Adaptive Sphere Classification via Conformal Geometric Algebra

This paper introduces the Adaptive Sphere Classifier (ASC), a supervised learning algorithm whose classification function is derived directly from the inner product formula of Conformal Geometric Algebra (CGA). The central observation is that the CGA containment relationship which determines whether a point lies inside, on, or outside a sphere in a five-dimensional conformal space can be recast as a differentiable error signal and used to drive an iterative learning rule that updates both the radius and centre of each class sphere in response to misclassifications. This positions ASC as the first CGA-based classifier with an online, error-driven learning mechanism, distinct from all prior work that relies exclusively on static quadratic optimisation. We derive the unified self-correcting prediction formula from first principles, establish its mathematical validity via three formal arguments, verify it numerically to four decimal places, and benchmark it against SVM (RBF), k-nearest neighbours, and a multi-layer perceptron on four standard datasets. The algorithm achieves 100% accuracy on spherically separable data and fails predictably on geometrically non-spherically-separable problems a theoretically expected outcome that points directly to a well-defined line of future work.