Exponential Divergence in Hierarchical Processes: A Renormalisation Group Derivation of β = D/(D+1)

The β-law β = D/(D+1) of the Axiomatica Universalis (Beyer 2026f) was derived in Beyer 2026v conditional on a rate-distortion cost structure L(β) = D²/β + 1/(1-β). The term D²/β follows from K(d) ~ d² (mechanistic). The term 1/(1-β) requires an exponential divergence D_KL ~ exp(D-d), which does not arise in the XOR-HPP with independent scales (where D_KL is quadratic). This paper introduces inter-scale couplings in the HPP Hamiltonian H = -Σ J_k σ_k - K Σ σ_k σₖ₊₁ and applies renormalisation group (RG) methods to show that coupled scales produce exponentially growing KL-divergence under RG iteration. Theorem W2 is proved analytically and numerically: D_KL(P_D ‖ PD-1) = K·2D-1, confirmed by transfer matrix (ratio → 1.000 exact for D ≥ 6). This closes the gap from Beyer 2026v and completes the derivation of Theorem V3 unconditionally.