Geometry of Complexities: A Unified Framework for Measuring, Observing, and Reconstructing Structural Complexity

We propose the foundations of a new mathematical framework — the Geometry of Complexities — which studies how the structural complexity of any object varies with the position of observation. The central result is a three-term decomposition: Sₜotal = Sdirect + Scurve + Sₒbs, where Sdirect is the absolute invariant complexity of the object measured by its internal projections, Scurve is the complexity revealed by complete multi-angle observation (zero for a single angle, maximal when all angles are available), and Sₒbs is the observer's fixed Euclidean contribution. Each object carries its own natural geometry of measurement — flat objects project in Euclidean space, curved objects in Riemannian space, fractal objects in fractional-dimension space. The Partial Loss Theorem recovers Heisenberg and Gödel as geometric corollaries. Six open problems are stated for future researchers.