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P1000-0654 · q-20261015-advance-20261116-r05-finite-sufficient-statistic-0054
Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], 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 (fiber_invariance_rows, sufficient, certificate).
G001 — Use complete theta labels=[a3, b3], samples=[30, 31, 32, 33], 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 (fiber_invariance_rows, sufficient, certificate).Conditional laws are defined only on positive-mass fibers; sufficiency needs every fiber to be parameter-invariant.
Before
Θ=[a3, b3]; Ω=[30, 31, 32, 33]
After
[[2/5, 1/10, 2/5, 1/10], [1/10, 2/5, 1/10, 2/5]]
Conditions: none
Each row is one parameter law and each column an outcome; reorder probabilities together with labels.
Rule: full_parametric_model · Uses: G001
Before
P_a3(Ω) := sum(2/5, 1/10, 2/5, 1/10)
After
P_a3(Ω) = 1
Conditions: none
Check this law's complete total mass first.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
P_b3(Ω) := sum(1/10, 2/5, 1/10, 2/5)
After
P_b3(Ω) = 1
Conditions: none
Check this law's complete total mass first.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
sample=30
After
T(30)=0
Conditions: none
Every sample has exactly one statistic value.
Rule: total_statistic_map · Uses: G001, S003
Before
sample=31
After
T(31)=1
Conditions: none
Every sample has exactly one statistic value.
Rule: total_statistic_map · Uses: G001, S004
Before
sample=32
After
T(32)=0
Conditions: none
Every sample has exactly one statistic value.
Rule: total_statistic_map · Uses: G001, S005
Before
sample=33
After
T(33)=1
Conditions: none
Every sample has exactly one statistic value.
Rule: total_statistic_map · Uses: G001, S006
Before
[(30, 0), (31, 1), (32, 0), (33, 1)]
After
fibers=[[0, [30, 32]], [1, [31, 33]]]
Conditions: none
Grouping by value produces disjoint fibers whose union is the complete carrier.
Rule: fiber_partition · Uses: G001, S007
Before
P_a3(T=0) := sum(2/5, 2/5)
After
P_a3(T=0) = 4/5
Conditions: none
First compute the conditioning event's full mass.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
P_a3(30|T=0) := div(2/5, 4/5)
After
P_a3(30|T=0) = 1/2
Conditions: none
Divide the same law's atom mass by its positive fiber mass.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
reconstruct atom 30 under a3 := mul(1/2, 4/5)
After
reconstruct atom 30 under a3 = 2/5
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
P_a3(32|T=0) := div(2/5, 4/5)
After
P_a3(32|T=0) = 1/2
Conditions: none
Divide the same law's atom mass by its positive fiber mass.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
reconstruct atom 32 under a3 := mul(1/2, 4/5)
After
reconstruct atom 32 under a3 = 2/5
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
conditional total (a3,0) := sum(1/2, 1/2)
After
conditional total (a3,0) = 1
Conditions: none
The complete conditional law must have total mass1.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
P_a3(T=1) := sum(1/10, 1/10)
After
P_a3(T=1) = 1/5
Conditions: none
First compute the conditioning event's full mass.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
P_a3(31|T=1) := div(1/10, 1/5)
After
P_a3(31|T=1) = 1/2
Conditions: none
Divide the same law's atom mass by its positive fiber mass.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
reconstruct atom 31 under a3 := mul(1/2, 1/5)
After
reconstruct atom 31 under a3 = 1/10
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
P_a3(33|T=1) := div(1/10, 1/5)
After
P_a3(33|T=1) = 1/2
Conditions: none
Divide the same law's atom mass by its positive fiber mass.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
reconstruct atom 33 under a3 := mul(1/2, 1/5)
After
reconstruct atom 33 under a3 = 1/10
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
conditional total (a3,1) := sum(1/2, 1/2)
After
conditional total (a3,1) = 1
Conditions: none
The complete conditional law must have total mass1.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
P_b3(T=0) := sum(1/10, 1/10)
After
P_b3(T=0) = 1/5
Conditions: none
First compute the conditioning event's full mass.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
P_b3(30|T=0) := div(1/10, 1/5)
After
P_b3(30|T=0) = 1/2
Conditions: none
Divide the same law's atom mass by its positive fiber mass.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
reconstruct atom 30 under b3 := mul(1/2, 1/5)
After
reconstruct atom 30 under b3 = 1/10
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
P_b3(32|T=0) := div(1/10, 1/5)
After
P_b3(32|T=0) = 1/2
Conditions: none
Divide the same law's atom mass by its positive fiber mass.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
reconstruct atom 32 under b3 := mul(1/2, 1/5)
After
reconstruct atom 32 under b3 = 1/10
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
conditional total (b3,0) := sum(1/2, 1/2)
After
conditional total (b3,0) = 1
Conditions: none
The complete conditional law must have total mass1.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
P_b3(T=1) := sum(2/5, 2/5)
After
P_b3(T=1) = 4/5
Conditions: none
First compute the conditioning event's full mass.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
P_b3(31|T=1) := div(2/5, 4/5)
After
P_b3(31|T=1) = 1/2
Conditions: none
Divide the same law's atom mass by its positive fiber mass.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
reconstruct atom 31 under b3 := mul(1/2, 4/5)
After
reconstruct atom 31 under b3 = 2/5
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
P_b3(33|T=1) := div(2/5, 4/5)
After
P_b3(33|T=1) = 1/2
Conditions: none
Divide the same law's atom mass by its positive fiber mass.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
reconstruct atom 33 under b3 := mul(1/2, 4/5)
After
reconstruct atom 33 under b3 = 2/5
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
conditional total (b3,1) := sum(1/2, 1/2)
After
conditional total (b3,1) = 1
Conditions: none
The complete conditional law must have total mass1.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
valid positive-mass laws for T=0: [[a3, [[30, 1/2], [32, 1/2]]], [b3, [[30, 1/2], [32, 1/2]]]]
After
parameter-invariant: Yes
Conditions: none
Compare full defined conditional laws; parameter invariance requires every sample probability to agree.
Rule: conditional_law_comparison · Uses: G001, S032
Before
valid positive-mass laws for T=1: [[a3, [[31, 1/2], [33, 1/2]]], [b3, [[31, 1/2], [33, 1/2]]]]
After
parameter-invariant: Yes
Conditions: none
Compare full defined conditional laws; parameter invariance requires every sample probability to agree.
Rule: conditional_law_comparison · Uses: G001, S033
Before
[[0, [[a3, [[30, 1/2], [32, 1/2]]], [b3, [[30, 1/2], [32, 1/2]]]], Yes], [1, [[a3, [[31, 1/2], [33, 1/2]]], [b3, [[31, 1/2], [33, 1/2]]]], Yes]]
After
all fibers invariant: Yes
Conditions: none
Sufficiency under this finite contract requires every fiber to pass, not just one.
Rule: finite_sufficiency_definition · Uses: G001, S034
S035 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.S034 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.S033 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.S032 · premise_and_result_audit — Recompute conditional total (b3,1) with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute reconstruct atom 33 under b3 with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute P_b3(33|T=1) with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute reconstruct atom 31 under b3 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute P_b3(31|T=1) with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute P_b3(T=1) with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute conditional total (b3,0) with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute reconstruct atom 32 under b3 with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute P_b3(32|T=0) with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute reconstruct atom 30 under b3 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute P_b3(30|T=0) with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute P_b3(T=0) with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute conditional total (a3,1) with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute reconstruct atom 33 under a3 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute P_a3(33|T=1) with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute reconstruct atom 31 under a3 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute P_a3(31|T=1) with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute P_a3(T=1) with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute conditional total (a3,0) with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute reconstruct atom 32 under a3 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute P_a3(32|T=0) with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute reconstruct atom 30 under a3 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute P_a3(30|T=0) with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute P_a3(T=0) with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.S007 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.S006 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.S005 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.S004 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.S003 · premise_and_result_audit — Recompute P_b3(Ω) with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute P_a3(Ω) with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.([[0, [[a3, [[30, 1/2], [32, 1/2]]], [b3, [[30, 1/2], [32, 1/2]]]], Yes], [1, [[a3, [[31, 1/2], [33, 1/2]]], [b3, [[31, 1/2], [33, 1/2]]]], Yes]], Yes, conditional_law_must_be_theta_independent_on_every_positive_mass_cell)
The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.
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.
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