Emergent Space, Time, and the Speed of Light from Binary Sequences

Abstract

We show that the complete kinematic structure of special relativity emerges from pairs of binary sequences of length $n$ under the single operation of bitwise XOR. A physical event is a pair of binary sequences encoding clock and ruler information, while their XOR difference defines an observer-independent worldline map. Giving equal information weight to the counts generating spacetime transformations (clock and ruler bits) we show that the Minkowski interval, Poincaré symmetry, and the dispersion relation $E^2-p^2=m^2$ follow directly. Finiteness of the speed of light follows from the time-ordering requirement of the worldline, while its universality, and frame-constancy follow from ${\mathbb{Z}}_2$ arithmetic, and are not independent postulates. Finite sequence length $n$ implies a discrete Planck-scale spacetime with exact Lorentz invariance, while the continuum limit $n\to \infty$ recovers standard special relativity. Relating this with the previous work in \cite{Powers:2021rfg}, we see that the binary-sequence framework provides a single substrate for both matter and spacetime. Particles are sequences with definite spin quantum numbers $(j,m_j)$, while spacetime displacements are maps between sequences. The framework yields testable predictions: the boost rapidity takes discrete values, and a single inter-frame map of length $n$ realizes Lorentz factors only up to ${\gamma }_{\max }=O(\sqrt{n})$, with arbitrarily large $\gamma$ recovered in the continuum limit. Most intriguingly, finite systems at the maximum of their information capacity (e.g. black holes, de Sitter space) must exhibit finite $n$ corrections. These results suggest that space and time might not be fundamental categories but emerge from discrete binary information.

17 pages main text plus appendices, one figure