Topological Persistence of Stellar Flaring Regimes in Magnetically Saturated M-Dwarf Stars

Stellar flaring in magnetically active M-dwarf stars is commonly modeled as a stochastic outcome of turbulent magnetic reconnection. In this study, we investigate whether flaring activity, once established in magnetically saturated M-dwarfs, instead exhibits persistent large-scale organization. Using a deep-time post-emergent diagnostic framework spanning one million years, we analyze the structural properties of flaring activity under non-equilibrium forcing. Across extensive robustness tests—including stochastic noise sweeps, symmetry-breaking perturbations, and null-structure controls—the system converges to a stable, low-dimensional flaring regime characterized by invariant topology, high causal density, and scale-invariant organization. These results demonstrate that stellar flaring in active M-dwarfs behaves as a topologically stable phase object rather than a transient stochastic process. This structural interpretation provides a unified explanation for long-term flare statistics, event clustering, and magnetic activity saturation, with implications for stellar magnetism, space weather, and exoplanet habitability.