Question 698: Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Original question and givens

Use complete theta labels=[a22, b22], samples=[220, 221, 222, 223], 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, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. 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=[a22, b22], samples=[220, 221, 222, 223], 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, 2, 3]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=identity. 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=[a22, b22]; Ω=[220, 221, 222, 223]

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

P_a22(Ω) := sum(2/5, 1/10, 2/5, 1/10)

P_a22(Ω) = 1

P_b22(Ω) := sum(1/10, 2/5, 1/10, 2/5)

P_b22(Ω) = 1

sample=220

T(220)=0

sample=221

T(221)=1

sample=222

T(222)=2

sample=223

T(223)=3

[(220, 0), (221, 1), (222, 2), (223, 3)]

fibers=[[0, [220]], [1, [221]], [2, [222]], [3, [223]]]

P_a22(T=0) := sum(2/5)

P_a22(T=0) = 2/5

P_a22(220|T=0) := div(2/5, 2/5)

P_a22(220|T=0) = 1

reconstruct atom 220 under a22 := mul(1, 2/5)

reconstruct atom 220 under a22 = 2/5

conditional total (a22,0) := sum(1)

conditional total (a22,0) = 1

P_a22(T=1) := sum(1/10)

P_a22(T=1) = 1/10

P_a22(221|T=1) := div(1/10, 1/10)

P_a22(221|T=1) = 1

reconstruct atom 221 under a22 := mul(1, 1/10)

reconstruct atom 221 under a22 = 1/10

conditional total (a22,1) := sum(1)

conditional total (a22,1) = 1

P_a22(T=2) := sum(2/5)

P_a22(T=2) = 2/5

P_a22(222|T=2) := div(2/5, 2/5)

P_a22(222|T=2) = 1

reconstruct atom 222 under a22 := mul(1, 2/5)

reconstruct atom 222 under a22 = 2/5

conditional total (a22,2) := sum(1)

conditional total (a22,2) = 1

P_a22(T=3) := sum(1/10)

P_a22(T=3) = 1/10

P_a22(223|T=3) := div(1/10, 1/10)

P_a22(223|T=3) = 1

reconstruct atom 223 under a22 := mul(1, 1/10)

reconstruct atom 223 under a22 = 1/10

conditional total (a22,3) := sum(1)

conditional total (a22,3) = 1

P_b22(T=0) := sum(1/10)

P_b22(T=0) = 1/10

P_b22(220|T=0) := div(1/10, 1/10)

P_b22(220|T=0) = 1

reconstruct atom 220 under b22 := mul(1, 1/10)

reconstruct atom 220 under b22 = 1/10

conditional total (b22,0) := sum(1)

conditional total (b22,0) = 1

P_b22(T=1) := sum(2/5)

P_b22(T=1) = 2/5

P_b22(221|T=1) := div(2/5, 2/5)

P_b22(221|T=1) = 1

reconstruct atom 221 under b22 := mul(1, 2/5)

reconstruct atom 221 under b22 = 2/5

conditional total (b22,1) := sum(1)

conditional total (b22,1) = 1

P_b22(T=2) := sum(1/10)

P_b22(T=2) = 1/10

P_b22(222|T=2) := div(1/10, 1/10)

P_b22(222|T=2) = 1

reconstruct atom 222 under b22 := mul(1, 1/10)

reconstruct atom 222 under b22 = 1/10

conditional total (b22,2) := sum(1)

conditional total (b22,2) = 1

P_b22(T=3) := sum(2/5)

P_b22(T=3) = 2/5

P_b22(223|T=3) := div(2/5, 2/5)

P_b22(223|T=3) = 1

reconstruct atom 223 under b22 := mul(1, 2/5)

reconstruct atom 223 under b22 = 2/5

conditional total (b22,3) := sum(1)

conditional total (b22,3) = 1

valid positive-mass laws for T=0: [[a22, [[220, 1]]], [b22, [[220, 1]]]]

parameter-invariant: Yes

valid positive-mass laws for T=1: [[a22, [[221, 1]]], [b22, [[221, 1]]]]

parameter-invariant: Yes

valid positive-mass laws for T=2: [[a22, [[222, 1]]], [b22, [[222, 1]]]]

parameter-invariant: Yes

valid positive-mass laws for T=3: [[a22, [[223, 1]]], [b22, [[223, 1]]]]

parameter-invariant: Yes

[[0, [[a22, [[220, 1]]], [b22, [[220, 1]]]], Yes], [1, [[a22, [[221, 1]]], [b22, [[221, 1]]]], Yes], [2, [[a22, [[222, 1]]], [b22, [[222, 1]]]], Yes], [3, [[a22, [[223, 1]]], [b22, [[223, 1]]]], Yes]]

all fibers invariant: Yes

each fiber is one sample point

every valid conditional is the same point mass

Reverse checks

• S046 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S045 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S044 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S043 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S042 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S041 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S040 · premise_and_result_audit — Recompute conditional total (b22,3) with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute reconstruct atom 223 under b22 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute P_b22(223|T=3) with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute P_b22(T=3) with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute conditional total (b22,2) with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute reconstruct atom 222 under b22 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute P_b22(222|T=2) with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute P_b22(T=2) with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute conditional total (b22,1) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute reconstruct atom 221 under b22 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute P_b22(221|T=1) with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute P_b22(T=1) with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute conditional total (b22,0) with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute reconstruct atom 220 under b22 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute P_b22(220|T=0) with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute P_b22(T=0) with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute conditional total (a22,3) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute reconstruct atom 223 under a22 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute P_a22(223|T=3) with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute P_a22(T=3) with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute conditional total (a22,2) with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute reconstruct atom 222 under a22 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute P_a22(222|T=2) with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute P_a22(T=2) with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute conditional total (a22,1) with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute reconstruct atom 221 under a22 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute P_a22(221|T=1) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute P_a22(T=1) with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute conditional total (a22,0) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute reconstruct atom 220 under a22 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P_a22(220|T=0) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute P_a22(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_b22(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a22(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

(identity, [[0, [[a22, [[220, 1]]], [b22, [[220, 1]]]], Yes], [1, [[a22, [[221, 1]]], [b22, [[221, 1]]]], Yes], [2, [[a22, [[222, 1]]], [b22, [[222, 1]]]], Yes], [3, [[a22, [[223, 1]]], [b22, [[223, 1]]]], Yes]], Yes, [], null, Yes, 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.