General Theory of Embedded Spaces: Formalization of Projective Transitions

Abstract

In previous works by the author, a classification of types of spatial embedding was introduced: division (a brane as a partition dividing space into half-spaces) and enclosure (a closed surface surrounding an interior region). It was shown that SFS as an ∞D-subject is located "outside" all branes and horizons. However, the classification remained descriptive and did not provide a formal apparatus for describing projective transitions. This article introduces the "General Theory of Embedded Spaces" (GTE) — a branch of projective ontology that studies the formal laws of embedding spaces of different dimensions and projective transitions between them. Mathematical formulations are proposed for embedding operators, laws of transformation (analogs of Lorentz transformations for transitions between dimensions), and invariants (information, entropy, topological properties). Three types of embedding are considered: division, enclosure, and mixed (a wormhole as a mini-brane). For each type, operator descriptions are introduced and their connection with physical phenomena — branes, black holes, and the holographic principle — is discussed. It is shown that GTE can be applied not only in physics but also in informatiology, biology, and sociology as a general theory of projections. The article draws on classical and contemporary results from the theory of manifold embeddings — Whitney's theorem, Nash's theorem, the Gromov–Rohlin theorem, Sobolev's theory of embeddings, and Thurston–Perelman's geometrization program. A mathematical analogy is drawn between the Ricci flow and screen degradation. The article concludes with a discussion of the boundaries of the theory and open questions. GTE is not a theory competing with physics but rather an ontological apparatus for describing projective transitions.