CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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 tangent space of an n-manifold has dimension n

If $M$ is a smooth $n$ -manifold and $p \in M$ , then $T_{p} M$ is an $n$ -dimensional real vector space.

Given: A smooth $n$ -manifold $M$ and a point $p \in M$ .

[L1]

In any smooth chart around $p$ , the coordinate derivations form a basis of $T_{p} M$ (Coordinate derivations form a basis of the tangent space).

technique · direct

1.1given

Choose a smooth chart around $p$ ; its coordinate basis has exactly $n$ vectors.

2.1L1step 1.1

By [L1], those $n$ coordinate derivations form a basis of $T_{p} M$ .

3.1step 2.1∎

Therefore $\dim T_{p} M = n$ .

Depends on