ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

Continuum approximation theorem versus finite-element experiments

Example

Let $H = L^{2} (\partial Ω \times \partial Ω)$ and $Y = L^{2} (Ω)$ for a bounded smooth EIT domain. Given any nonempty finite sample set $S = {x_{1}, \dots, x_{N}} \subset H$ , any $x_{*} \in H ∖ S$ , and any $y_{*} \in Y ∖ {0}$ , there are continuous maps $F_{0}, F_{1} : H \to Y$ that agree at every sample input and differ by $y_{*}$ at $x_{*}$ . Therefore exact fit on finitely many inputs alone cannot identify an arbitrary continuous inverse map. A continuum FNO approximation theorem concerns the full $L^{2}$ kernel input and $L^{2}$ conductivity output on a compact set; a finite-element experiment reports a finite discretization and finite train/test sample. Neither finite-data fit nor a discretized operator matrix supplies the continuum theorem's uniform bound or a statistical generalization guarantee.

Example references

The continuum compact-set existence statement and its precise scope.

de Hoop et al., Theorem 4.8 and §5 numerical experiments

Facts & Assumptions

Given: a nonempty finite $S \subset H$ , a point $x_{*} \notin S$ , and $y_{*} \neq 0$ in $Y$ .

[F1]

The continuum theorem takes inputs in $L^{2} (\partial Ω \times \partial Ω)$ and outputs in $L^{2} (Ω)$ , and bounds the reconstruction error for every $γ \in X_{d} (R)$ and $η \in K_{δ}$ , where $K_{δ} = {η \in K : ∥ η ∥_{2} \leq δ}$ for compact $K$ . Step 1.1 of its proof shows that these inputs form the compact sum ${κ_{γ} : γ \in X_{d} (R)} + K_{δ}$ . The theorem asserts no training or finite-sample risk guarantee. (Noise-robust existence of generalized FNO approximants for EIT)

Proof

Given: the finite sample set $S$ , the unsampled point $x_{*}$ and $y_{*} \neq 0$ .

1.1givenconstruct

The finite set $S$ is closed in the Hilbert metric and $d (x_{*}, S) > 0$ . Define $φ (x) = d (x, S) / d (x_{*}, S)$ , $F_{0} (x) = 0$ and $F_{1} (x) = φ (x) y_{*}$ . Distance to a closed set is 1-Lipschitz, so both maps are continuous. For every $x_{i} \in S$ , $φ (x_{i}) = 0$ , hence $F_{0} (x_{i}) = F_{1} (x_{i}) = 0$ ; at $x_{*}$ , $F_{1} (x_{*}) - F_{0} (x_{*}) = y_{*}$ .

2.1F1step 1.1given∎

The construction shows that finite exact sample values do not determine an arbitrary continuous map on the ambient Hilbert input space. The distinction in the Example follows from the domains of the two claims: the continuum theorem quantifies over a compact subset of the full kernel $L^{2}$ space, whereas a finite-element experiment only evaluates finitely many discretized inputs and outputs. The finite-map construction is not asserted to belong to the generalized-FNO class or to contradict the continuum theorem.

Depends on

Noise-robust existence of generalized FNO approximants for EIT

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

Maarten V. de Hoop, Nikola B. Kovachki, Matti Lassas and Nicholas H. Nelsen, Extension and neural operator approximation of the electrical impedance tomography inverse map (standard reference, not scraped)