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 perfect-correlation control removes all variance

Example

Let $X_{i}$ be IID uniform on $[0, 1]$ , take $f (x) = h (x) = x$ , and use the known control mean $θ = 1 / 2$ . The fixed coefficient $β = 1$ gives an exact estimate with no Monte Carlo variance.

Facts & Assumptions

Given: The uniform law on $[0, 1]$ , $f = h = id$ , and the known mean $θ = 1 / 2$ .

[F1]

Unbiasedness and optimal coefficients for fixed controls: The scalar minimizing coefficient is $Cov (f (X), h (X)) / Var (h (X))$ when $Var (h (X)) > 0$ .

[F2]

Control-variate estimators: ${\hat{μ}}_{β} = n^{- 1} \sum_{i} [f (X_{i}) - β (h (X_{i}) - θ)]$ .

Verification

Given: The specified $f, h, θ$ and IID $X_{i} \sim Unif [0, 1]$ .

1.1F1algebra

Direct integration gives $E X = 1 / 2$ and $Var (X) = \int_{0}^{1} (x - 1 / 2)^{2} d x = 1 / 12 > 0$ , so [F1] gives $β^{*} = 1$ .

2.1F2step 1.1given∎

For every sample value, $X_{i} - (X_{i} - 1 / 2) = 1 / 2$ . By [F2], ${\hat{μ}}_{1} = 1 / 2$ identically, hence its variance is zero and its expectation is the target $1 / 2$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.