Question 655: Sufficiency Requires Theta Invariance in Every Positive-Mass Fiber

Original question and givens

Use complete theta labels=[a4, b4], samples=[40, 41, 42, 43], 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=[a4, b4], samples=[40, 41, 42, 43], 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

Θ=[a4, b4]; Ω=[40, 41, 42, 43]

[[2/5, 1/10, 2/5, 1/10], [1/10, 2/5, 1/10, 2/5]]

P_a4(Ω) := sum(2/5, 1/10, 2/5, 1/10)

P_a4(Ω) = 1

P_b4(Ω) := sum(1/10, 2/5, 1/10, 2/5)

P_b4(Ω) = 1

sample=40

T(40)=0

sample=41

T(41)=1

sample=42

T(42)=0

sample=43

T(43)=1

[(40, 0), (41, 1), (42, 0), (43, 1)]

fibers=[[0, [40, 42]], [1, [41, 43]]]

P_a4(T=0) := sum(2/5, 2/5)

P_a4(T=0) = 4/5

P_a4(40|T=0) := div(2/5, 4/5)

P_a4(40|T=0) = 1/2

reconstruct atom 40 under a4 := mul(1/2, 4/5)

reconstruct atom 40 under a4 = 2/5

P_a4(42|T=0) := div(2/5, 4/5)

P_a4(42|T=0) = 1/2

reconstruct atom 42 under a4 := mul(1/2, 4/5)

reconstruct atom 42 under a4 = 2/5

conditional total (a4,0) := sum(1/2, 1/2)

conditional total (a4,0) = 1

P_a4(T=1) := sum(1/10, 1/10)

P_a4(T=1) = 1/5

P_a4(41|T=1) := div(1/10, 1/5)

P_a4(41|T=1) = 1/2

reconstruct atom 41 under a4 := mul(1/2, 1/5)

reconstruct atom 41 under a4 = 1/10

P_a4(43|T=1) := div(1/10, 1/5)

P_a4(43|T=1) = 1/2

reconstruct atom 43 under a4 := mul(1/2, 1/5)

reconstruct atom 43 under a4 = 1/10

conditional total (a4,1) := sum(1/2, 1/2)

conditional total (a4,1) = 1

P_b4(T=0) := sum(1/10, 1/10)

P_b4(T=0) = 1/5

P_b4(40|T=0) := div(1/10, 1/5)

P_b4(40|T=0) = 1/2

reconstruct atom 40 under b4 := mul(1/2, 1/5)

reconstruct atom 40 under b4 = 1/10

P_b4(42|T=0) := div(1/10, 1/5)

P_b4(42|T=0) = 1/2

reconstruct atom 42 under b4 := mul(1/2, 1/5)

reconstruct atom 42 under b4 = 1/10

conditional total (b4,0) := sum(1/2, 1/2)

conditional total (b4,0) = 1

P_b4(T=1) := sum(2/5, 2/5)

P_b4(T=1) = 4/5

P_b4(41|T=1) := div(2/5, 4/5)

P_b4(41|T=1) = 1/2

reconstruct atom 41 under b4 := mul(1/2, 4/5)

reconstruct atom 41 under b4 = 2/5

P_b4(43|T=1) := div(2/5, 4/5)

P_b4(43|T=1) = 1/2

reconstruct atom 43 under b4 := mul(1/2, 4/5)

reconstruct atom 43 under b4 = 2/5

conditional total (b4,1) := sum(1/2, 1/2)

conditional total (b4,1) = 1

valid positive-mass laws for T=0: [[a4, [[40, 1/2], [42, 1/2]]], [b4, [[40, 1/2], [42, 1/2]]]]

parameter-invariant: Yes

valid positive-mass laws for T=1: [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]]

parameter-invariant: Yes

[[0, [[a4, [[40, 1/2], [42, 1/2]]], [b4, [[40, 1/2], [42, 1/2]]]], Yes], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 1/2]]]], Yes]]

all fibers invariant: Yes

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 (b4,1) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute reconstruct atom 43 under b4 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute P_b4(43|T=1) with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute reconstruct atom 41 under b4 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute P_b4(41|T=1) with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute P_b4(T=1) with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute conditional total (b4,0) with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute reconstruct atom 42 under b4 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute P_b4(42|T=0) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute reconstruct atom 40 under b4 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute P_b4(40|T=0) with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute P_b4(T=0) with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute conditional total (a4,1) with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute reconstruct atom 43 under a4 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute P_a4(43|T=1) with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute reconstruct atom 41 under a4 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute P_a4(41|T=1) with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute P_a4(T=1) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute conditional total (a4,0) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute reconstruct atom 42 under a4 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute P_a4(42|T=0) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute reconstruct atom 40 under a4 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P_a4(40|T=0) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute P_a4(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_b4(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a4(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, [[a4, [[40, 1/2], [42, 1/2]]], [b4, [[40, 1/2], [42, 1/2]]]], Yes], [1, [[a4, [[41, 1/2], [43, 1/2]]], [b4, [[41, 1/2], [43, 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.