Causal Co-Scaling of Distance and Time in a Fractal S³ Reconstruction Model

Causal Co-Scaling of Distance and Time in a Fractal S³ Reconstruction Model presents a compact structural principle for Planck-normalized reconstruction models: distance, reconstruction time, and the speed of light form one inseparable causal triad rather than three independent quantities. Instead of attempting to derive c from topology or geometry alone, the note introduces the Causal Co-Scaling Axiom, requiring every admissible reconstruction hierarchy to preserve the invariant relation R = ct, c = R/t, and t = R/c. If carrier radii evolve as Rₙ = ℓₚφⁿ, causal consistency requires reconstruction times to scale identically, tₙ = tₚφⁿ, ensuring that Rₙ/tₙ = c at every reconstruction level. The manuscript therefore reframes the problem from deriving the numerical value of c to preserving the causal structure of the model itself. The paper also establishes a clear distinction between a pre-photon reconstruction regime and the later photon-supported cosmological expansion regime. Internal reconstruction indices such as N = 207* are treated as model-internal reference scales, not as observational definitions of cosmological epochs. Compact S³ geometry is used consistently as the carrier manifold, with the correct volume V(S³) = 2π²R³, leading naturally to the density scaling ρₑ(N) ∝ φ⁻³ᴺ and a complementary notion of reconstruction capacity, which increases as distinguishable cells become available at fixed resolution. This separation between energy density and reconstructible distinguishability provides a clean conceptual bridge between geometric expansion and information-theoretic capacity without identifying the two quantities. Equally important is the manuscript's methodological discipline. It explicitly rejects claims to derive the speed of light, replace observational cosmology, or identify golden-ratio scaling with established cosmological epochs. The previously obtained null result for observational searches of golden-ratio log-periodicity is incorporated directly into the framework: the φ-hierarchy is retained only as a structural admissibility ansatz, not as an empirically confirmed cosmological law. The result is a concise research note that isolates one precise mathematical principle—causal co-scaling—and positions it as a consistent building block within the broader compact S³ reconstruction programme, emphasizing causal closure, Planck normalization, and structural consistency rather than speculative cosmological claims.