LemmaStatement: Literature-sourcedProof: 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.

The printed diameter-bounded activation class is empty

Statement

For each $q \geq 1$ , the class ${B A}_{q}$ in the displayed activation-class definition is empty.

Statement references

Kovachki et al., Definition 7: global bound indexed by $C \geq {diam}_{2} (K)$

Facts & Assumptions

Given: An integer $q \geq 1$ and the definition of ${B A}_{q}$ .

[F1]

Membership in ${B A}_{q}$ requires, for every compact $K \subset R^{q}$ , every $ε > 0$ , and every $C \geq {diam}_{2} (K)$ , a network satisfying both the identity-approximation and global-bound inequalities (Activation classes for neural-operator universality).

Proof

Proof strategy: direct.

1.1givenF1choose

Suppose $σ \in {B A}_{q}$ . Choose $x_{0} \in R^{q}$ with $x_{0} \neq 0$ , set $K = {x_{0}}$ , choose $C = 0$ , and choose $0 < ε < ∣ x_{0} ∣_{2}$ . Since a singleton has diameter zero, these choices satisfy the quantifiers in [F1].

2.1givenstep 1.1F1contradiction∎

The required network would satisfy $∣ f (x) ∣_{2} \leq 0$ for every $x \in R^{q}$ , hence $f (x_{0}) = 0$ . It would also satisfy $∣ f (x_{0}) - x_{0} ∣_{2} \leq ε$ , which becomes $∣ x_{0} ∣_{2} \leq ε$ , contrary to the choice of $ε$ . Therefore no $σ$ belongs to ${B A}_{q}$ .

Remark

The source's immediately adjacent assertion that ReLU belongs to this class cannot follow from the printed definition: the contradiction above applies to every activation. The global-output-bound branch of the source's Theorem 11 is therefore vacuous as printed. A change from diameter to a bound involving the location of $K$ would be a different definition and requires owner reconciliation.

Depends on

Activation classes for neural-operator universality

Used by

Neural operators approximate continuous maps uniformly on compacta Theorem

Dependency tree · two levels

2 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

Kovachki et al., Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs (standard reference, not scraped)