An Elementary Universal Step Function for Translating Programming Constructs into Pure Mathematics

This paper introduces a five-parameter Universal Step Function (USF) constructed using floor, ceiling, and absolute-value operations. The USF evaluates to 1 when a specified inequality holds and 0 otherwise, functioning as a real-valued expression within classical analysis rather than a meta-notational shorthand. Laws are established for strict/non-strict inequalities, Boolean operations, and number-theoretic conditions, enabling the closed-form translation of indexed lists, conditional branches, bounded loops, and recursive functions. Furthermore, the paper demonstrates direct algebraic applications including Abel summation and Euler-Maclaurin summation, where the indicator participates as a literal factor. This includes an explicit recovery of the squarefree-counting asymptotic Q(N) = 6N/π^2 + O(√N). In formal logic frameworks and system environments lacking native Boolean-to-numeric coercion, the USF eliminates the additional definitional steps required by the traditional Iverson bracket.