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

This paper formally demonstrates why the scale-invariant shortcut architecture developed for the Collatz conjecture rigorously extends from finite computational bounds (10¹⁰⁰⁰⁰) to the infinite limit (N → ∞). By evaluating the bounded interval I_k = [1, 10^k], we establish that layer averages μ(k) = (L_1+L_2)/2 converge deterministically to 5 · 10ᵏ⁻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.