CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 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.

For $n \geq 2$ , the sphere $S^{n - 1}$ is path-connected and connected

For $n \geq 2$ , the unit sphere $S^{n - 1} \subseteq R^{n}$ is path-connected and connected.

Given: $n \geq 2$ and points $x, y \in S^{n - 1}$ .

[L1]

[L2]

[L3]

technique · constructive

1.1L1L2construct

Choose a path $γ : [0, 1] \to R^{n} ∖ {0}$ from $x$ to $y$ as in [L1], and define $η : = ρ \circ γ$ , where $ρ (z) = z / ∥ z ∥_{2}$ .

2.1L2step 1.1

The map $η$ is continuous and takes values in $S^{n - 1}$ by [L2]. Since $ρ (x) = x$ and $ρ (y) = y$ , it joins $x$ to $y$ in the sphere.

Depends on