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P1000-0652 · q-20261015-advance-20261116-r05-finite-sufficient-statistic-0052
Use complete theta labels=[a1, b1], samples=[10, 11, 12, 13], 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=[a1, b1], samples=[10, 11, 12, 13], 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=[a1, b1]; Ω=[10, 11, 12, 13]
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_a1(Ω) := sum(2/5, 1/10, 2/5, 1/10)
After
P_a1(Ω) = 1
Conditions: none
Check this law's complete total mass first.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
P_b1(Ω) := sum(1/10, 2/5, 1/10, 2/5)
After
P_b1(Ω) = 1
Conditions: none
Check this law's complete total mass first.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
sample=10
After
T(10)=0
Conditions: none
Every sample has exactly one statistic value.
Rule: total_statistic_map · Uses: G001, S003
Before
sample=11
After
T(11)=1
Conditions: none
Every sample has exactly one statistic value.
Rule: total_statistic_map · Uses: G001, S004
Before
sample=12
After
T(12)=0
Conditions: none
Every sample has exactly one statistic value.
Rule: total_statistic_map · Uses: G001, S005
Before
sample=13
After
T(13)=1
Conditions: none
Every sample has exactly one statistic value.
Rule: total_statistic_map · Uses: G001, S006
Before
[(10, 0), (11, 1), (12, 0), (13, 1)]
After
fibers=[[0, [10, 12]], [1, [11, 13]]]
Conditions: none
Grouping by value produces disjoint fibers whose union is the complete carrier.
Rule: fiber_partition · Uses: G001, S007
Before
P_a1(T=0) := sum(2/5, 2/5)
After
P_a1(T=0) = 4/5
Conditions: none
First compute the conditioning event's full mass.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
P_a1(10|T=0) := div(2/5, 4/5)
After
P_a1(10|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 10 under a1 := mul(1/2, 4/5)
After
reconstruct atom 10 under a1 = 2/5
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
P_a1(12|T=0) := div(2/5, 4/5)
After
P_a1(12|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 12 under a1 := mul(1/2, 4/5)
After
reconstruct atom 12 under a1 = 2/5
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
conditional total (a1,0) := sum(1/2, 1/2)
After
conditional total (a1,0) = 1
Conditions: none
The complete conditional law must have total mass1.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
P_a1(T=1) := sum(1/10, 1/10)
After
P_a1(T=1) = 1/5
Conditions: none
First compute the conditioning event's full mass.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
P_a1(11|T=1) := div(1/10, 1/5)
After
P_a1(11|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 11 under a1 := mul(1/2, 1/5)
After
reconstruct atom 11 under a1 = 1/10
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
P_a1(13|T=1) := div(1/10, 1/5)
After
P_a1(13|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 13 under a1 := mul(1/2, 1/5)
After
reconstruct atom 13 under a1 = 1/10
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
conditional total (a1,1) := sum(1/2, 1/2)
After
conditional total (a1,1) = 1
Conditions: none
The complete conditional law must have total mass1.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
P_b1(T=0) := sum(1/10, 1/10)
After
P_b1(T=0) = 1/5
Conditions: none
First compute the conditioning event's full mass.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
P_b1(10|T=0) := div(1/10, 1/5)
After
P_b1(10|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 10 under b1 := mul(1/2, 1/5)
After
reconstruct atom 10 under b1 = 1/10
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
P_b1(12|T=0) := div(1/10, 1/5)
After
P_b1(12|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 12 under b1 := mul(1/2, 1/5)
After
reconstruct atom 12 under b1 = 1/10
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
conditional total (b1,0) := sum(1/2, 1/2)
After
conditional total (b1,0) = 1
Conditions: none
The complete conditional law must have total mass1.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
P_b1(T=1) := sum(2/5, 2/5)
After
P_b1(T=1) = 4/5
Conditions: none
First compute the conditioning event's full mass.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
P_b1(11|T=1) := div(2/5, 4/5)
After
P_b1(11|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 11 under b1 := mul(1/2, 4/5)
After
reconstruct atom 11 under b1 = 2/5
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
P_b1(13|T=1) := div(2/5, 4/5)
After
P_b1(13|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 13 under b1 := mul(1/2, 4/5)
After
reconstruct atom 13 under b1 = 2/5
Conditions: none
Multiply by fiber mass to check the original atom probability.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
conditional total (b1,1) := sum(1/2, 1/2)
After
conditional total (b1,1) = 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: [[a1, [[10, 1/2], [12, 1/2]]], [b1, [[10, 1/2], [12, 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: [[a1, [[11, 1/2], [13, 1/2]]], [b1, [[11, 1/2], [13, 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, [[a1, [[10, 1/2], [12, 1/2]]], [b1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[a1, [[11, 1/2], [13, 1/2]]], [b1, [[11, 1/2], [13, 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 (b1,1) with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute reconstruct atom 13 under b1 with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute P_b1(13|T=1) with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute reconstruct atom 11 under b1 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute P_b1(11|T=1) with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute P_b1(T=1) with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute conditional total (b1,0) with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute reconstruct atom 12 under b1 with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute P_b1(12|T=0) with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute reconstruct atom 10 under b1 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute P_b1(10|T=0) with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute P_b1(T=0) with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute conditional total (a1,1) with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute reconstruct atom 13 under a1 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute P_a1(13|T=1) with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute reconstruct atom 11 under a1 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute P_a1(11|T=1) with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute P_a1(T=1) with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute conditional total (a1,0) with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute reconstruct atom 12 under a1 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute P_a1(12|T=0) with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute reconstruct atom 10 under a1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute P_a1(10|T=0) with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute P_a1(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_b1(Ω) with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute P_a1(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, [[a1, [[10, 1/2], [12, 1/2]]], [b1, [[10, 1/2], [12, 1/2]]]], Yes], [1, [[a1, [[11, 1/2], [13, 1/2]]], [b1, [[11, 1/2], [13, 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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