Question 676: Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Original question and givens

Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[3, 5], [1, 10], [1, 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. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

• G001 — Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[3, 5], [1, 10], [1, 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. Mode=nonsufficient. Give (mode, fiber_invariance_rows, sufficient, zero_mass_cells, nonsufficiency_witness_or_null, injective_statistic, constant_statistic, theta_laws_equal, constant_equivalence_if_applicable, factorization_or_quality_claim, boundary).

Definition and scope

One positive-mass fiber with differing conditional laws refutes sufficiency. An injective statistic is sufficient; a constant one exactly when all laws agree; zero-mass cells stay null.

• mko-finite-sufficient-statistic

Step-by-step derivation

Θ=[a0, b0]; Ω=[0, 1, 2, 3]

[[3/5, 1/10, 1/5, 1/10], [1/10, 2/5, 1/10, 2/5]]

P_a0(Ω) := sum(3/5, 1/10, 1/5, 1/10)

P_a0(Ω) = 1

P_b0(Ω) := sum(1/10, 2/5, 1/10, 2/5)

P_b0(Ω) = 1

sample=0

T(0)=0

sample=1

T(1)=1

sample=2

T(2)=0

sample=3

T(3)=1

[(0, 0), (1, 1), (2, 0), (3, 1)]

fibers=[[0, [0, 2]], [1, [1, 3]]]

P_a0(T=0) := sum(3/5, 1/5)

P_a0(T=0) = 4/5

P_a0(0|T=0) := div(3/5, 4/5)

P_a0(0|T=0) = 3/4

reconstruct atom 0 under a0 := mul(3/4, 4/5)

reconstruct atom 0 under a0 = 3/5

P_a0(2|T=0) := div(1/5, 4/5)

P_a0(2|T=0) = 1/4

reconstruct atom 2 under a0 := mul(1/4, 4/5)

reconstruct atom 2 under a0 = 1/5

conditional total (a0,0) := sum(3/4, 1/4)

conditional total (a0,0) = 1

P_a0(T=1) := sum(1/10, 1/10)

P_a0(T=1) = 1/5

P_a0(1|T=1) := div(1/10, 1/5)

P_a0(1|T=1) = 1/2

reconstruct atom 1 under a0 := mul(1/2, 1/5)

reconstruct atom 1 under a0 = 1/10

P_a0(3|T=1) := div(1/10, 1/5)

P_a0(3|T=1) = 1/2

reconstruct atom 3 under a0 := mul(1/2, 1/5)

reconstruct atom 3 under a0 = 1/10

conditional total (a0,1) := sum(1/2, 1/2)

conditional total (a0,1) = 1

P_b0(T=0) := sum(1/10, 1/10)

P_b0(T=0) = 1/5

P_b0(0|T=0) := div(1/10, 1/5)

P_b0(0|T=0) = 1/2

reconstruct atom 0 under b0 := mul(1/2, 1/5)

reconstruct atom 0 under b0 = 1/10

P_b0(2|T=0) := div(1/10, 1/5)

P_b0(2|T=0) = 1/2

reconstruct atom 2 under b0 := mul(1/2, 1/5)

reconstruct atom 2 under b0 = 1/10

conditional total (b0,0) := sum(1/2, 1/2)

conditional total (b0,0) = 1

P_b0(T=1) := sum(2/5, 2/5)

P_b0(T=1) = 4/5

P_b0(1|T=1) := div(2/5, 4/5)

P_b0(1|T=1) = 1/2

reconstruct atom 1 under b0 := mul(1/2, 4/5)

reconstruct atom 1 under b0 = 2/5

P_b0(3|T=1) := div(2/5, 4/5)

P_b0(3|T=1) = 1/2

reconstruct atom 3 under b0 := mul(1/2, 4/5)

reconstruct atom 3 under b0 = 2/5

conditional total (b0,1) := sum(1/2, 1/2)

conditional total (b0,1) = 1

valid positive-mass laws for T=0: [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]]

parameter-invariant: No

valid positive-mass laws for T=1: [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]]

parameter-invariant: Yes

[[0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]]

all fibers invariant: No

[0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No]

one positive-mass conditional-law disagreement refutes sufficiency

Reverse checks

• S036 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• 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 (b0,1) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute reconstruct atom 3 under b0 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute P_b0(3|T=1) with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute reconstruct atom 1 under b0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute P_b0(1|T=1) with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute P_b0(T=1) with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute conditional total (b0,0) with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute reconstruct atom 2 under b0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute P_b0(2|T=0) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute reconstruct atom 0 under b0 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute P_b0(0|T=0) with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute P_b0(T=0) with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute conditional total (a0,1) with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute reconstruct atom 3 under a0 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute P_a0(3|T=1) with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute reconstruct atom 1 under a0 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute P_a0(1|T=1) with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute P_a0(T=1) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute conditional total (a0,0) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute reconstruct atom 2 under a0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute P_a0(2|T=0) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute reconstruct atom 0 under a0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P_a0(0|T=0) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute P_a0(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_b0(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a0(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

(nonsufficient, [[0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], [1, [[a0, [[1, 1/2], [3, 1/2]]], [b0, [[1, 1/2], [3, 1/2]]]], Yes]], No, [], [0, [[a0, [[0, 3/4], [2, 1/4]]], [b0, [[0, 1/2], [2, 1/2]]]], No], No, No, No, Yes, No, identity_is_sufficient_constant_is_sufficient_exactly_when_theta_laws_match_no_factorization_or_estimator_quality_claim)

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.