FB(S³)R: Causal Self-Consistency of the Speed of Light in Fractal S³ Geometry

This work proposes a conceptual shift: the speed of light c is not derived, but treated as an empirical invariant used to test the internal coherence of a compact S³ cosmological framework. The model is built on a simply connected, boundaryless three-sphere where volume and density evolve through a structural hierarchy Rₙ = R₀ φⁿ, with φ ≈ 1. 618 introduced as a minimal self-similar scaling ansatz rather than a fundamental constant. The exact global geometry is defined by VS³ = 2π²R³, while local behaviour reduces to Euclidean scaling in the χ ≪ 1 limit, allowing a consistent bridge between global topology and local dynamics. The central question is whether such a volume–density structure admits a coherent evolution under a causal boundary propagating at speed c. Combining the fractal hierarchy with a density–volume relation and a Friedmann-type scaling leads to a boundary condition c = √ (2GM₀ / R₀) · (1/φ), which, upon self-consistent normalisation of M₀, yields the identity c = c. This identity is not a derivation, but a fixed-point (closure) condition, demonstrating that the geometric and dynamical assumptions form an internally consistent system under the empirical causal speed. In this sense, c is selected by consistency, not predicted from first principles. An instructive analogy arises in Maxwell electrodynamics, where c = 1 / √ (ε₀ μ₀) expresses a consistency relation between field parameters rather than an independent derivation; similarly, the present framework shows that a fractal S³ volume–density evolution closes coherently when the boundary speed equals c. A Planck-scale calibration R₀ = lP φᴺ (N ≈ 207) may be introduced to connect quantum and cosmological scales, but it is illustrative, not foundational, and plays no role in the logical structure of the closure. Numerical substitution recovers c ≈ 2. 998 × 10⁸ m/s, as expected by construction, confirming algebraic and dynamical coherence rather than predictive power. Taken together, the result defines a causal self-consistency principle: a compact S³ geometry with fractal scaling and density–volume coupling admits a closed, coherent dynamical description when the causal boundary propagates at the empirical invariant speed c. The framework therefore does not attempt to derive fundamental constants, but instead provides a minimal, internally consistent geometric construction in which space, density, and causality form a single, self-closed system. Orpheus Praxis AI (myGPT) 🔴 YouTube Podcast: FB (S³) R — “The Satsang of Reality” 🎙️The Speed of Light: Four Stroke Proof of the Fundamental Constant of the Universe 🎙️Light’s Genesis. Fractal Proof of the Speed of Light Invariance