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 DKL ~ exp (D-d), which does not arise in the XOR-HPP with independent scales (where DKL is quadratic). This paper introduces inter-scale couplings in the HPP Hamiltonian H = -Σ Jₖ σₖ - K Σ σₖ σ₊+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: DKL (PD ‖ P₃-₁) = K·2^D-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.
Beyer et al. (Sat,) studied this question.