Question 688: Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Original question and givens

Use complete theta labels=[a12, b12], samples=[120, 121, 122, 123], 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=[a12, b12], samples=[120, 121, 122, 123], 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=[a12, b12]; Ω=[120, 121, 122, 123]

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

P_a12(Ω) := sum(3/5, 1/10, 1/5, 1/10)

P_a12(Ω) = 1

P_b12(Ω) := sum(1/10, 2/5, 1/10, 2/5)

P_b12(Ω) = 1

sample=120

T(120)=0

sample=121

T(121)=1

sample=122

T(122)=0

sample=123

T(123)=1

[(120, 0), (121, 1), (122, 0), (123, 1)]

fibers=[[0, [120, 122]], [1, [121, 123]]]

P_a12(T=0) := sum(3/5, 1/5)

P_a12(T=0) = 4/5

P_a12(120|T=0) := div(3/5, 4/5)

P_a12(120|T=0) = 3/4

reconstruct atom 120 under a12 := mul(3/4, 4/5)

reconstruct atom 120 under a12 = 3/5

P_a12(122|T=0) := div(1/5, 4/5)

P_a12(122|T=0) = 1/4

reconstruct atom 122 under a12 := mul(1/4, 4/5)

reconstruct atom 122 under a12 = 1/5

conditional total (a12,0) := sum(3/4, 1/4)

conditional total (a12,0) = 1

P_a12(T=1) := sum(1/10, 1/10)

P_a12(T=1) = 1/5

P_a12(121|T=1) := div(1/10, 1/5)

P_a12(121|T=1) = 1/2

reconstruct atom 121 under a12 := mul(1/2, 1/5)

reconstruct atom 121 under a12 = 1/10

P_a12(123|T=1) := div(1/10, 1/5)

P_a12(123|T=1) = 1/2

reconstruct atom 123 under a12 := mul(1/2, 1/5)

reconstruct atom 123 under a12 = 1/10

conditional total (a12,1) := sum(1/2, 1/2)

conditional total (a12,1) = 1

P_b12(T=0) := sum(1/10, 1/10)

P_b12(T=0) = 1/5

P_b12(120|T=0) := div(1/10, 1/5)

P_b12(120|T=0) = 1/2

reconstruct atom 120 under b12 := mul(1/2, 1/5)

reconstruct atom 120 under b12 = 1/10

P_b12(122|T=0) := div(1/10, 1/5)

P_b12(122|T=0) = 1/2

reconstruct atom 122 under b12 := mul(1/2, 1/5)

reconstruct atom 122 under b12 = 1/10

conditional total (b12,0) := sum(1/2, 1/2)

conditional total (b12,0) = 1

P_b12(T=1) := sum(2/5, 2/5)

P_b12(T=1) = 4/5

P_b12(121|T=1) := div(2/5, 4/5)

P_b12(121|T=1) = 1/2

reconstruct atom 121 under b12 := mul(1/2, 4/5)

reconstruct atom 121 under b12 = 2/5

P_b12(123|T=1) := div(2/5, 4/5)

P_b12(123|T=1) = 1/2

reconstruct atom 123 under b12 := mul(1/2, 4/5)

reconstruct atom 123 under b12 = 2/5

conditional total (b12,1) := sum(1/2, 1/2)

conditional total (b12,1) = 1

valid positive-mass laws for T=0: [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]]

parameter-invariant: No

valid positive-mass laws for T=1: [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]]

parameter-invariant: Yes

[[0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]]

all fibers invariant: No

[0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 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 (b12,1) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute reconstruct atom 123 under b12 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute P_b12(123|T=1) with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute reconstruct atom 121 under b12 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute P_b12(121|T=1) with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute P_b12(T=1) with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute conditional total (b12,0) with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute reconstruct atom 122 under b12 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute P_b12(122|T=0) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute reconstruct atom 120 under b12 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute P_b12(120|T=0) with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute P_b12(T=0) with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute conditional total (a12,1) with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute reconstruct atom 123 under a12 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute P_a12(123|T=1) with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute reconstruct atom 121 under a12 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute P_a12(121|T=1) with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute P_a12(T=1) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute conditional total (a12,0) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute reconstruct atom 122 under a12 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute P_a12(122|T=0) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute reconstruct atom 120 under a12 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P_a12(120|T=0) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute P_a12(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_b12(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a12(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, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 1/2]]]], No], [1, [[a12, [[121, 1/2], [123, 1/2]]], [b12, [[121, 1/2], [123, 1/2]]]], Yes]], No, [], [0, [[a12, [[120, 3/4], [122, 1/4]]], [b12, [[120, 1/2], [122, 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.