Simple Verification and Implementation of Observer Error Dynamics Linearization: A Pascal's Triangle--Hessian Matrix Criterion

What the math gives to ML

The paper provides a constructive structural test for whether a single-output nonlinear system admits observer-error linearization: a Pascal-triangle pattern in selected Hessian entries together with an identically vanishing lower-right Hessian block replaces expensive Lie-bracket involutivity checks. When the condition holds, the system can be transformed into an observer canonical form with companion matrix A and output-dependent forcing, so a linear observer gain produces exactly linear error dynamics. The transferable mechanism is a Hessian-structured latent state-space model with an observer correction module. Its main falsifiable prediction is that latent reconstruction errors decay exponentially at a rate determined by the eigenvalues of A-LC, rather than by the nonlinear forcing.