The Signature Dynamics Framework: An Algorithmic Generator for Metallic Signatures in the Universal Metallic Family

This paper introduces the Signature Dynamics framework, a discrete dynamical system that algorithmically generates metallic signatures for Diophantine approximation by the Universal Metallic Family α(n, N) = n + √(n² + N). For a target constant T > 0, the map Φ_T(n) = round(T² · n) generates an orbit L(T) whose consecutive terms (n_{m+1}, n_m) are, by construction, valid metallic signatures of T. We prove a complete trichotomy for rational T and an exact linear recurrence for quadratic-irrationa