass fiber must sum to 1.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
P_b2(Ω) := sum(1/10, 2/5, 1/10, 2/5)
After
P_b2(Ω) = 1
Conditions: none
Verify normalization of the probability row for this parameter.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
P_b2(T=0) := sum(1/10, 1/10)
After
P_b2(T=0) = 1/5
Conditions: none
Sum only the masses in fiber [20, 22].
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
P_b2(X=20|T=0) := div(1/10, 1/5)
After
P_b2(X=20|T=0) = 1/2
Conditions: none
The numerator is the atom mass 1/10 under the same law and the denominator is the fiber mass 1/5.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
reconstruct P_b2(X=20) := mul(1/2, 1/5)
After
reconstruct P_b2(X=20) = 1/10
Conditions: none
Multiply by the fiber mass to recover that atom's original probability.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
P_b2(X=22|T=0) := div(1/10, 1/5)
After
P_b2(X=22|T=0) = 1/2
Conditions: none
The numerator is the atom mass 1/10 under the same law and the denominator is the fiber mass 1/5.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
reconstruct P_b2(X=22) := mul(1/2, 1/5)
After
reconstruct P_b2(X=22) = 1/10
Conditions: none
Multiply by the fiber mass to recover that atom's original probability.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
conditional total (b2,0) := sum(1/2, 1/2)
After
conditional total (b2,0) = 1
Conditions: none
The full conditional law on a positive-mass fiber must sum to 1.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
P_b2(T=1) := sum(2/5, 2/5)
After
P_b2(T=1) = 4/5
Conditions: none
Sum only the masses in fiber [21, 23].
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
P_b2(X=21|T=1) := div(2/5, 4/5)
After
P_b2(X=21|T=1) = 1/2
Conditions: none
The numerator is the atom mass 2/5 under the same law and the denominator is the fiber mass 4/5.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
reconstruct P_b2(X=21) := mul(1/2, 4/5)
After
reconstruct P_b2(X=21) = 2/5
Conditions: none
Multiply by the fiber mass to recover that atom's original probability.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
P_b2(X=23|T=1) := div(2/5, 4/5)
After
P_b2(X=23|T=1) = 1/2
Conditions: none
The numerator is the atom mass 2/5 under the same law and the denominator is the fiber mass 4/5.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
reconstruct P_b2(X=23) := mul(1/2, 4/5)
After
reconstruct P_b2(X=23) = 2/5
Conditions: none
Multiply by the fiber mass to recover that atom's original probability.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
conditional total (b2,1) := sum(1/2, 1/2)
After
conditional total (b2,1) = 1
Conditions: none
The full conditional law on a positive-mass fiber must sum to 1.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
all parameter × fiber cells audited
After
[[a2, 0, 4/5, [[20, 1/2], [22, 1/2]]], [a2, 1, 1/5, [[21, 1/2], [23, 1/2]]], [b2, 0, 1/5, [[20, 1/2], [22, 1/2]]], [b2, 1, 4/5, [[21, 1/2], [23, 1/2]]]]
Conditions: none
Return every cell in parameter order and then statistic-value order; retain the zero-mass branch rule.
Rule: complete_conditional_table · Uses: G001, S028
S029 · premise_and_result_audit — This task computes the conditional table; it makes no estimator-quality claim and does not invoke factorization.S028 · premise_and_result_audit — Recompute conditional total (b2,1) with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute reconstruct P_b2(X=23) with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute P_b2(X=23|T=1) with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute reconstruct P_b2(X=21) with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute P_b2(X=21|T=1) with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute P_b2(T=1) with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute conditional total (b2,0) with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute reconstruct P_b2(X=22) with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute P_b2(X=22|T=0) with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute reconstruct P_b2(X=20) with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute P_b2(X=20|T=0) with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute P_b2(T=0) with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute P_b2(Ω) with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute conditional total (a2,1) with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute reconstruct P_a2(X=23) with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute P_a2(X=23|T=1) with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute reconstruct P_a2(X=21) with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute P_a2(X=21|T=1) with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute P_a2(T=1) with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute conditional total (a2,0) with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute reconstruct P_a2(X=22) with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute P_a2(X=22|T=0) with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute reconstruct P_a2(X=20) with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute P_a2(X=20|T=0) with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute P_a2(T=0) with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute P_a2(Ω) with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — A fiber stays in the partition even when it has zero mass under a parameter.S001 · premise_and_result_audit — T must have exactly one value at every sample point.([[a2, 0, 4/5, [[20, 1/2], [22, 1/2]]], [a2, 1, 1/5, [[21, 1/2], [23, 1/2]]], [b2, 0, 1/5, [[20, 1/2], [22, 1/2]]], [b2, 1, 4/5, [[21, 1/2], [23, 1/2]]]], zero_mass_cells_remain_undefined)
The full cellwise conditional laws are listed; zero-mass cells must retain null.
Review date: 2026-10-24
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
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