Emergent Space, Time, and Lorentz Symmetry from Binary Sequences

Abstract

We construct an information theory framework in which the fundamental objects are binary sequences of length $n$, equipped with the bitwise XOR operation. The only physical observables are counts of XOR-generated symbol classes, while the exact locations of symbols are inaccessible. Averaging over those locations ultimately yields the Minkowski interval as an invariant object that maximizes (information) entropy. Correlations between two binary sequences are base-4 sequences that we label as ``events'', and events are connected with maps. The entire kinematic structure of Special Relativity is recovered under minimal assumptions, i.e. counts that represent space and time increments carry equal informational weight. The central claim is that Lorentz symmetry is the typical large-n behaviour of XOR counting. At finite $n$ the framework yields a discrete rapidity spectrum, a bound $\gamma_{\max} = O(\sqrt{n})$, and interval fluctuations of relative size $O(n^{-1/2})$, with standard special relativity recovered as $n \to \infty$. However, the light cone and one null coordinate are exact for every microscopic configuration, so the symmetry group is undeformed and dispersion relations are unmodified. Thus, the theory, though discrete, implies no Lorentz violation of the standard phenomenological kind. Ultra-high-energy cosmic rays already require $n \gtrsim 10^{23}$; interferometry excludes the variant in which $n$ scales linearly with system size, leaving a holographic area law. The most striking prediction of the framework concerns systems at the maximum of their information capacity, where $n$ is necessarily finite: black holes and de Sitter space. There the corrections are of order one within a Planck proper length of the horizon, regardless of the horizon's size, and the spacetime description fails altogether at the endpoint of black-hole evaporation, where $n$ itself is of order unity.