Question 689: Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Original question and givens

Use complete theta labels=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. 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=[a13, b13], samples=[130, 131, 132, 133], aligned exact probability rows=[[[1, 2], 0, [1, 2], 0], [0, [1, 2], 0, [1, 2]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=zero_mass. 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=[a13, b13]; Ω=[130, 131, 132, 133]

[[1/2, 0, 1/2, 0], [0, 1/2, 0, 1/2]]

P_a13(Ω) := sum(1/2, 0, 1/2, 0)

P_a13(Ω) = 1

P_b13(Ω) := sum(0, 1/2, 0, 1/2)

P_b13(Ω) = 1

sample=130

T(130)=0

sample=131

T(131)=1

sample=132

T(132)=0

sample=133

T(133)=1

[(130, 0), (131, 1), (132, 0), (133, 1)]

fibers=[[0, [130, 132]], [1, [131, 133]]]

P_a13(T=0) := sum(1/2, 1/2)

P_a13(T=0) = 1

P_a13(130|T=0) := div(1/2, 1)

P_a13(130|T=0) = 1/2

reconstruct atom 130 under a13 := mul(1/2, 1)

reconstruct atom 130 under a13 = 1/2

P_a13(132|T=0) := div(1/2, 1)

P_a13(132|T=0) = 1/2

reconstruct atom 132 under a13 := mul(1/2, 1)

reconstruct atom 132 under a13 = 1/2

conditional total (a13,0) := sum(1/2, 1/2)

conditional total (a13,0) = 1

P_a13(T=1) := sum(0, 0)

P_a13(T=1) = 0

P_a13(T=1)=0

conditional=null (undefined)

P_b13(T=0) := sum(0, 0)

P_b13(T=0) = 0

P_b13(T=0)=0

conditional=null (undefined)

P_b13(T=1) := sum(1/2, 1/2)

P_b13(T=1) = 1

P_b13(131|T=1) := div(1/2, 1)

P_b13(131|T=1) = 1/2

reconstruct atom 131 under b13 := mul(1/2, 1)

reconstruct atom 131 under b13 = 1/2

P_b13(133|T=1) := div(1/2, 1)

P_b13(133|T=1) = 1/2

reconstruct atom 133 under b13 := mul(1/2, 1)

reconstruct atom 133 under b13 = 1/2

conditional total (b13,1) := sum(1/2, 1/2)

conditional total (b13,1) = 1

valid positive-mass laws for T=0: [[a13, [[130, 1/2], [132, 1/2]]]]

parameter-invariant: Yes

valid positive-mass laws for T=1: [[b13, [[131, 1/2], [133, 1/2]]]]

parameter-invariant: Yes

[[0, [[a13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[b13, [[131, 1/2], [133, 1/2]]]], Yes]]

all fibers invariant: Yes

[[a13, 1, 0, null], [b13, 0, 0, null]]

zero-mass cells do not supply a fabricated conditional law

Reverse checks

• S028 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S027 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S026 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S025 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S024 · premise_and_result_audit — Recompute conditional total (b13,1) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute reconstruct atom 133 under b13 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute P_b13(133|T=1) with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute reconstruct atom 131 under b13 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute P_b13(131|T=1) with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute P_b13(T=1) with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S017 · premise_and_result_audit — Recompute P_b13(T=0) with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S015 · premise_and_result_audit — Recompute P_a13(T=1) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute conditional total (a13,0) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute reconstruct atom 132 under a13 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute P_a13(132|T=0) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute reconstruct atom 130 under a13 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P_a13(130|T=0) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute P_a13(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_b13(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a13(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

(zero_mass, [[0, [[a13, [[130, 1/2], [132, 1/2]]]], Yes], [1, [[b13, [[131, 1/2], [133, 1/2]]]], Yes]], Yes, [[a13, 1, 0, null], [b13, 0, 0, null]], null, 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.