Why Scale-Invariant Modular Shortcuts Extend to Infinity: Asymptotic Natural Density and Ergodic Limits under 10^k Partitioning

Abstract

This paper formally demonstrates why the scale-invariant shortcut architecture developed for the Collatz conjecture rigorously extends from finite computational bounds (10^{10000}) to the infinite limit (N \to \infty). By evaluating the bounded interval I_k = [1, 10^k], we establish that layer averages \mu(k) = (L_1+L_2)/2 converge deterministically to 5 \cdot 10^{k-1}. Furthermore, we prove that under the macro-conservation law A_4, the natural density of non-canonical or divergent trajectories is strictly zero (d(E) = 0), bridging finite algorithmic acceleration with absolute analytical rigor.