ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 $P=\mathbb R^n\setminus\{0\}$ with $n\ge1$ , the radial deformation retraction onto $S^{n-1}$ is

$H(x,t)=\left((1-t)+\frac{t}{\lVert x\rVert_2}\right)x.$

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 $n\ge1$ , 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 $x/\lVert x\rVert_2$ , and fixes norm-one vectors (For $n\ge1$ , the map $H(x,t)=((1-t)+t/\lVert x\rVert_2)x$ is continuous on $(\mathbb{R}^n\setminus\{0\})\times[0,1]$ , 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 $n\ge1$ , radial normalisation is a deformation retraction of $\mathbb{R}^n\setminus\{0\}$ onto $S^{n-1}$ ).

Verification

technique · direct

1.1

Substituting $t=0$ gives $H(x,0)=x$ , and substituting $t=1$ gives $H(x,1)=x/\lVert x\rVert_2$ .

algebra

1.2

If $s\in S^{n-1}$ then $\lVert s\rVert_2=1$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 67 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)