Holographic Algebraic Geometry: Mathematical Ontology of Intrinsic Duality Structure — On the Natural Realization of the Holographic Principle Without Bulk-Boundary Partition and Anti-de Sitter Background

This paper proposes and elaborates a novel paradigm for realizing the holographic principle. Unlike the conventional bulk-boundary duality (e.g., AdS/CFT correspondence) which relies heavily on a negatively curved Anti-de Sitter background and a static conformal boundary, the proposed framework reconstructs holography as an intrinsic bilateral topological property of algebraic-geometric structures, independent of any external boundary or prescribed spacetime curvature. The mathematical foundation is built upon the modular quantum flag manifold algebra M = (O_q(SU_q(3)/T)'', φ, Δ, J, σ_t, D_F). The Kähler complex structure J induces a 3+3 skew-dual decomposition of the tangent bundle, forming a Hermitian-coupled pair consisting of a 3-dimensional macroscopic infrared subspace and a 3-dimensional microscopic ultraviolet intrinsic subspace. The core conclusion is as follows: the holographic principle does not require an external boundary as an information carrier. The spectral inversion symmetry Λ ↔ -Λ of the modular operator already realizes the algebraic equivalence between local observable information and global topological information. The Tomita modular flow σ_t = Δⁱᵗ extends static geometric holography into dynamical evolving holography. This paper demonstrates that the traditional bulk-boundary holographic duality (AdS/CFT) is a special low-energy limiting case of the proposed intrinsic holographic algebraic geometry, rather than the only fundamental realization of the holographic principle.