ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

The hyperbolic plane is hyperbolic

Example

The hyperbolic plane $H^{2}$ is a hyperbolic geodesic metric space.

Facts & Assumptions

Given: The standard geodesic metric on $H^{2}$ .

[A1]

Classical hyperbolic geometry gives a uniform slimness constant for geodesic triangles in $H^{2}$ .

[L1]

A geodesic metric space is hyperbolic exactly when all geodesic triangles are $δ$ -slim for some $δ \geq 0$ (Delta-slim triangles and hyperbolic spaces).

Verification

technique · direct

1.1

By [A1], geodesic triangles in $H^{2}$ are uniformly slim.

givenA1

2.1

Therefore [L1] shows that $H^{2}$ is hyperbolic.

L1step 1.1∎

Depends on

Delta-slim triangles and hyperbolic spaces

Used by

Closed surface groups are hyperbolic Example

Dependency tree · two levels

2 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

Clara Löh, Geometric Group Theory, Section 6.2.1 (standard reference, not scraped)