ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Same target with adaptive and target-independent architectures

Example

Assume the Axiom of Choice. For $f (x) = x^{2} / 2$ on $[0, 1]$ , which belongs to the unit ball of $W^{1, \infty} ((0, 1))$ , the target-independent theorem supplies one architecture that works for the whole unit ball with size $O (ε^{- 1} \log (1 / ε))$ , while the adaptive theorem supplies an architecture chosen for this $f$ with size $O (1 / [ε \log (1 / ε)])$ and depth $6$ . The ratio of these two upper bounds is $O ((\log (1 / ε))^{- 2})$ as $ε ↓ 0$ .

Example references

Yarotsky, Theorems 1–2: target-independent and adaptive rates

Telgarsky, Chapter 5: depth and approximation context

Facts & Assumptions

Given: The Axiom of Choice, $f (x) = x^{2} / 2$ on $[0, 1]$ , and $0 < ε < 1 / 2$ .

[F1]

The $W^{1, \infty}$ norm is the maximum of the function and weak-derivative essential suprema (Integer-order Sobolev spaces and their norms).

[F2]

Every target in the unit $W^{1, \infty}$ ball has a target-independent architecture of size $O (ε^{- 1} (1 + \log (1 / ε)))$ (Target-independent ReLU rate on the unit Sobolev ball).

[F3]

Every target in the unit $W^{1, \infty}$ ball in one dimension has an adaptive depth-six architecture of size $O (1 / [ε \log (1 / ε)])$ (Target-dependent depth-six approximation of Lipschitz functions).

[A1]

The Axiom of Choice is assumed by both approximation theorems (The Axiom of Choice).

Verification

1.1givenF1algebra

The derivative of $f$ is $f^{'} (x) = x$ , so $∥ f ∥_{L^{\infty}} = 1 / 2$ and $∥ f^{'} ∥_{L^{\infty}} = 1$ . By [F1], $∥ f ∥_{W^{1, \infty}} = 1$ . Directly, $∣ x^{2} - y^{2} ∣ / 2 = ∣ x - y ∣ ∣ x + y ∣ / 2 \leq ∣ x - y ∣$ on $[0, 1]$ , so $f$ is unit-Lipschitz.

2.1step 1.1F2F3A1algebra∎

Under [A1], apply [F2] and [F3] to this same target. Writing $t = \log (1 / ε)$ , the target-independent size is bounded by $C_{1} ε^{- 1} (1 + t)$ and the adaptive size by $C_{2} / (ε t)$ . The quotient of these explicit upper-bound functions is $(C_{2} / C_{1}) / [t (1 + t)] = O (t^{- 2})$ , which tends to zero. Upper bounds alone give no ratio of the actual architecture sizes or of optimal sizes. This comparison makes no claim about optimization or training.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Dmitry Yarotsky, Error bounds for approximations with deep ReLU networks, Theorems 1–2 (standard reference, not scraped)
Matus Telgarsky, Deep learning theory lecture notes, Chapter 5 (standard reference, not scraped)