Progressive Cumulative Counting – Intersections Between Bases and Algebraic Structure (Version 6)

This document presents Version 6 (V6) of the Progressive Cumulative Counting framework and studies intersections between sequences defined on different bases. For each base B ≥ 2, the sequence a_B(n) is constructed through a two-phase process: an initial linear phase generating consecutive odd numbers, followed by a structural transition after which the sequence evolves cumulatively as a(n) = a(n−1) + n. For n ≥ B − 1 the sequence admits the closed-form expression a_B(n) = n(n+1)/2 + K(B) where K(B) = (−B² + 7B − 8)/2. The main result characterizes intersections between two sequences associated with bases B1 and B2. For indices n ≥ B1 − 1 and m ≥ B2 − 1 the equality aB1(n) = aB2(m) is equivalent to the factorized equation (m − n)(m + n + 1) = (B2 − B1)(B1 + B2 − 7). This relation fully determines the possible intersections between sequences. In particular: • if B1 + B2 ≠ 7, only finitely many intersections may occur • if B1 + B2 = 7, the sequences coincide and infinitely many intersections appear The result reveals a simple algebraic structure governing intersections between base-dependent cumulative sequences.