TheoremStatement: Literature-sourcedProof: 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.

Brouwer fixed point theorem

Statement

Every continuous map $f : D^{n} \to D^{n}$ of the closed unit ball has a fixed point, for every integer $n \geq 0$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

For every integer $n \geq 1$ , there is no continuous retraction $D^{n} \to S^{n - 1}$ of the closed unit ball onto its boundary. (No retraction from a disk onto its boundary)

[F2]

For $n \geq 1$ , a continuous fixed-point-free map $f : D^{n} \to D^{n}$ would produce a continuous retraction $r : D^{n} \to S^{n - 1}$ by following the ray from $f (x)$ through $x$ to the boundary. (A fixed point free ball map produces a boundary retraction)

Proof

1.1

When $n = 0$ , $D^{0}$ is a singleton, and its unique point is fixed by every self-map.

given

2.1

For $n \geq 1$ , a fixed-point-free map would give a continuous boundary retraction by F2. F1 excludes precisely such a retraction in every positive dimension. Therefore a fixed point exists.

F1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Miller, Algebraic Topology I lecture notes, Theorem 10.7 and proof, p.24 (standard reference, not scraped)