Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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 lies outside the analytic activation assumptions

Example

Let σ(t)=max⁡{0,t} be the ReLU activation. It is not differentiable at 0, so it is not analytic and does not satisfy the smooth-activation assumptions of the two finite-width convergence theorems.

Facts & Assumptions

Given: The ReLU function and the activation hypotheses used for the stated finite-width theorems.

[F1]

The theorems assume an analytic activation with a Lipschitz derivative. Fully connected and residual networks with the Du Gram kernel

Verification

Given: σ(t)=max⁡{0,t}.

1.1givenalgebra

For t<0, σ(t)=0, so its left derivative at 0 is 0; for t>0, σ(t)=t, so its right derivative at 0 is 1. The two one-sided derivatives differ, and the derivative at 0 does not exist.

2.1F1step 1.1∎

Every analytic real function is differentiable at every point of its domain. Since ReLU is not differentiable at 0, it is not analytic; it also fails the Lipschitz-derivative condition in [F1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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