A Discrete Projective-Geometric Interpretation of Spacetime: Temporal Bifurcation and Relativistic Phenomena in the ABOS Framework

This work presents a discrete projective-geometric interpretation of spacetime, developed as a geometric extension of the ABOS framework (see: 10.5281/zenodo.18002324).The model is based on a combined angular and linear discretization of a normalized circular domain, supplemented by a distinguished geodesic (the α-geodesic) that naturally separates different kinematic and geometric regimes. Within this construction, relativistic effects such as time dilation, the existence of a limiting velocity, and the equivalence between acceleration and gravitation emerge as direct geometric consequences rather than as postulated principles. The degeneration of local grid cells near the α-geodesic defines a natural boundary of applicability for planar coordinate representations, providing a geometric interpretation of event horizons without invoking singularities. The model remains consistent with the observable predictions of Special and General Relativity, while offering an alternative geometric intuition based on discrete projective structures. In particular, accelerated motion corresponds to an increasing phase inclination within the grid, and gravitational effects are interpreted as local distortions of the same geometric structure. A phenomenological interpretation is also discussed. Near the α-geodesic, the effective metric exhibits strong projective amplification: infinitesimal phase intervals correspond to large coordinate separations. From the perspective of an embedded observer, this may be perceived as an extreme spatial compression and a forced continuation into a new spacetime chart—a process analogous to an “infinitely magnifying microscope.” This description is intended as an intuitive aid rather than a physical force-based mechanism. This work does not propose a replacement for Special or General Relativity. Instead, it provides a discrete geometric framework in which their key kinematic and gravitational effects arise naturally from the structure of spacetime itself. Additional note.As a supplementary illustrative example, a classical mechanical phenomenon known as the Dzhanibekov effect (the tennis racket instability) is referenced in the accompanying materials. This effect exhibits a well-defined bifurcation with phase inversion and alternating stable regimes separated by an unstable transition. While no direct dynamical modeling is attempted, the qualitative structure of this phenomenon—particularly the observed phase inversion and recurrent temporal proportions (e.g., approximately 1:2.677)—is presented as an independent example of bifurcation behavior consistent with the phase inversion principle discussed in this work. The example is included for conceptual illustration only and does not serve as empirical validation of the proposed geometric framework. The article is released under the CC0 (Creative Commons Zero) license and is placed in the public domain for unrestricted use.