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.

Note: The core idea and underlying mechanism presented in this work were developed independently by the author. The mathematical formalization was refined with the support of assistive tools.