ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

A Lipschitz graph has finite Hausdorff length

Example

Assume the Axiom of Countable Choice. If $f : [0, 1] \to R$ is $L$ -Lipschitz, $0 \leq L < \infty$ , its graph $Γ = {(x, f (x)) : 0 \leq x \leq 1}$ satisfies

$1 \leq H^{1} (Γ) \leq \sqrt{1 + L^{2}}, \dim_{H} Γ = 1.$

Facts & Assumptions

Given: The objects, conventions, and hypotheses in the statement above.

[F1]

Under the standing Countable Choice hypothesis, an $M$ -Lipschitz map with $M > 0$ multiplies $H^{1}$ by at most $M$ . Lipschitz maps control Hausdorff measure

[F2]

Under the standing Countable Choice hypothesis, for every subset $A \subseteq R$ , $H^{1} (A) = λ_{1}^{*} (A)$ . One-dimensional Hausdorff measure on the line is Lebesgue outer measure

[F3]

Finite positive $H^{1}$ implies dimension one. Hausdorff dimension is the unique critical exponent

Verification

1.1F1F2

The graph map $g (x) = (x, f (x))$ satisfies $∣ g (x) - g (y) ∣^{2} = ∣ x - y ∣^{2} + ∣ f (x) - f (y) ∣^{2} \leq (1 + L^{2}) ∣ x - y ∣^{2}$ . Its Lipschitz constant is at most $\sqrt{1 + L^{2}}$ , which is positive even for $L = 0$ . Since the unit interval has Lebesgue length one, [F2] gives $H^{1} ([0, 1]) = 1$ and hence the upper measure bound.

2.1F1F2F3step 1.1∎

The coordinate projection $π : Γ \to [0, 1]$ is $1$ -Lipschitz and onto. Therefore $1 = H^{1} ([0, 1]) \leq H^{1} (Γ)$ . Both bounds show finite positive measure, and thus dimension one. For $L = 0$ both measure bounds equal one.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Fremlin 264G and 264Xf(i) (specialised Lipschitz graph) (standard reference, not scraped)