This paper presents ∴ ⁞ ∞ as a formal notation for the recursive structure of signal interaction in shared systems. Each mark is derived through reduction of a complete formal system: ATI (Alignment, Threshold, Infinity) establishes that sequence determines outcome (∴); Recursive Field Dynamics establishes that fields cross thresholds producing emergent states outside the span of inputs (⁞); Symbolic Systems Engineering establishes that symbolic environments carry meaning forward recursively without terminal state (∞). The three marks in sequence constitute a complete recursive loop. This paper traces the reduction of each proof to its irreducible mark, demonstrates their interaction as a single system, identifies universal instantiation across organizational, economic, computational, healthcare, and governance domains, and names the formal claim the stack collectively proves about the structure of intention, signal interaction, and emergent outcomes in shared environments.
D.L. Gee-Kay (Wed,) studied this question.