A Proposal for Addressing Division by Zero - From an operability question to a structural transposition framework

This research preprint presents the current state of a mathematical research program that began with a concrete question: whether division by zero could be given an operable treatment without invalidating ordinary arithmetic. The investigation progressively led to a broader structural problem: a scalar result does not necessarily preserve the information carried by the operation that produced it. The proposed framework therefore distinguishes scalar value from representational location by introducing an enlarged direct-sum state space. Within this space, zero-associated singular events are modeled through an elementary transposition operator ΦΦ, which transfers a conserved value from one representational component to another while preserving ordinary local arithmetic, including x⋅0=0 The paper develops the elementary definitions of this structural framework, connects ordinary division to redistribution between representational components, introduces a formal treatment of zero-denominator objects outside the ordinary field quotient, and studies conservation, elementary reversibility, projections, and the distinction between local and global state descriptions. The present work does not assert the scalar identity a/0=a and does not modify classical field arithmetic. It proposes instead a minimal structural hypothesis whose algebraic consistency, composition laws, extensions, and mathematical usefulness can be investigated independently.