Projective Curvature theory, Part I : Tangent Direction Curvature as Geometric Field

We introduce Projective Curvature Theory, a geometrically grounded framework in whichgravitational and fluctuation phenomena are described in terms of curvature of tangent directions.The formalism is based on a projective completion of the lines of physical variation (timeand position), in which the addition of a point at infinity induces a natural notion of curvaturefield r(x, t). Within this setting, diffraction-like effects arise as geometric manifestationsof tangent-direction curvature, leading to a reformulation of uncertainty relations in curvature-dependentform. In particular, the framework predicts a modification of fluctuation bounds inregimes of high curvature. The present work develops the geometric structure and derives theassociated relations, providing the foundation for a curvature-based description of gravitationaland inertial phenomena explored in a companion paper.