Question 687: Nonsufficiency, Zero Mass, Identity and Constant Statistics Need Separate Checks

Original question and givens

Use complete theta labels=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. 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=[a11, b11], samples=[110, 111, 112, 113], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 0, 0, 0]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Mode=constant. 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=[a11, b11]; Ω=[110, 111, 112, 113]

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

P_a11(Ω) := sum(2/5, 1/10, 2/5, 1/10)

P_a11(Ω) = 1

P_b11(Ω) := sum(1/10, 2/5, 1/10, 2/5)

P_b11(Ω) = 1

sample=110

T(110)=0

sample=111

T(111)=0

sample=112

T(112)=0

sample=113

T(113)=0

[(110, 0), (111, 0), (112, 0), (113, 0)]

fibers=[[0, [110, 111, 112, 113]]]

P_a11(T=0) := sum(2/5, 1/10, 2/5, 1/10)

P_a11(T=0) = 1

P_a11(110|T=0) := div(2/5, 1)

P_a11(110|T=0) = 2/5

reconstruct atom 110 under a11 := mul(2/5, 1)

reconstruct atom 110 under a11 = 2/5

P_a11(111|T=0) := div(1/10, 1)

P_a11(111|T=0) = 1/10

reconstruct atom 111 under a11 := mul(1/10, 1)

reconstruct atom 111 under a11 = 1/10

P_a11(112|T=0) := div(2/5, 1)

P_a11(112|T=0) = 2/5

reconstruct atom 112 under a11 := mul(2/5, 1)

reconstruct atom 112 under a11 = 2/5

P_a11(113|T=0) := div(1/10, 1)

P_a11(113|T=0) = 1/10

reconstruct atom 113 under a11 := mul(1/10, 1)

reconstruct atom 113 under a11 = 1/10

conditional total (a11,0) := sum(2/5, 1/10, 2/5, 1/10)

conditional total (a11,0) = 1

P_b11(T=0) := sum(1/10, 2/5, 1/10, 2/5)

P_b11(T=0) = 1

P_b11(110|T=0) := div(1/10, 1)

P_b11(110|T=0) = 1/10

reconstruct atom 110 under b11 := mul(1/10, 1)

reconstruct atom 110 under b11 = 1/10

P_b11(111|T=0) := div(2/5, 1)

P_b11(111|T=0) = 2/5

reconstruct atom 111 under b11 := mul(2/5, 1)

reconstruct atom 111 under b11 = 2/5

P_b11(112|T=0) := div(1/10, 1)

P_b11(112|T=0) = 1/10

reconstruct atom 112 under b11 := mul(1/10, 1)

reconstruct atom 112 under b11 = 1/10

P_b11(113|T=0) := div(2/5, 1)

P_b11(113|T=0) = 2/5

reconstruct atom 113 under b11 := mul(2/5, 1)

reconstruct atom 113 under b11 = 2/5

conditional total (b11,0) := sum(1/10, 2/5, 1/10, 2/5)

conditional total (b11,0) = 1

valid positive-mass laws for T=0: [[a11, [[110, 2/5], [111, 1/10], [112, 2/5], [113, 1/10]]], [b11, [[110, 1/10], [111, 2/5], [112, 1/10], [113, 2/5]]]]

parameter-invariant: No

[[0, [[a11, [[110, 2/5], [111, 1/10], [112, 2/5], [113, 1/10]]], [b11, [[110, 1/10], [111, 2/5], [112, 1/10], [113, 2/5]]]], No]]

all fibers invariant: No

the only fiber is Ω with probability1

conditional sample law=P_theta; sufficient iff all rows agree

Reverse checks

• S031 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S030 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S029 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S028 · premise_and_result_audit — Recompute conditional total (b11,0) with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute reconstruct atom 113 under b11 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute P_b11(113|T=0) with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute reconstruct atom 112 under b11 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute P_b11(112|T=0) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute reconstruct atom 111 under b11 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute P_b11(111|T=0) with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute reconstruct atom 110 under b11 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute P_b11(110|T=0) with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute P_b11(T=0) with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute conditional total (a11,0) with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute reconstruct atom 113 under a11 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute P_a11(113|T=0) with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute reconstruct atom 112 under a11 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute P_a11(112|T=0) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute reconstruct atom 111 under a11 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute P_a11(111|T=0) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute reconstruct atom 110 under a11 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P_a11(110|T=0) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute P_a11(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_b11(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a11(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

(constant, [[0, [[a11, [[110, 2/5], [111, 1/10], [112, 2/5], [113, 1/10]]], [b11, [[110, 1/10], [111, 2/5], [112, 1/10], [113, 2/5]]]], No]], No, [], null, No, Yes, 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.