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

This work presents a discrete projective–geometric interpretation of spacetime, developed as a structural extension of the ABOS framework (Alpha–Beta–Omega–Sigma geometry; see: 10.5281/zenodo.18002324).

The model is based on a combined angular and linear discretization of a normalized circular domain, yielding a non-uniform grid whose geometric tension increases with phase inclination. A distinguished geodesic — the α-geodesic — emerges uniquely from the incompatibility of the minimal symmetry set Π₀ = {3, 4, 5}, and serves as a natural boundary between planar and non-planar projective regimes.

Within this construction, relativistic effects — such as time dilation, a limiting velocity, and the equivalence between acceleration and gravitation — arise as direct geometric consequences rather than postulated principles. Crucially, the framework also contains a fixed point of geometric anticipation (P), where the start of the next event cycle (t₀′) is encoded before the current one concludes. This endows the model with predictive capacity: bifurcations are not merely described post factum, but anticipated via invariant projective relations.

The degeneration of grid cells near the α-geodesic defines a boundary of coordinate applicability — interpreted here as a geometric horizon without physical singularities. Continuation beyond this boundary is described as a projective renormalization, accompanied by a change of geometric polarity (ABOS-20), consistent with the irreversibility imposed by the non-removable phase defect Δₛ ≈ 4.43°.

The model remains fully consistent with the observable predictions of Special and General Relativity, while offering an alternative interpretation in which:

• acceleration corresponds to increasing phase inclination,
• gravitation corresponds to local grid distortion,
• horizons arise from chart degeneration, not metric collapse,
• time emerges as accumulated phase mismatch.

A phenomenological analogy is provided: near the α-geodesic, infinitesimal phase intervals map to large coordinate separations — akin to viewing through an “infinitely magnifying microscope.” This is intended as an intuitive aid, not a dynamical mechanism.

As a supplementary illustration, the classical Dzhanibekov (tennis racket) effect is referenced for its qualitative similarity: a recurrent bifurcation with phase inversion and a ≈1 : 2.677 temporal ratio — matching the ABOS Ωₕ regime. This example is included only for conceptual resonance, not as empirical validation.

The article is released under the CC0 (Creative Commons Zero) license and placed in the public domain for unrestricted use, adaptation, and integration.