GUE Spacing from Cauchy–Poincaré Invariance

This work bridges two frameworks: the coarse-grained quantum dynamics of Agon–Balasubramanian–Kasko–Lawrence, where UV effects are suppressed by 1/ΔEUV1/ Eₔₕ 1/ΔEUV under time averaging, and the Möbius TST iteration framework, where each map application generates a sign-reversed dipole via pullback. The core observation is that not all dipoles cancel under coarse-graining. Those straddling an inclusion boundary — where one pole is inside the contour and the other outside — survive as irreversible defects. We define the irreversibility accumulator InIₙ In, which counts these surviving boundary crossings up to step nn n. Inserting this as a source term into the UV–IR coupling gives a modified master equation. The standard 1/ΔEUV1/ Eₔₕ 1/ΔEUV suppression now competes with the growth of InIₙ In. The UV survival condition is simply that μIn/ΔEUVₙ / Eₔₕ μIn/ΔEUV stays bounded away from zero. The result splits into two regimes. In composite-like configurations, InIₙ In saturates and UV decoupling is restored — the system reaches equilibrium and further observations add no information. In prime-like configurations, InIₙ In grows linearly with nn n, overwhelming the UV suppression. The UV signature persists in the IR density matrix, and UV–IR entanglement grows without bound. The equivalence is: primality corresponds to failure of scale decoupling, compositeness corresponds to successful decoupling. The Abel summation identity for the signed residue sum is reinterpreted as the algebraic expression of this multi-scale filtering, and the primitive path density ρ (n) ∼1/log⁡n (n) 1/ n ρ (n) ∼1/logn emerges as the survival rate of effective poles under successive coarse-graining.