Alfic Algebra IV: Inter-System Isolation and the Sigma Notation

his paper extends Alfic Algebra (Papers I–III) to address the structural relationship between numbers belonging to different systems. Building exclusively on the foundational transformation x = x−1 and the Alfic operations established in previous papers, we prove that different systems in Alfic Algebra are genuinely isolated — no algebraic operation passes between them. The Sigma notation (Σ) is introduced not as an operation but as a recording of this isolation: numbers from different systems stand alongside each other unchanged. A rule for combining Sigma with Alfic addition is established: numbers sharing the same system merge through Alfic addition; numbers from different systems remain recorded separately. The paper identifies discrete rotation as a third structural category — neither classical continuity nor Alfic isolation — and marks it as an open question requiring independent construction. The paper is presented in both Arabic and English.