ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

Empirical interpolation does not determine test predictions

Example

Take one training input $x = 0$ with label $y = 0$ , and one test input $x_{*} = 1$ . Define $g_{0} (x) = 0$ and $g_{1} (x) = x$ , and define the one-point empirical quadratic loss by $L_{S} (g) = \frac{1}{2} (g (0) - y)^{2}$ . Both predictors have empirical loss zero, while their test predictions are $g_{0} (1) = 0$ and $g_{1} (1) = 1$ . Therefore agreement with the training labels alone does not determine test predictions.

Example references

Du et al., empirical training objective, equation (1)

Toronto, Theory of Deep Learning, Lecture 6, slides 31–43, separation of training interpolation and generalization questions

Empirical quadratic training loss

Facts & Assumptions

Given: The single training example, test input, two predictors, and empirical quadratic loss.

[F1]

The empirical objective is the sum of squared training residuals and does not include a test input absent from the training sample. Fully connected and residual networks with the Du Gram kernel

Verification

Given: $x = 0$ , $y = 0$ , $x_{*} = 1$ , $g_{0} (x) = 0$ , and $g_{1} (x) = x$ .

1.1F1givenalgebra

At the training point, $g_{0} (0) = g_{1} (0) = 0 = y$ , so both one-point empirical losses $\frac{1}{2} (g_{j} (0) - y)^{2}$ are zero.

2.1givenstep 1.1algebra∎

At the test point, $g_{0} (1) = 0$ and $g_{1} (1) = 1$ , so the two predictors agree on the entire training sample but have different test predictions. This verifies the stated logical limitation of empirical interpolation.

Depends on

Fully connected and residual networks with the Du Gram kernel

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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.

Sources

Du et al., empirical training objective, equation (1) (standard reference, not scraped)
University of Toronto, Theory of Deep Learning, Lecture 6, slides 31–43 (standard reference, not scraped)