Finite-Memory Stochastic Cosmology: A Log-Oscillatory Dark Energy Model with Geometric Noise Cutoff (v3.2)

We present a finite-memory stochastic cosmology framework in which the dark energy equation of state exhibits damped log-oscillations driven by self-consistent stochastic noise. The effective equation of state is parametrized as w (z) = -1 + A exp (-z/z_τ) cosω ln (1+z) + δ, where A is the oscillation amplitude, ω is a frequency in logarithmic redshift, and z_τ encodes a finite memory time τ. At the microscopic level, the model is derived from two interacting scalar fields subjected to Ornstein-Uhlenbeck noise with intensity proportional to the Gibbons-Hawking temperature, modulated by a geometric window function that suppresses noise at high redshift. Model 2. 1. 1 restricts the amplitude to A ≤ 0. 03 and regulates noise intensity via a sigmoidal cutoff S (z) activated around zc ~ 4. This ensures compatibility with current constraints from type Ia supernovae, BAO, and the CMB, while retaining non-trivial temporal structure in w (z) at low redshift. The model is explicitly falsifiable through Bayesian comparison with ΛCDM using public data (Pantheon+ and follow-up datasets), and is accompanied by an open validation pipeline. Version 3. 2 includes: - Complete mathematical derivation with all citations- Three publication-quality figures- Bayesian validation protocol- LaTeX source code and bibliography Author website: https: //ernestocisneros. art/cosmology-physicsGitHub: https: //github. com/cisnerosmusic This work received no external funding and is released for independent testing, criticism, and extension.