Open Causal Memory Gravity IV: Almost-Lambda Cosmology: A Reduced Condensate Model with Testable Deviations

This paper develops the cosmological sector of the Causal Memory Gravity (CMG) framework and investigates its consequences for late-time cosmic evolution. Starting from a reduced condensate model of memory dynamics, the work derives modified Friedmann equations that interpolate smoothly between standard CDM behavior and controlled deviations driven by microscopic memory effects. The analysis shows that an effective cosmological constant emerges dynamically from the evolution of the memory condensate, leading to an ``almost--'' cosmology that remains consistent with current observational data. The model is confronted with supernova, baryon acoustic oscillation, and Hubble parameter measurements, and its parameter space is constrained using Markov Chain Monte Carlo methods. Numerical solutions demonstrate stable cosmic expansion histories and predict small, testable departures from the standard model of cosmology. These deviations are linked to the underlying causal network structure and provide potential observational windows into emergent gravitational dynamics. This work establishes the phenomenological bridge between the microscopic foundations of CMG and observable cosmology, forming a key step in the broader program of deriving spacetime and gravity from discrete causal memory networks.