ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A data-dependent rule dominates a constant rule

Example

Let $Θ = {0, 1}$ and observe $X \in {0, 1}$ with $P_{θ} ({x}) = 1$ when $x = θ$ and zero otherwise. Under squared-error loss, the constant rule $d_{0} (x) = 0$ has risk $0$ at $θ = 0$ and $1$ at $θ = 1$ . The data rule $d_{1} (x) = x$ has risk zero at both parameters, so it dominates $d_{0}$ .

Facts & Assumptions

Given: The finite model, two decision rules, squared-error loss, and domination definition in the Example.

[F1]

A probability measure has total mass one (Probability measures and probability spaces).

[F2]

A statistic or estimator is a measurable data map (Statistics and estimators).

[F3]

Risk is the expected loss under the sampling law (Loss functions and risk).

[F4]

Domination requires weakly lower risk at every parameter and strict improvement at one parameter (Risk domination and admissibility).

[F5]

Finite expectations are probability-weighted sums (Expectation agrees with the published finite weighted sum).

Verification

technique · direct finite verification

1.1givenF1F2

For each $θ \in {0, 1}$ , let $P_{θ}$ be the point mass at $θ$ . In any disjoint sequence of events at most one event contains $θ$ , so this law is countably additive and has total mass one; it is a probability measure by [F1]. Both maps $d_{0} (x) = 0$ and $d_{1} (x) = x$ are measurable on the finite full-power-set spaces by [F2].

1.2F3F5algebra

Under squared-error loss, the only data value with positive $P_{θ}$ -mass is $x = θ$ . The finite-sum formula [F5] and risk definition [F3] therefore give $R (θ, d_{0}) = θ^{2} = θ$ and $R (θ, d_{1}) = (θ - θ)^{2} = 0$ .

2.1F3F4step 1.2∎

The risks satisfy $R (θ, d_{1}) \leq R (θ, d_{0})$ at both parameters, and the inequality is strict at $θ = 1$ . Thus $d_{1}$ dominates the constant rule $d_{0}$ by [F4].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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