Progressive Cumulative Counting with Base-10 Transition and Layered Structure (Version 4)

This document presents Version 4 (V4) of the Progressive Cumulative Counting sequence, defined by a two-phase deterministic process. In the initial phase (1 <= n <= 9), the sequence generates consecutive odd numbers:a(n) = 2n - 1. After the transition at n = 9, the sequence becomes cumulative:a(n) = a(n-1) + n for n >= 10. From this point onward, the sequence admits a closed-form expression:a(n) = n(n+1)/2 - 19. Thus, after a finite initial phase, the sequence coincides with a shifted triangular number sequence, providing a simple example of a discrete system with a structural transition from linear to quadratic growth. This version refines previous formulations and clarifies the mathematical structure of the sequence.