Determinant Characteristics and Argument-Principle Certification for Visible Poles in Meromorphic Continuation

What the math gives to ML

The paper turns exterior-pole recovery into finite exponential-sum identification: Taylor or Fourier coefficients decompose into a rapidly decaying background plus a small number of exponential modes whose bases are reciprocal pole locations. This suggests a principled way to initialize and compress state-space components in neural sequence models, where long-range behavior is often governed by a few slowly decaying or oscillatory modes. The transferable asset is not merely Prony fitting, but the combination of shifted determinant pencils for proposing modes and argument-principle-style contour tests for rejecting unstable or noise-induced modes before they are inserted into a trainable model.