LemmaStatement: AI-adaptedProof: 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.

In source rank coordinates, the remaining components depend only on the rank coordinates

Statement

Assume the hypotheses of A nonzero rank minor supplies the source coordinates for the constant-rank theorem and that $f$ has constant rank $r$ near $a$ . In the source coordinates $y = Φ (x)$ , shrink to a rectangular neighbourhood $P \times Q \subseteq R^{r} \times R^{m - r}$ and write $g = f \circ Φ^{- 1}, g (u, v) = (u, h (u, v)) .$ Then $h (u, v)$ is independent of $v$ . When $r = 0$ , $g$ is locally constant; when $r = m$ , the $v$ block is empty; and when $r = n$ , the $h$ block is empty.

Facts & Assumptions

Given: A $C^{k}$ map of constant rank $r$ near $a$ and the local coordinates $Φ$ from the source-coordinate lemma.

[L1]

The source-coordinate map is a local $C^{k}$ diffeomorphism, the chain rule computes the derivative of $g = f \circ Φ^{- 1}$ , and a nonzero $(r + 1)$ -rowed minor forces rank at least $r + 1$ (A nonzero rank minor supplies the source coordinates for the constant-rank theorem, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, A matrix has rank at least $r$ exactly when it has a nonzero $r$ -rowed minor).

[L2]

A differentiable map with zero derivative on a nonempty connected open Euclidean set is constant (A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant).

Proof

technique · direct

1.1givenL1

Shrink the coordinate image to a product of open rectangles $P \times Q$ around $Φ (a)$ . By [L1], $g$ is $C^{k}$ , has constant rank $r$ , and its first $r$ components are the coordinates $u$ .

2.1step 1.1L1algebra

If $r < n$ and $r < m$ , any nonzero partial derivative $\partial h_{i} / \partial v_{j}$ would join the identity $r \times r$ block of $D g$ to form a nonzero $(r + 1)$ -rowed minor, contradicting constant rank $r$ by [L1]. Hence all derivatives of $v \mapsto h (u, v)$ vanish.

3.1step 2.1L2

For each fixed $u$ , the rectangle $Q$ is connected and [L2] makes $v \mapsto h (u, v)$ constant. If $r = 0$ , the same argument applies to all components on the connected rectangle; if $r = m$ or $r = n$ , the relevant block is empty and the conclusion is immediate.

4.1step 3.1∎

Thus, after shrinking, $g (u, v) = (u, h (u))$ in every rank regime.

In source rank coordinates, the remaining components depend only on the rank coordinates

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources