Progressive Cumulative Counting: Base Generalization and Structural Symmetry (Version 5)

This document presents Version 5 (V5) of the Progressive Cumulative Counting framework, extending the construction to an entire family of sequences parameterized by the base B. For each base B ≥ 2, the sequence a_B(n) is defined by a two-phase process: an initial linear phase generating consecutive odd numbers, followed by a structural transition at n = B - 1, after which the sequence evolves cumulatively as a(n) = a(n-1) + n. For n ≥ B - 1, the sequence admits a closed-form expression:a_B(n) = n(n+1)/2 + K(B),where K(B) = (-B^2 + 7B - 8)/2. A structural symmetry emerges: if two bases B1 and B2 satisfy B1 + B2 = 7, then the corresponding sequences coincide for all n ≥ max(B1 - 1, B2 - 1). This establishes a direct connection between base-dependent constructions and shifted triangular number sequences, revealing a simple but nontrivial algebraic structure.