Mathematical Continuum Limit: Rulial Multiway Graphs to FEF Fractional Dynamics

This paper provides a rigorous mathematical derivation demonstrating how discrete multiway state updates on a hypergraph with long-range spatial or branchial shortcuts converge to the continuous Fractional Schrödinger Equation utilized in the Franquero Entropy Framework (FEF). By analyzing a discrete master equation on a multiway graph, I show that when non-local causal updates are introduced, the transition probability distribution develops heavy algebraic tails characterized by the power law W(r) ~ [C_(α,d) * a^α] / |r|^(d + α). Taking the spatial/branchial Fourier transform and applying the continuous time and space limit (a → 0, τ → 0), the discrete hypergraph parameters must satisfy the fractional scaling relation D_α = lim(a→0, τ→0) [A_α * a^α] / τ to obtain a finite continuum limit. Applying the inverse Fourier transform converts the spectral power -|k|^α back to the continuous spatial domain via Riesz fractional derivatives, (-∇²)^(α/2), recovering the non-unitary FEF Time-Dependent Schrödinger Equation: iħ ∂ψ(x,t)/∂t = [ D_α(-ħ²∇²)^(α/2) + V(x) - iħ Γ_ent(ψ) ] ψ(x,t) This mapping establishes a direct physical interpretation between discrete rulial systems and the FEF, where long-range hypergraph rewrites generate fractional dispersion (driving a +3.27° angular deflection shift at α = 1.6), multiway causal bottlenecking emerges as entropic damping, and branchial path divergence physically manifests as fringe visibility loss.