Relaxation Dynamics in Moduli Space: An Effective Mechanism for Compactification Selection (TSDC)

This work proposes and analyzes the Dynamic Compactification Selection Theory (TSDC), an effective model for studying the time evolution of geometric degrees of freedom in string compactifications with α ′ corrections included.Motivated by curvature-squared (Gauss-Bonnet) terms and dilaton stabilization, we construct an effective potential whose gradient flow governs the coupled dynamics of a dominant geometric modulus and the dilaton.Using methods from dynamical systems theory-stability analysis, a global Lyapunov function, and LaSalle's invariance theorem-we show that the reduced twodimensional model possesses a single stable attractor, while the toroidal configuration serves as a dynamical repulsor.Numerical simulations demonstrate robust convergence to the attractor across a wide range of initial conditions and indicate that the dissipative structure persists in preliminary higher-dimensional extensions.These results establish Phase I of TSDC as a coherent and mathematically well-posed proof of concept, suggesting that α ′ corrections may induce non-trivial selection mechanisms within the moduli space.The paper concludes by outlining the limitations of the effective model and presenting a research program to extend the framework toward realistic Calabi-Yau compactifications, including fluxes, branes, and full moduli dynamics.