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.

An uncorrelated control has no variance benefit

Example

Let $A$ and $B$ be independent fair signs, let $X = (A, B)$ , and set $f (A, B) = A$ , $h (A, B) = B$ , and $θ = 0$ . The optimal fixed control coefficient is zero, so this control does not reduce variance.

Facts & Assumptions

Given: The four equiprobable outcomes $(A, B) \in {- 1, 1}^{2}$ .

[F1]

Unbiasedness and optimal coefficients for fixed controls: The scalar optimum is covariance divided by control variance when that variance is positive.

[F2]

Control-variate estimators: The fixed-coefficient estimate subtracts $β (\overset{ˉ}{h} - θ)$ from $\overset{ˉ}{f}$ .

Verification

Given: $f = A$ , $h = B$ , $E A = E B = 0$ , and the four outcomes each have probability $1 / 4$ .

1.1givenalgebra

Summing the four table values gives $E f = E h = 0$ , $Var (h) = 1$ , $Var (f) = 1$ , and $Cov (f, h) = E [A B] = 0$ .

2.1F1F2step 1.1∎

By [F1], $β^{*} = 0$ and the minimum variance of one corrected draw equals $Var (f) = 1$ . Thus [F2] reduces to the ordinary sample mean, with variance $1 / n$ ; this control gives no reduction.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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