ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

A quadratic remainder at L2 radius n^(-1/4)

Example

Let $Ψ (f) = ∥ f ∥_{2}^{2}$ at $f_{0} = 1$ . The derivative is nonzero, but the quadratic remainder is not negligible at perturbations of size $n^{- 1 / 4}$ : for any unit vector $v \in L^{2}$ and integers $n \geq 1$ , $h_{n} = n^{- 1 / 4} v$ has $\sqrt{n}$ -scaled remainder equal to $1$ . This sequence does not satisfy the stronger radius premise in Nonlinear-functional BvM limit under fast L2 contraction.

Facts & Assumptions

Given: $Ψ (f) = ⟨ f, f ⟩_{L^{2}}$ , $f_{0} = 1$ , and $∥ v ∥_{2} = 1$ .

[F1]

Nonlinear-functional BvM limit under fast L2 contraction: its Taylor remainder must be uniformly quadratic near $f_{0}$ , and its posterior $L^{2}$ radius must be $o (n^{- 1 / 2})$ for the stated BvM conclusion.

Verification

Given: The functional and unit vector in the Example.

1.1givenalgebra

Expanding the inner product gives $Ψ (f_{0} + h) = ∥ f_{0} ∥_{2}^{2} + 2 ⟨ f_{0}, h ⟩_{L^{2}} + ∥ h ∥_{2}^{2}$ . Thus the derivative is $D Ψ_{f_{0}} [h] = 2 ⟨ 1, h ⟩_{L^{2}}$ , which is nonzero, and the remainder equals $∥ h ∥_{2}^{2}$ .

2.1F1givenstep 1.1algebra∎

For integers $n \geq 1$ and $h_{n} = n^{- 1 / 4} v$ , $∥ h_{n} ∥_{2} = n^{- 1 / 4}$ and $\sqrt{n} ∥ h_{n} ∥_{2}^{2} = 1$ . Hence the scaled quadratic remainder does not tend to zero along this sequence, so the linear approximation needed in [F1] is not justified at this radius.

Depends on

Nonlinear-functional BvM limit under fast L2 contraction

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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