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.

Small regularization can amplify coefficient error

Example

Take $n = 1$ , $y = 1$ , original Gram matrix $K = [0]$ , positive-semidefinite approximation $\tilde{K} = [d]$ with fixed $d > 0$ , and regularization shift $τ > 0$ . Then $∥ K - \tilde{K} ∥_{2} = d$ , the coefficient vectors are $a_{K} = 1 / τ$ and $a_{\tilde{K}} = 1 / (d + τ)$ , and their absolute difference is $d / (τ (d + τ))$ , which diverges as $τ ↓ 0$ .

Example references

Regularized kernel prediction system: The coefficient vector solves $(G + τ I) a_{G} = y$ .

Kernel ridge predictions are stable under Gram-matrix perturbations: The general coefficient perturbation bound scales as $δ ∥ y ∥_{2} / τ^{2}$ .

Facts & Assumptions

Given: $n = 1$ , $y = 1$ , $K = [0]$ , $\tilde{K} = [d]$ with $d > 0$ , and $τ > 0$ .

[A1]

The scalar matrices and parameters are those in the Example.

[F1]

Regularized kernel prediction system: $a_{G} = (G + τ I)^{- 1} y$ and $u_{G} = G a_{G}$ .

[F2]

Kernel ridge predictions are stable under Gram-matrix perturbations: if both Gram matrices are positive semidefinite and differ in norm by $δ$ , then $∣ a_{K} - a_{\tilde{K}} ∣ \leq δ ∣ y ∣ / τ^{2}$ .

Verification

Given: The scalar data in [A1].

1.1givenA1F1algebra

Both $K$ and $\tilde{K}$ are positive semidefinite, and their spectral-norm difference is $d$ . Their shifted systems are the positive scalars $τ$ and $d + τ$ , so [F1] gives $a_{K} = 1 / τ$ and $a_{\tilde{K}} = 1 / (d + τ)$ .

2.1givenA1F2step 1.1algebra∎

Subtraction gives $∣ a_{K} - a_{\tilde{K}} ∣ = ∣ \frac{1}{τ} - \frac{1}{d + τ} ∣ = \frac{d}{τ (d + τ)}$ . For $0 < τ \leq d$ , this is at least $1 / (2 τ)$ , so it diverges as $τ ↓ 0$ . The general upper bound in [F2] is $d / τ^{2}$ ; it remains a valid scale-sensitive certificate and gives no uniform coefficient bound as $τ$ approaches zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, Chapter 6 §6.2.2 (standard reference, not scraped)