The Fundamentals of Zulfia Geometry: Beyond Classical Geometry ..... No metric. No distance. No convexity. No differentiability....

AbstractFor decades, geometry has reduced relation to x − u. This collapses geometricalanatomy into distance and forces all relations to appear flat by definition.We resolve this with the Unavoidable Decomposition Principle (UDP).Every relation decomposes asx = u + η(x, u) + ∆(x, u), (1)where η(x, u) is the primary direction (direction component) and ∆(x, u) is thestructural correction (correction component). UDP follows from arithmetic alone.No metric. No convexity. No differentiability.From UDP we derive the Arithmetic Force Equation (AFE):∥x − u∥ = ∥η(x, u) + ∆(x, u)∥. (2)AFE separates distance from structure and defines curvature as deviation fromstraight-line motion: κ(x, u) := ∥∆(x, u)∥/∥η(x, u)∥.We define the Z-mapping Z(x, u) := (η(x, u), ∆(x, u)) as the anatomy of arelation. This establishes Zulfia Geometry: a metric-free, assumption-free Ge-ometry of Deviativity with applications in physics, optimization, and generalsystems. The fundamental feature of Zulfia Geometry is scale invariance:the same formalism describes both micro and macro systems withoutassumptions.