TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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 Euclidean constant-rank normal form

Statement

Let $k \geq 1$ , let $f : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ , and suppose $f$ has constant rank $r$ on a neighbourhood of $a$ . There are local $C^{k}$ coordinate diffeomorphisms $α$ at $a$ and $β$ at $f (a)$ , both sending the distinguished point to $0$ , such that $β \circ f \circ α^{- 1} (u, v) = (u, 0)$ for $(u, v) \in R^{r} \times R^{m - r}$ near $0$ . The final zero lies in $R^{n - r}$ ; every zero-dimensional block is omitted.

Facts & Assumptions

Given: The stated $C^{k}$ map, point, and constant rank $r$ .

[L1]

Source rank coordinates make $f \circ Φ^{- 1}$ equal to $(u, h (u))$ after shrinking, and a differentiable map with zero derivative on a connected open set is constant (A nonzero rank minor supplies the source coordinates for the constant-rank theorem, In source rank coordinates, the remaining components depend only on the rank coordinates, A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant).

[L2]

Finite sums, differences, componentwise maps, and composites of $C^{k}$ Euclidean maps are $C^{k}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

Proof

technique · direct

1.1givenL1

Translate the source and target distinguished points to $0$ and apply [L1], obtaining $g (u, v) = (u, h (u))$ on a product neighbourhood.

2.1step 1.1L2construct

Define the target shear $β (z, w) = (z, w - h (z))$ . It is a $C^{k}$ diffeomorphism with explicit inverse $(z, w) \mapsto (z, w + h (z))$ by [L2].

3.1step 1.1step 2.1L1

The composite satisfies $β (g (u, v)) = (u, 0)$ . When $r = 0$ , [L1] makes $f$ locally constant before the target translation; when $r = m$ or $r = n$ , the empty blocks make the same displayed formula literal.

4.1step 3.1∎

Taking $α$ to be the translated source coordinate map gives the asserted local normal form.

The Euclidean constant-rank normal form

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources