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.

Local-linear regression reproduces affine responses

Example

Assume AC as in Regression target and chosen version. Let $X, Y$ be real random variables with $E ∣ Y ∣ < \infty$ and $Y = a + b X$ almost surely. Choose the regular conditional law $Q (t, d y) = δ_{a + b t} (d y)$ for every $t \in R$ ; then the chosen regression-function version is $m (t) = a + b t$ at every point. For any observed sample and any $x, h$ whose degree-one local moment matrix is nonsingular, the local-linear estimate equals $a + b x$ , even when the design points are asymmetric around $x$ .

Facts & Assumptions

Given: The integrable affine-response model, its specified pointwise regression-function version, and a nonsingular degree-one local moment matrix.

[F1]

For degree one, the estimator is the intercept from the weighted least-squares coefficient vector $(Z^{T} W Z)^{- 1} Z^{T} W Y$ . Local-polynomial estimator.

[F2]

The chosen version $m (t)$ represents $E [Y ∣ X = t]$ in the almost-sure regression-target sense. Regression target and chosen version.

[A1]

AC is inherited only through the selected version in [F2]; the finite-dimensional calculation is choice-free.

[A2]

The specified conditional law $Q (t, d y) = δ_{a + b t} (d y)$ gives $m (t) = a + b t$ for every $t$ , not just almost surely under the law of $X$ .

Verification

Given: The data satisfy $Y_{i} = a + b X_{i}$ , the selected conditional law is as in the Example, and the sample moment matrix is nonsingular.

1.1givenF1A1algebra

With $u_{i} = (x - X_{i}) / h$ , the design row is $(1, u_{i})$ and $X_{i} = x - h u_{i}$ , so $Y_{i} = (a + b x) - b h u_{i}$ . Thus the response vector is exactly $Z γ$ for $γ = (a + b x, - b h)^{T}$ .

2.1givenF1F2A2step 1.1algebra∎

The weighted least-squares formula gives $\hat{β} = (Z^{T} W Z)^{- 1} Z^{T} W Z γ = γ$ . By the chosen-version convention in [F2], the specified conditional law in [A2] fixes $m (x) = a + b x$ at this evaluation point. Thus its intercept is $a + b x = m (x)$ , proving exact reproduction of that chosen version.

Depends on

Used by

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Sources

Whitney K. Newey, Locally Linear Regression, MIT course notes (standard reference, not scraped)
Ryan Tibshirani, Nonparametric Regression: Nearest Neighbors and Kernels (standard reference, not scraped)