ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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 geometric loops $t \mapsto (\cos 2 π n t, \sin 2 π n t)$ have degree $n$

Example

For every integer $n$ , the based geometric loop

$γ_{n} (t) = (\cos 2 π n t, \sin 2 π n t)$

has degree $n$ , where degree is transported from the quotient-circle model by the based homeomorphism.

Facts & Assumptions

Given: An integer $n$ and the quotient-circle dictionary.

[L1]

$[t] \mapsto (\cos 2 π t, \sin 2 π t)$ is a homeomorphism from $R / Z$ to the unit circle and sends $[0]$ to $(1, 0)$ ( $[t] \mapsto (\cos 2 π t, \sin 2 π t)$ is a homeomorphism from $R / Z$ to the unit circle).

[L2]

$\deg (ω_{n}) = n$ for every integer $n$ ( $\deg (ω_{n}) = n$ for every integer $n$ ).

[L3]

For every integer $n$ , $ω_{n} (t) = [n t]$ (The standard circle loops $ω_{n} (t) = [n t]$ for $n \in Z$ ).

Verification

technique · direct

1.1L1L3algebra

Under the homeomorphism of [L1], the loop in [L3] has image $t \mapsto (\cos (2 π n t), \sin (2 π n t)) = γ_{n} (t)$ .

2.1step 1.1L2∎

The quotient loop has degree $n$ by [L2], so the transported degree of its geometric image $γ_{n}$ is also $n$ . At $n = 0$ this is the constant loop, and the same computation covers every negative $n$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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

Allen Hatcher, Algebraic Topology, Ch. 1, Section 1.1 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Ch. 1, Section 5 (standard reference, not scraped)
Jonathan Wise, Math 6210 Lecture Notes, Week 3, Section 3.4 (standard reference, not scraped)

The geometric loops t↦(cos⁡2πnt,sin⁡2πnt)t↦(cos2πnt,sin2πnt) have degree nn