Spectral Neutrality and Stability Selection in Projection-Origin Geometry (Projection-Origin Gravity 9)

This work investigates the spectral neutrality structure of a discrete time-evolution map arising in projection-origin geometry. Previous studies in this framework established the emergence of weak-field geometric structure, Bianchi-compatible evolution, constraint propagation, and a dynamically stable transverse–traceless perturbative sector. The present study focuses on the spectral properties of the linearized evolution operator associated with the discrete update map. By linearizing the time-step operator and estimating its spectral radius through a power-iteration-based norm estimate, we identify a localized neutral region in the deformation parameter β where the spectral radius remains close to unity. Deviations from this region lead to exponential amplification or decay in the dynamical evolution. A systematic scan of the parameter space reveals a stability window centered near β ≈ 0. The width of this neutral region depends on the discrete time step and becomes increasingly localized as the step size is reduced. Time-evolution experiments confirm that the sign of the observed growth rates is broadly consistent with the sign of the spectral deviation of the linearized operator. Finally, the neutral spectral branch is compared with the Bianchi closure condition derived in earlier work. The results indicate that the spectrally neutral evolution branch coincides with the branch that minimizes the Bianchi residual, suggesting a dynamical selection mechanism linking spectral stability with conservation-compatible geometric evolution. The analysis provides a stability-based interpretation of slicing selection within projection-origin geometry without invoking Einstein field equations. Note: Parts of the manuscript were linguistically and structurally refined with the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own. Note: This work represents Version 1.0 of an ongoing research program on the Order-Projection Principle (OPP). Minor typographical corrections and clarifications may appear in later versions. The core conceptual claims remain unchanged.