The ϕ-Framework: A Hyperbolic Geometric Approach to Gravity and Light Propagation

We propose a geometric framework in which all physical measurements, including spatial trajectories, temporal intervals, and effective forces, emerge from the observer-relative hierarchical structure of matter, quantified by a hyperbolic twist parameter ϕ. In this framework, light propagation is governed by ϕ-geodesics, which reduce to standard GR geodesics in the ϕ → 0 limit. Gravitational lensing appears as a relational, observer-dependent effect, arising from path dilation through ϕ-gradients rather than from a local force acting on photons. Gravity emerges as a structural inclination of matter to form low-entropy, hierarchicallycompatible configurations, producing effective attraction between massive bodies. Apparent time delays, including the Shapiro delay and gravitational redshift, are consequences of differences in ϕ between source and observer; the photon itself does not lose energy. The effective speed of light is locally invariant, but globally appears modified in observer coordinates due to ϕ-gradients, analogous to a refractive index. Micro-scale stability (N = 3) and macro-scale gravitational effects are unified through a single ϕ-variable acting across hierarchical scales. This framework unifies light deflection, time dilation, and apparent mass anomalies under a single geometric principle: the operational mapping of hierarchical space to observer coordinates through hyperbolic geometry.