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.

Zero training residual and positive test error

Example

Take one fixed-design observation with $X = [1]$ , true fitted mean $μ = 0$ , and training response $Y = ε$ . Let $ε^{'}$ be an independent fresh response error with the same mean-zero, variance- $σ^{2}$ law, where $σ^{2} > 0$ . Least squares fits the one-dimensional column space exactly: its training residual is zero, its expected error for estimating the mean is $σ^{2}$ , and its expected squared error for predicting the fresh response is $2 σ^{2}$ .

Facts & Assumptions

Given: A one-point fixed design $X = [1]$ , $μ = 0$ , integrable centered square-integrable errors $ε, ε^{'}$ with variance $σ^{2} > 0$ , and independence of the training and fresh errors.

[F1]

The fitted vector is the orthogonal projection of the observed response onto the design column space (Fixed-design least squares as orthogonal projection).

[F2]

For a rank-one projection and centered variance- $σ^{2}$ noise, the expected squared fitted-value error is $σ^{2}$ (Exact squared-error risk of an isotropic projection fit).

[F3]

Variance is the expected square of the centered variable (Moments, variance, and covariance on a probability space).

[F4]

Independent integrable random variables have factorized product expectations (Independent random elements, Expectations factor over finite products of independent random variables).

[F5]

For a real estimator of a real target, mean squared error is the expected squared difference; more generally risk is expected loss (Bias and mean squared error, Loss functions and risk).

Verification

1.1F1given

Since $col (X) = R$ , [F1] gives $\hat{μ} = Y = ε$ . Therefore the training residual $∣ Y - \hat{μ} ∣^{2}$ is zero for every outcome.

1.2F2F5given

By [F2], the expected squared error for estimating the true mean is $E ∣ \hat{μ} - μ ∣^{2} = σ^{2}$ . This is the scalar MSE and squared-loss risk by [F5].

1.3F3F4givenalgebra

The test-response error is $Y^{'} - \hat{μ} = ε^{'} - ε$ . Expanding its square and using [F3] and [F4], $E ∣ ε^{'} - ε ∣^{2} = E [(ε^{'})^{2}] + E [ε^{2}] - 2 E [ε^{'}] E [ε] = 2 σ^{2}$ .

2.1F5step 1.1step 1.2step 1.3∎

The training residual, mean-estimation error, and fresh-response test error are respectively $0$ , $σ^{2}$ , and $2 σ^{2}$ in this model, so these quantities are distinct.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 2, §7 (standard reference, not scraped)
Sylvain Arlot, Model/estimator selection, Lecture on model selection, §1 (standard reference, not scraped)