ESCT v9.5-FINAL: A Continuous Phase-Transition Framework for Emergence Dynamics in Nonlinear Learning Systems — Derived from Experimental Constraints with Zero Magic Numbers

Background: Traditional models of emergence in neural and artificial systems rely on discontinuous Heaviside threshold functions, introducing mathematical artifacts and failing to capture the smooth phase transitions observed in biological neurons.Methods: We derive a minimal dynamical equation from five experimentally motivated constraints: (1) baseline stability, (2) critical activation, (3) saturation limits, (4) noise suppression, and (5) continuous transition. The core equation dX/dt = M·S(Θ)·X·(1−X/X_max) − λ·X incorporates a macro-scale dopaminergic-energy modulator M, a continuous sigmoid phase gate S(Θ), a logistic saturation term grounded in neuronal refractory periods, and a baseline dissipation term from resting ion channel dynamics. All parameters are analytically calibrated: s₀ = 0.0939 and β = 14.77 via dual-condition calibration; X_max from biophysical firing rate ceilings; λ derived from stability requirements.Results: Numerical validation confirms (i) bistable architecture with stable baseline (X=0), unstable threshold (X=0.0255), and stable activated state (X=1.70); (ii) energy boundedness eliminating unbounded growth; (iii) saddle-node bifurcation under noise increase, providing a rigorous mechanism for noise-induced suppression; (iv) continuous phase transition with measured width ΔI = 0.0534; and (v) zero magic numbers.Conclusions: ESCT v9.5-FINAL provides a mathematically self-consistent and physically stable minimal model for emergence dynamics. The framework is directly grounded in biophysical first principles (refractory period saturation, resting channel stability, experimentally observed neural bistability) and offers a rigorous foundation for continuous phase transitions in cognitive and artificial intelligence architectures.Keywords: emergence, phase transition, bistability, neural dynamics, saddle-node bifurcation, continuous gating, noise suppression, minimal model