Speaker
Description
We present a non-perturbative framework in which two compatible 4D spatial geometries emerge from sheer arithmetic properties of growing set of positive integers. The process of adding integers conceived as time, Lagrange's four-square theorem provides the dimensional foundation. Integer distances between integers in a ring algebra admit decomposition into four coordinates, with multiple decompositions related by Hurwitz quaternion rotations. The 24 unit elements of the Hurwitz integers — eight with integer coefficients, sixteen with half-integer coefficients — generate the binary tetrahedral group without postulation.
Discrete symmetries yield the Standard Model U(1) and SU(2) gauge holonomy directly through quaternion products around closed vertex loops, providing a natural arithmetic realization of Wilson loops without requiring continuous gauge field. The structure invites investigation whether the framework extends to SU(3) through the action of the sixteen half-integer Hurwitz elements on the three independent spatial directions.
There are also two metric signatures emerging from number-theoretic universality: the all-positive Euclidean form describing spatial distances across simultaneity along S3, and the Minkowski signature describing observations into the past from a specific point on S3. They arise as distinct metric projections of the same integer distance, endowing Wick rotation a geometric rather than formal meaning. All results are exact discrete calculations with no series truncation or renormalization required. We discuss the relationship between this discrete arithmetic structure and continuum spacetime geometry.