ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Parallel first-order decay observations depend only on the sum of two rates

Example

A species $A$ disappears by two parallel first-order routes with positive rates $k_{1}, k_{2}$ : its concentration obeys $A^{'} (t) = - (k_{1} + k_{2}) A (t)$ and $A (0) = A_{0} > 0$ . Only $A$ is observed. No collection of these observations separates $k_{1}$ from $k_{2}$ ; the stacked sensitivity Jacobian has rank at most one.

Facts & Assumptions

Given: A finite nonempty set $T$ of observation times in $[0, \infty)$ , $A_{0} > 0$ , parameters $k_{1}, k_{2} > 0$ , the scalar rate equation $A^{'} = - (k_{1} + k_{2}) A$ , $A (0) = A_{0}$ , and the observation map that records $A (t)$ for $t \in T$ .

[F1]

Finite nonlinear observation designs, local identifiability, and weighted information: local identifiability requires the joint observation map to be injective on a neighborhood; its stacked sensitivity Jacobian is formed by placing the observation Jacobian rows together.

[F2]

The weighted information matrix is positive definite exactly at full sensitivity rank: for positive design weights and positive observation precisions, the local information matrix is positive definite exactly when the stacked sensitivity Jacobian has trivial kernel.

[F3]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : if a real function is continuous on a closed interval and differentiable inside, its endpoint increment is its derivative at an interior point times the interval length.

Verification

1.1givenF1F3choosealgebra

Put $λ = k_{1} + k_{2}$ . The function $A (t) = A_{0} e^{- λ t}$ satisfies the rate equation and initial value. If $\tilde{A}$ is any other differentiable solution, then $(e^{λ t} \tilde{A} (t))^{'} = 0$ ; applying [F3] on every compact time interval shows this product is constant, and its value at $0$ is $A_{0}$ . Thus the observation map is $F (k_{1}, k_{2}) = (A_{0} e^{- (k_{1} + k_{2}) t})_{t \in T}$ . Now fix any neighborhood $U$ of $(k_{1}, k_{2})$ inside $(0, \infty)^{2}$ . Since $U$ is open, it contains a ball of some radius $r > 0$ . Choose $0 < ε < \min {k_{2}, r / 2}$ . Then $(k_{1} + ε, k_{2} - ε)$ is positive, distinct from $(k_{1}, k_{2})$ , and lies in that ball. The sums of the rates agree, so these two parameters have identical observations. Every neighborhood therefore contains a distinct parameter with identical observations; the model is not locally identifiable by [F1], and hence is not globally identifiable.

2.1givenstep 1.1F2algebra∎

For each $t \in T$ , differentiating the observation gives the row $- A_{0} t e^{- (k_{1} + k_{2}) t} (1, 1)$ . Every row is a scalar multiple of $(1, 1)$ , so the stacked Jacobian has rank at most one and has the nonzero kernel vector $(1, - 1)$ . By [F2], every supported design's information matrix is singular. This rank failure agrees with, but is weaker than, the exact aliases proved in step 1.1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Henrik Madsen, Optimal Experimental Design for Grey-box Models, Ch. 3 (standard reference, not scraped)
Leah R. McLean et al., Estimability Analysis and Optimal Design in Dynamic Multi-scale Models of Cardiac Electrophysiology, §2.1 (standard reference, not scraped)