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 planar segment has Hausdorff measure equal to length

Example

Assume the Axiom of Countable Choice. For $p, q \in R^{2}$ , the segment $[p, q] = {(1 - t) p + t q : 0 \leq t \leq 1}$ satisfies

$H^{1} ([p, q]) = ∣ p - q ∣ .$

Its dimension is one when $p \neq q$ and zero when $p = q$ .

Facts & Assumptions

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

[F1]

Under the standing Countable Choice hypothesis, isometries preserve Hausdorff measure, and ambient and subspace outer values agree. Similarities scale Hausdorff measure exactly

[F2]

Under the standing Countable Choice hypothesis, hausdorff one-measure on the line equals Lebesgue outer measure. One-dimensional Hausdorff measure on the line is Lebesgue outer measure

[F3]

Finite positive measure at exponent one gives dimension one. Hausdorff dimension is the unique critical exponent

[F4]

Under the standing Countable Choice hypothesis, every at most countable set has dimension zero. Hausdorff dimension is monotone and countably stable

Verification

1.1

If $ℓ = ∣ q - p ∣ > 0$ , the map $u \mapsto p + u (q - p) / ℓ$ is an isometry from $[0, ℓ]$ onto $[p, q]$ , since the distance between its images is $∣ u - v ∣$ . Hence the segment has $H^{1}$ equal to the interval length $ℓ$ .

F1F2

2.1

For $ℓ > 0$ this value is finite and positive, so the dimension is one. If $ℓ = 0$ , the segment is the singleton ${p}$ : its own singleton cover costs zero at exponent one and it is countable, giving dimension zero. Both closed endpoints are present in the parametrisation.

F3F4step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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; Falconer §1.4 p.12 (standard reference, not scraped)