Alphabeta MathHow 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.
Rank at one point need not determine nearby rank
Statement
False claim: if a smooth map has rank at one point, then it has rank on some neighbourhood of that point.
Facts & Assumptions
Given: The smooth map , .
The derivative of is (For a natural the function is differentiable everywhere with derivative ; for it is the constant , with derivative ; for a natural the function is differentiable at every with derivative ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, A total derivative computes every directional derivative, and its matrix is the Jacobian).
Local constancy of rank requires a neighbourhood conclusion stronger than a single-point rank computation (A smooth map of locally maximal rank has locally constant rank).
Refutation
By [L1], , so the rank at is .
For every , the derivative is , so the rank at is . Thus every neighbourhood of contains points of rank .
Therefore the rank at does not persist locally, which refutes the claim and shows why the extra maximal-rank hypothesis in [L2] matters.
Depends on
- A smooth map of locally maximal rank has locally constant rank
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- A total derivative computes every directional derivative, and its matrix is the Jacobian
Used by
- A rank drop at one point need not persist locally Counterexample
Dependency tree · two levels
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Sources
- John M. Lee, Introduction to Smooth Manifolds, Maps of Constant Rank (standard reference, not scraped)