ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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 torus as a product smooth manifold

Example

The torus

$T^{2} : = S^{1} \times S^{1}$

is a smooth $2$ -manifold with the product smooth structure.

Facts & Assumptions

Given: The circle $S^{1}$ with its two-chart smooth atlas and the product $T^{2} = S^{1} \times S^{1}$ .

[F1]

The circle is a smooth $1$ -manifold (The circle from two stereographic charts).

[F2]

Products of smooth manifolds carry canonical product smooth structures (Products of smooth manifolds have a canonical product smooth structure).

Verification

technique · direct

1.1

By [F1], each factor $S^{1}$ is a smooth $1$ -manifold.

2.1

Applying [F2] to the two circle factors gives a canonical product smooth [F2, step 1.1] structure on $S^{1} \times S^{1}$ , making it a smooth $(1 + 1)$ -manifold. This is the torus $T^{2}$ .

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)
Rob van der Vorst, Introduction to differentiable manifolds, §1 (standard reference, not scraped)