Effective Description Equivalence in Horizon Monism: A Projection-Based Correspondence Between Noncommutativity and Lorentzian Curvature

We formulate the Effective Description Equivalence Principle (EDEP) within Horizon Monism, a relational framework based on a single scale-free relational totality and embedded finite reference horizons. The principle proposes that noncommutative algebraic and Lorentzian geometric structures may arise as distinct but compatible effective descriptions of the same underlying relational organization, rather than constituting independent ontological primitives. To develop an explicit mathematical realization, we introduce a projection-based model of horizon-relative compression and geometric expansion. We first establish that compressing mutually commuting operators onto a finite accessible subspace can generate noncommuting effective operators, with the resulting commutator determined by couplings through the complementary subspace. For a smooth family of projection operators, we then construct the induced connection and derive the curvature identity F^P(X,Y)=P[\partial_XP,\partial_YP]P. Under additional localization, nondegeneracy, and physical transport-identification conditions, the projected connection coincides with the Levi-Civita connection of an effective Lorentzian geometry. An explicit four-dimensional de Sitter realization demonstrates that the same projection structure can support nonvanishing Riemann curvature and satisfy the vacuum Einstein equations with a positive cosmological constant. These results establish a conditional mathematical correspondence between projection-induced operator noncommutativity and geometric curvature, without identifying arbitrary noncommutative algebras with curved spacetimes. We discuss the implications for horizon-relative algebra–geometry reconstruction, the relation to the HM-14 coordination model, and the possibility of selecting compatible infrared gravitational and ultraviolet gauge-theoretic realizations. The framework provides a mathematically explicit research direction for emergent spacetime and quantum gravity while leaving the dynamical selection of physical projector families and the full recovery of Standard Model interactions as open problems.