ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-07-31

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 radial homotopy is checked explicitly on punctured Euclidean space and on the unit sphere

Example

For $P = R^{n} ∖ {0}$ with $n \geq 1$ , the radial deformation retraction onto $S^{n - 1}$ is

$H (x, t) = ((1 - t) + \frac{t}{∥ x ∥_{2}}) x .$

At $t = 0$ it is the identity, at $t = 1$ it is radial normalisation, and every point of the unit sphere remains fixed.

Facts & Assumptions

Given: A natural $n \geq 1$ , a point $x \in P$ , a parameter $t \in I$ , and a point $s \in S^{n - 1}$ .

[L1]

The radial formula is continuous on $P \times I$ , is nonzero there, starts at $x$ , ends at $x / ∥ x ∥_{2}$ , and fixes norm-one vectors (For $n \geq 1$ , the map $H (x, t) = ((1 - t) + t / ∥ x ∥_{2}) x$ is continuous on $(R^{n} ∖ {0}) \times [0, 1]$ , starts at $x$ , ends at radial normalisation, fixes the unit sphere, and never reaches $0$ ).

[L2]

This map and radial normalisation form a deformation retraction of $P$ onto $S^{n - 1}$ (For $n \geq 1$ , radial normalisation is a deformation retraction of $R^{n} ∖ {0}$ onto $S^{n - 1}$ ).

Verification

technique · direct

1.1

Substituting $t = 0$ gives $H (x, 0) = x$ , and substituting $t = 1$ gives $H (x, 1) = x / ∥ x ∥_{2}$ .

algebra

1.2

If $s \in S^{n - 1}$ then $∥ s ∥_{2} = 1$ , so $H (s, t) = ((1 - t) + t) s = s$ for all $t \in I$ .

algebra

2.1

Continuity and avoidance of the origin are supplied by [L1]. Thus steps 1.1 and 1.2 explicitly verify the endpoint and fixed-sphere clauses of the deformation retraction [L2].

step 1.1step 1.2L1L2∎

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

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)
MAT 530 Topology lecture notes (Stony Brook University) (standard reference, not scraped)