ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

A zero Jacobian does not prevent local identifiability of the cubic map

Example

Let $Θ = R$ and $F (θ) = θ^{3}$ . At $θ_{0} = 0$ , the Jacobian has rank zero, yet the observation map is injective on every neighborhood of $0$ .

Facts & Assumptions

Given: The map $F (θ) = θ^{3}$ on $Θ = R$ , with $θ_{0} = 0$ .

[F1]

Finite nonlinear observation designs, local identifiability, and weighted information: local identifiability at $θ_{0}$ means injectivity on some open neighborhood; the scalar Jacobian is $F^{'} (θ_{0})$ .

Verification

1.1givenalgebra

Differentiation gives $F^{'} (θ) = 3 θ^{2}$ , so $F^{'} (0) = 0$ and the Jacobian has rank $0 < 1$ .

2.1givenF1algebra∎

If $u^{3} = v^{3}$ , then $(u - v) (u^{2} + u v + v^{2}) = 0$ . The second factor is $(u + v / 2)^{2} + 3 v^{2} / 4$ , which is nonnegative and vanishes only when $u = v = 0$ . Hence either $u = v$ directly or the second factor vanishes and again $u = v = 0$ . Thus $F$ is injective on $R$ , in particular on a neighborhood of $0$ , so local identifiability holds despite the rank-deficient Jacobian.

Depends on

Finite nonlinear observation designs, local identifiability, and weighted information

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Leah R. McLean et al., Estimability Analysis and Optimal Design in Dynamic Multi-scale Models of Cardiac Electrophysiology, §2.1 (standard reference, not scraped)