Prime Structure as a Topological Obstruction to Scale Decoupling

This work bridges two frameworks: the coarse-grained quantum dynamics of Agon–Balasubramanian–Kasko–Lawrence, where UV effects are suppressed by 1/ΔEUV1/Δ EUV 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 InI_n 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/Δ EUV 1/ΔEUV suppression now competes with the growth of InI_n In. The UV survival condition is simply that μIn/ΔEUV_n / Δ EUV μIn/ΔEUV stays bounded away from zero. The result splits into two regimes. In composite-like configurations, InI_n In saturates and UV decoupling is restored — the system reaches equilibrium and further observations add no information. In prime-like configurations, InI_n 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⁡nρ(n) ~ 1/log n ρ(n)∼1/logn emerges as the survival rate of effective poles under successive coarse-graining.