Optimized Analytic Polynomial Transition Model with Spectral Data Validation

This study introduces an optimized analytic polynomial transition model inspired by functional transitions between linear and nonlinear systems. Utilizing spectral vibrational data acquired from a millikelvin spectrometer setup, we propose a new methodology for modeling transitions between f(x) = x and f(x) = x² with a continuous, smooth analytical behavior. A key innovation is the use of a weighted hyperbolic tangent function to control the transition, allowing parameter optimization via error minimization techniques. The optimized model reduces the average percent error from classical piecewise methods by over 9%, achieving an average percent error of 0.50%, which is significant for modeling physical behaviors under quantum mechanical conditions. This paper outlines the theoretical foundation, algorithmic construction, experimental validation, and implications for quantum vibrational analysis.The author acknowledges the use of OpenAI’s language model for assistance in drafting certain sections of this workl