Exact Cardinality And Nonredundant Parametrization Of Character-Polynomial Codes

What the math gives to ML

The paper identifies a concrete non-injectivity mechanism in character-polynomial evaluations over extension fields: additive characters observe only the base-field trace, while Frobenius-related monomial exponents lie in the same cyclotomic coset. This gives an explicit quotient of polynomial parameter space and a nonredundant parametrization by cyclotomic-coset representatives. A transferable neural-network use is a finite-field Fourier or phase embedding layer whose learnable coefficients are stored only on trace-visible orbit representatives, eliminating exactly redundant parameters and preventing optimization from moving in zero-function directions. The main payoff is a compact deterministic embedding for categorical IDs or structured discrete states; experiments can compare parameter count, conditioning, and accuracy against an unconstrained character-feature layer.