Question 672: Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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P1000-0672 · q-20261015-advance-20261116-r05-finite-sufficient-statistic-0072

Original question and givens

Use complete theta labels=[a21, b21], samples=[210, 211, 212, 213], 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=[a21, b21], samples=[210, 211, 212, 213], 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).

Definition and scope

Conditional laws are defined only on positive-mass fibers; sufficiency needs every fiber to be parameter-invariant.

mko-finite-sufficient-statistic

Step-by-step derivation

S001 · implies

Before

Θ=[a21, b21]; Ω=[210, 211, 212, 213]

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.

Step identifiers and dependencies

Rule: full_parametric_model · Uses: G001

S002 · calculation

Before

P_a21(Ω) := sum(2/5, 1/10, 2/5, 1/10)

After

P_a21(Ω) = 1

Conditions: none

Check this law's complete total mass first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

P_b21(Ω) := sum(1/10, 2/5, 1/10, 2/5)

After

P_b21(Ω) = 1

Conditions: none

Check this law's complete total mass first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

sample=210

After

T(210)=0

Conditions: none

Every sample has exactly one statistic value.

Step identifiers and dependencies

Rule: total_statistic_map · Uses: G001, S003

S005 · implies

Before

sample=211

After

T(211)=1

Conditions: none

Every sample has exactly one statistic value.

Step identifiers and dependencies

Rule: total_statistic_map · Uses: G001, S004

S006 · implies

Before

sample=212

After

T(212)=0

Conditions: none

Every sample has exactly one statistic value.

Step identifiers and dependencies

Rule: total_statistic_map · Uses: G001, S005

S007 · implies

Before

sample=213

After

T(213)=1

Conditions: none

Every sample has exactly one statistic value.

Step identifiers and dependencies

Rule: total_statistic_map · Uses: G001, S006

S008 · implies

Before

[(210, 0), (211, 1), (212, 0), (213, 1)]

After

fibers=[[0, [210, 212]], [1, [211, 213]]]

Conditions: none

Grouping by value produces disjoint fibers whose union is the complete carrier.

Step identifiers and dependencies

Rule: fiber_partition · Uses: G001, S007

S009 · calculation

Before

P_a21(T=0) := sum(2/5, 2/5)

After

P_a21(T=0) = 4/5

Conditions: none

First compute the conditioning event's full mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

P_a21(210|T=0) := div(2/5, 4/5)

After

P_a21(210|T=0) = 1/2

Conditions: none

Divide the same law's atom mass by its positive fiber mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

reconstruct atom 210 under a21 := mul(1/2, 4/5)

After

reconstruct atom 210 under a21 = 2/5

Conditions: none

Multiply by fiber mass to check the original atom probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

P_a21(212|T=0) := div(2/5, 4/5)

After

P_a21(212|T=0) = 1/2

Conditions: none

Divide the same law's atom mass by its positive fiber mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

reconstruct atom 212 under a21 := mul(1/2, 4/5)

After

reconstruct atom 212 under a21 = 2/5

Conditions: none

Multiply by fiber mass to check the original atom probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

conditional total (a21,0) := sum(1/2, 1/2)

After

conditional total (a21,0) = 1

Conditions: none

The complete conditional law must have total mass1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

P_a21(T=1) := sum(1/10, 1/10)

After

P_a21(T=1) = 1/5

Conditions: none

First compute the conditioning event's full mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

P_a21(211|T=1) := div(1/10, 1/5)

After

P_a21(211|T=1) = 1/2

Conditions: none

Divide the same law's atom mass by its positive fiber mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

reconstruct atom 211 under a21 := mul(1/2, 1/5)

After

reconstruct atom 211 under a21 = 1/10

Conditions: none

Multiply by fiber mass to check the original atom probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

P_a21(213|T=1) := div(1/10, 1/5)

After

P_a21(213|T=1) = 1/2

Conditions: none

Divide the same law's atom mass by its positive fiber mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

reconstruct atom 213 under a21 := mul(1/2, 1/5)

After

reconstruct atom 213 under a21 = 1/10

Conditions: none

Multiply by fiber mass to check the original atom probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

conditional total (a21,1) := sum(1/2, 1/2)

After

conditional total (a21,1) = 1

Conditions: none

The complete conditional law must have total mass1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

P_b21(T=0) := sum(1/10, 1/10)

After

P_b21(T=0) = 1/5

Conditions: none

First compute the conditioning event's full mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

P_b21(210|T=0) := div(1/10, 1/5)

After

P_b21(210|T=0) = 1/2

Conditions: none

Divide the same law's atom mass by its positive fiber mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

reconstruct atom 210 under b21 := mul(1/2, 1/5)

After

reconstruct atom 210 under b21 = 1/10

Conditions: none

Multiply by fiber mass to check the original atom probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

P_b21(212|T=0) := div(1/10, 1/5)

After

P_b21(212|T=0) = 1/2

Conditions: none

Divide the same law's atom mass by its positive fiber mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

reconstruct atom 212 under b21 := mul(1/2, 1/5)

After

reconstruct atom 212 under b21 = 1/10

Conditions: none

Multiply by fiber mass to check the original atom probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

conditional total (b21,0) := sum(1/2, 1/2)

After

conditional total (b21,0) = 1

Conditions: none

The complete conditional law must have total mass1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

P_b21(T=1) := sum(2/5, 2/5)

After

P_b21(T=1) = 4/5

Conditions: none

First compute the conditioning event's full mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

P_b21(211|T=1) := div(2/5, 4/5)

After

P_b21(211|T=1) = 1/2

Conditions: none

Divide the same law's atom mass by its positive fiber mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

reconstruct atom 211 under b21 := mul(1/2, 4/5)

After

reconstruct atom 211 under b21 = 2/5

Conditions: none

Multiply by fiber mass to check the original atom probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

P_b21(213|T=1) := div(2/5, 4/5)

After

P_b21(213|T=1) = 1/2

Conditions: none

Divide the same law's atom mass by its positive fiber mass.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

reconstruct atom 213 under b21 := mul(1/2, 4/5)

After

reconstruct atom 213 under b21 = 2/5

Conditions: none

Multiply by fiber mass to check the original atom probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

conditional total (b21,1) := sum(1/2, 1/2)

After

conditional total (b21,1) = 1

Conditions: none

The complete conditional law must have total mass1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · implies

Before

valid positive-mass laws for T=0: [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]]

After

parameter-invariant: Yes

Conditions: none

Compare full defined conditional laws; parameter invariance requires every sample probability to agree.

Step identifiers and dependencies

Rule: conditional_law_comparison · Uses: G001, S032

S034 · implies

Before

valid positive-mass laws for T=1: [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]]

After

parameter-invariant: Yes

Conditions: none

Compare full defined conditional laws; parameter invariance requires every sample probability to agree.

Step identifiers and dependencies

Rule: conditional_law_comparison · Uses: G001, S033

S035 · implies

Before

[[0, [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 1/2]]]], Yes]]

After

all fibers invariant: Yes

Conditions: none

Sufficiency under this finite contract requires every fiber to pass, not just one.

Step identifiers and dependencies

Rule: finite_sufficiency_definition · Uses: G001, S034

Reverse checks

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 (b21,1) with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute reconstruct atom 213 under b21 with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute P_b21(213|T=1) with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute reconstruct atom 211 under b21 with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute P_b21(211|T=1) with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute P_b21(T=1) with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute conditional total (b21,0) with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute reconstruct atom 212 under b21 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute P_b21(212|T=0) with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute reconstruct atom 210 under b21 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute P_b21(210|T=0) with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute P_b21(T=0) with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute conditional total (a21,1) with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute reconstruct atom 213 under a21 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute P_a21(213|T=1) with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute reconstruct atom 211 under a21 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute P_a21(211|T=1) with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute P_a21(T=1) with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute conditional total (a21,0) with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute reconstruct atom 212 under a21 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute P_a21(212|T=0) with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute reconstruct atom 210 under a21 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute P_a21(210|T=0) with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute P_a21(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_b21(Ω) with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute P_a21(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.

Result

([[0, [[a21, [[210, 1/2], [212, 1/2]]], [b21, [[210, 1/2], [212, 1/2]]]], Yes], [1, [[a21, [[211, 1/2], [213, 1/2]]], [b21, [[211, 1/2], [213, 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.

This question review and source binding

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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input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: b358c3fb5206856bb9077a1cc067054d3c523990d9e833854f7aa96daa95fd22
candidate record SHA-256: ff392a56d42b9bb3724d74bd38a187b0e5a5ea69579bfa8827eece2f63463259
semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
Lean: NOT_RUN · human review: NOT_PERFORMED

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