Spectral dynamics of the causal horizon and optimal growth of the discrete universe

This is the second paper in the Factorization by Function Zeros (FZF) series. While the preceding work established that the effective cosmological constant (Λ_eff) emerges from the dilution of a minimum spectral cost over a finite causal volume, it treated the volume size as a given input. This work addresses the complementary question: what is the dynamic origin of the vast causal scale associated with the current universe? We propose that the expansion of the causal horizon is governed by a discrete transfer dynamics, described by a linear operator whose dominant eigenvalue corresponds to the Golden Ratio (φ). This growth rate is shown to be structurally optimal, maximizing the arithmetic admissibility of the causal lattice without inducing spectral collapse or resonance. Key Findings: Golden Ratio Dynamics: The causal horizon evolves according to a recursive Fibonacci-like process, where the transfer matrix determines the generation of new admissible states. The Golden Ratio (φ ≈ 1.618) emerges naturally as the attractor for the asymptotic growth rate, representing the "Golden Spectral Frontier" of arithmetic stability. Dynamic Scaling Law: A direct scaling law for the cosmological constant is derived: Λ_eff(k) ~ φ^(-2k) where k ∈ ℤ is a discrete index parameterizing the temporal iteration of the causal horizon. Identification of the Causal Epoch: By relating this law to the observed cosmological hierarchy (Λ ~ 10^-122), we identify the current epoch of the universe as corresponding to the causal step k ≈ 295 ± 2. Solution to the Coincidence Problem: Within this framework, the so-called cosmological coincidence problem is reinterpreted not as a statistical improbability, but as a determination of the current evolutionary state within a deterministic trajectory governed by the spectral efficiency of causal growth. Methodology: The paper utilizes the FZF formalism on a discrete Lorentzian lattice, applying concepts from Number Theory (Diophantine approximations, Pisot numbers) and Dynamical Systems (Anosov diffeomorphisms) to cosmology.