Recursive Causal Closure: Dimension, Phase, Temporal Naturality, and Lorentzian Geometry

Abstract

Why does physical reality exhibit three spatial dimensions, complex quantum phases, and Lorentzian spacetime geometry? Could these seemingly different structures share a deeper origin in the way causal processes combine and persist? Recursive Causal Closure (RCC) explores this possibility by starting from finite causal operations, without assuming spacetime, wavefunctions, or a universal clock. Its central idea is simple: when two different causal histories connect the same events, their comparison can retain a non-trivial geometric structure. This residual structure becomes the starting point for recursive closure. Under explicit conditions of isotropic recapture, the framework selects a three-dimensional carrier whose elementary closures generate complex phases. The direction of causal succession provides a natural orientation for these phases, while the absence of a preferred temporal origin leads, under suitable persistence conditions, to autonomous evolution. The same construction also supports coherent sums over alternative causal histories and a minimal Lorentzian causal geometry. An explicit realization involving Thomas–Wigner rotations illustrates how complex phase and relativistic geometry can emerge as two aspects of a common closure structure. RCC does not claim to derive quantum mechanics or general relativity in full. Instead, it identifies a compact mathematical architecture connecting causality, dimension, phase, and spacetime geometry, while making the remaining physical assumptions and unresolved bridges explicit. The central question is whether some of the structures we usually take as fundamental might instead be consequences of causal closure itself.