ExampleConstruction: Literature-sourcedVerification: 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.

Degree of the circle power map

Example

For every $d \in Z$ , the continuous map $p_{d} : S^{1} \to S^{1}$ , $p_{d} (z) = z^{d}$ , has degree $d$ when both circles have their counterclockwise orientations. This includes $d = 0$ and negative exponents.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

Let $f : S^{n} \to S^{n}$ be continuous, $n \geq 1$ , with source and target orientations fixed. If $f^{- 1} (y)$ is finite, then $\deg (f) = \sum_{x \in f^{- 1} (y)} \deg_{x} f .$ The sum over an empty fibre is $0$ . (Global sphere degree is the sum of local degrees)

[F2]

On $S^{n} \subset R^{n + 1}$ for $n \geq 1$ , the identity, a constant map, any single coordinate reflection, and the antipodal map have degrees $1$ , $0$ , $- 1$ , and $(- 1)^{n + 1}$ respectively. (Degree of identity constant reflection and antipodal sphere maps)

Verification

1.1F1F2construct

If $d > 0$ , the fibre of $1$ consists of the $d$ roots $e^{2 π i k / d}$ , $0 \leq k < d$ . In positively oriented angular coordinates centered at such a point and at $1$ , the local map is $t \mapsto d t$ . The homotopy $t \mapsto ((1 - s) d + s) t$ , $0 \leq s \leq 1$ , never sends a nonzero sufficiently small $t$ to zero, so it preserves the local pair and identifies the local action with that of the identity. Each local degree is therefore $+ 1$ , and the finite-fibre formula gives $\deg p_{d} = d$ .

2.1F1F2step 1.1∎

If $d = - m < 0$ , the same $m$ -point fibre has angular formula $t \mapsto - m t$ . Homotoping its positive factor $m$ to $1$ leaves the reflection $t \mapsto - t$ . Complex conjugation is a coordinate reflection on $S^{1}$ , of degree $- 1$ ; its singleton fibre and the local-sum formula give local sign $- 1$ . Thus $\deg p_{- m} = - m$ . For $d = 0$ , $p_{0} = 1$ is constant and has degree zero. In particular $p_{1}$ is the identity.

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

Hatcher, Algebraic Topology, Example 2.32, p.137 (standard reference, not scraped)