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.

The observation theta squared is locally identifiable but globally aliased

Example

Let $Θ = R$ and $F (θ) = θ^{2}$ . The parameter $θ_{0} = 1$ is locally identifiable, but the full parameter map is not globally identifiable on $Θ$ .

Facts & Assumptions

Given: $F (θ) = θ^{2}$ on the open domain $R$ and the parameter $θ_{0} = 1$ .

[F1]

Finite nonlinear observation designs, local identifiability, and weighted information: local identifiability means injectivity on a neighborhood, while global identifiability means injectivity on the full parameter domain.

[F2]

Full column rank of the sensitivity Jacobian implies local identifiability: a C1 scalar observation map with nonzero derivative at $θ_{0}$ is locally identifiable there.

Verification

1.1givenF2algebra

The map is C1 and $F^{'} (1) = 2 \neq 0$ , so its one-dimensional Jacobian has full column rank. By [F2], $F$ is injective on some open neighborhood of $1$ , proving local identifiability.

2.1givenF1algebra∎

The distinct parameters $1$ and $- 1$ both lie in $Θ$ and satisfy $F (1) = 1 = F (- 1)$ . Hence $F$ is not injective on all of $Θ$ , so it is not globally identifiable by [F1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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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)