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.

ReLU and logistic sigmoid activation examples

Example

Assume Countable Choice. The rectified linear activation $σ (t) = \max {t, 0}$ and the logistic sigmoid $τ (t) = 1 / (1 + e^{- t})$ both satisfy the pointwise locally bounded activation hypotheses and are not polynomials almost everywhere. Under the corresponding single-hidden-layer universality theorem, each gives uniform approximation of continuous functions on every compact subset of $R^{d}$ , with no width bound asserted.

Example references

Leshno et al., Theorem 1: nonpolynomial activation universality

Goodfellow et al., Deep Learning, §6.4.1: common activation examples and scope

Facts & Assumptions

Given: $σ (t) = \max {t, 0}$ and $τ (t) = 1 / (1 + e^{- t})$ .

[F1]

The pointwise locally bounded subclass consists of activations in $M$ that are bounded at every point on each compact interval (Locally essentially bounded activations with null discontinuity closure).

[F2]

The universality theorem supplies pointwise compact-uniform density for pointwise locally bounded, nonpolynomial-almost-everywhere activations (Single-hidden-layer universal approximation).

[A1]

Countable Choice is the explicit premise of the cited universality theorem (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

1.1givenF1algebra

ReLU and logistic sigmoid are continuous, hence their discontinuity sets are empty; ReLU is bounded on each compact interval and sigmoid is globally bounded between zero and one. Thus both lie in the pointwise locally bounded activation class.

2.1step 1.1algebra

Neither is a polynomial almost everywhere. If a continuous function equals a polynomial almost everywhere on $R$ , their difference is continuous and must vanish everywhere, since any nonzero value would persist on an open interval of positive measure. ReLU vanishes on $(- \infty, 0)$ but is positive on $(0, \infty)$ ; the sigmoid is bounded and nonconstant. These properties exclude a polynomial identity for each.

3.1step 2.1F2A1∎

Applying the pointwise $M_{p t}$ branch of [F2] separately to each activation proves the stated compact-uniform approximation property. Both activations and their networks are continuous, so they are already Borel representatives when the theorem's $L^{p}$ clause is used. The theorem is an existence statement only and supplies no uniform width bound.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

31 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

Leshno, Lin, Pinkus, and Schocken, Multilayer Feedforward Networks With a Non-Polynomial Activation Function Can Approximate Any Function, Theorem 1 (standard reference, not scraped)
Ian Goodfellow, Yoshua Bengio, and Aaron Courville, Deep Learning, §6.4.1 (standard reference, not scraped)