Two-Branch Algebraic Geometry from Intrinsic Spectral Relations: A Perspective Beyond Space-Based Geometry

Abstract We present a framework in which a second-order algebraic relation is interpreted as a genuine geometric structure arising internally within an algebra, rather than as a curve embedded in an ambient space. In contrast to classical algebraic geometry, where geometry is defined as a set of points satisfying polynomial equations in a coordinate space, the present approach treats algebraic identities themselves as generators of geometric stratification. The factorization of the minimal polynomial induces a canonical two-branch decomposition through idempotent projectors, yielding a stratified algebraic geometry without coordinates, metrics, or differentiable structure. This construction clarifies how such an intrinsic two-layer geometry differs from classical differential and algebraic geometries, while remaining conceptually compatible with them.