Spinor Trigonometry and Exacalculus: Mathematical Tools for Discrete Systems Exhibiting Double-Cover Geometry

We present a mathematical framework extending classical trigonometry and calculus to accommodate systems whose state representation naturally lives on a double cover of ordinary rotations. While standard circular trigonometry suffices for most applications, certain specialized systems—notably fermionic quantum states (modeled in SU (2) rather than SO(3)) and selected helical/topological constructions—are more naturally expressed using spinor-valued functions with 4π closure in the rotation parameter. We develop a “spinor trigonometry” formalism for quaternionic and octonionic exponentials, introduce a product-based “Exacalculus” notation (product integrals and multiplicative derivatives) following Wallis and Volterra, and illustrate how these tools package double-cover geometry and multiplicative accumulation in a way that can simplify select calculations. Spinor trigonometry integrates the classical triangle (additive/linear), circle (multiplicative/rotational), and spiral/helix (iterative/progressive) motifs, with the circle appearing as a projection of a lifted SU (2) trajectory.