The Mechanism of Scale-Invariance

The paper’s Theorem 1 states that an analytic SISO system is invariant with respect to a one-parameter Lie group of input transformations if and only if it admits the normal form with û(t)=π_{−p̂(t)}(u(t)), p̂′(t)=h_e(z(t),û(t)), z′(t)=f_z(z(t),û(t)), y(t)=h_z(z(t)), and it provides a constructive coordinate change via the PDEs E_ρδ_{z,i}=0 and E_ρδ_p=1 (using the infinitesimal generator E_ρ), exactly as in the candidate solution. The definitions of invariance and equivariance used by the paper match those assumed by the candidate, and the sufficiency (“normal form ⇒ invariance”) and necessity (“invariance ⇒ normal form”) directions coincide in substance: both arguments rectify the state action to translations and deduce the key identity F̂(z,p̂+q;π_q(u))=F̂(z,p̂;u), yielding the normal form upon setting q=−p̂ (paper: Eq. (4) and Theorem 1; candidate: Step B, push-forward identity). The candidate adds a detailed construction from invariance to equivariance (which the paper references to prior work) and is more explicit about locality/regular points of E_ρ. Minor omissions in the paper include not making the local/regular set caveat explicit and invoking an undefined “irreducible” assumption in one line of the proof; neither affects the core result. Overall, both are correct with substantially the same proof strategy, with the candidate offering slightly sharper quantifiers and technical caveats. Key matches: paper’s invariance/equivariance definitions and normal form (Eq. (2), (3), (4)) and Theorem 1 with the PDE recipe for δ (Eρδ_z=0, Eρδ_p=1) align with the candidate’s statements and construction.