Observability of complex systems via conserved quantities

The paper’s Theorem 4 claims that, given conserved quantities with an invertible ∂G/∂s and full-rank ∂G/∂r, there exists ĝ(r) making the system observable. Its proof invokes the implicit function theorem to assert a global s = ψ(r) (silently suppressing the dependence on the conserved value c) and then concludes injectivity of a differential embedding Φ̂ via a Jacobian product. This step is not justified: (i) the IFT only gives a local ψ on each invariant level set G = c; (ii) the paper equates observability with injectivity of a finite-order differential embedding without additional assumptions; and (iii) the rank argument cannot generally yield local injectivity unless extra dimensional/rank hypotheses are imposed. By contrast, the model’s solution provides a clean and correct local argument on an invariant level set: it uses s = Φ_c(r) from IFT, shows that g(s) and ĝ(r) coincide along trajectories on that leaf, and transfers observability by equality of output maps restricted to that set. Hence the paper’s proof is incomplete as stated, while the model’s local statement is correct.