ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Duplicate inputs make the unquotiented NTK Gram matrix singular

Example

Let $X$ be a set, let $k : X \times X \to R$ be a symmetric positive-semidefinite kernel, fix $x \in X$ , and use two observations at that same input. Their Gram matrix is $K = κ (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ , where $κ = k (x, x) \geq 0$ . For labels $y = (1, - 1)^{T}$ and initial prediction $f_{0} = (0, 0)^{T}$ , consider the two-point empirical least-squares kernel flow

$\dot{f} (t) = - \frac{1}{2} K (f (t) - y), f (0) = f_{0} .$

The vector $v_{-} = (1, - 1)^{T} / \sqrt{2}$ is a zero-eigenvalue mode of $K$ , and the residual component in this mode remains unchanged, so the flow cannot reach $y$ . The unquotiented Gram matrix has rank at most one. In particular, assume AC, fix $n_{0} \geq 1$ , $L \geq 2$ , $β > 0$ , and a globally Lipschitz nonpolynomial activation $σ : R \to R$ , and use the width-normalized fully connected network with fan-in scaling $1 / \sqrt{n_{ℓ}}$ , bias scale $β$ , scalar output, and independent standard-normal initialization. For any $x \in S^{n_{0} - 1}$ , its sequential hidden-width limit kernel $K_{\infty}$ has $K_{\infty} (x, x) > 0$ by the sphere-positivity proposition, but the Gram matrix for two observations at $x$ is still singular. This does not conflict with that proposition, which asserts strict positivity only for finite lists of pairwise distinct sphere points.

Example references

Jacot, Gabriel, and Hongler, §5, finite-data kernel flow and zero modes

Repeated inputs give repeated rows and columns in a scalar kernel Gram matrix

Zero-eigenvalue residual modes stay constant

The sequential limiting NTK is strictly positive definite on distinct sphere inputs

Facts & Assumptions

Given: A set $X$ , a symmetric positive-semidefinite scalar kernel $k : X \times X \to R$ , and two identical input observations at $x \in X$ , so $κ = k (x, x) \geq 0$ ; labels $(1, - 1)^{T}$ ; initial prediction $(0, 0)^{T}$ ; and, for the sphere comparison, AC, $n_{0} \geq 1$ , $L \geq 2$ , $β > 0$ , a globally Lipschitz nonpolynomial activation, the stated width-normalized scalar-output network, and $x \in S^{n_{0} - 1}$ .

[F1]

A kernel Gram matrix on an input list has entries given by evaluating the kernel at each ordered input pair; in particular, repeating the input makes all four entries equal to $k (x, x)$ .

Verification

technique · compute the two-by-two Gram eigenvectors and inspect the inconsistent target component

1.1F1givenalgebra

Repeating the input makes every matrix entry equal to $κ$ , so $K = κ (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ . Multiplication gives $K v_{-} = 0$ for $v_{-} = (1, - 1)^{T} / \sqrt{2}$ . Thus $K$ has rank at most one and the target $y = (1, - 1)^{T} = \sqrt{2} v_{-}$ lies entirely in its nullspace.

2.1givenstep 1.1algebra∎

Set $r_{-} (t) = v_{-}^{T} (f (t) - y)$ . The flow equation, symmetry of $K$ , and $K v_{-} = 0$ give ${\dot{r}}_{-} (t) = - \frac{1}{2} v_{-}^{T} K (f (t) - y) = - \frac{1}{2} (K v_{-})^{T} (f (t) - y) = 0$ . Since $r_{-} (0) = - \sqrt{2}$ , this residual component stays nonzero for every $t \geq 0$ , so $f (t) \neq y$ at every finite time and the flow cannot converge to $y$ . This is consistent with the sphere proposition because its strictness conclusion requires pairwise distinct inputs.

Source notes

[F2] For the source-normalized limiting NTK flow, least-squares residual components in zero-eigenvalue modes remain constant (Least-squares NTK flow decays mode by mode and fits distinct sphere data); the more general two-point claim here is checked directly above.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Arthur Jacot, Franck Gabriel, and Clément Hongler, Neural Tangent Kernel: Convergence and Generalization in Neural Networks (standard reference, not scraped)