Projective Duality in Descriptive Syntax for Engineering Design From Ideal Circles to Band-Limited Stability

This technical report introduces a novel design framework for gear architecture inspired by the In'ei Spiral — a topological form arising from bounded non-closure and failed centering. By applying projective duality in descriptive syntax, the gear's tooth profile is redesigned to accommodate structural fluctuations rather than idealizing circular motion. The report presents theoretical background, design principles, and early-stage modeling visuals that explore how asymmetry can encode and absorb real-world vibrational drift in rotational systems.