Reset Dynamics, Paper 8: Address Calculus — Differential Operators as Address and Dimension Operators

This is Paper 8 in the Reset Dynamics series. **Title:** Address Calculus — Differential Operators as Address and Dimension Operators **Abstract:**We introduce Address Calculus in its operator formulation, where differential and integral operations are reinterpreted as algebraic operators acting on address space. In contrast to classical calculus based on limits and infinitesimals, we define dx as a fundamental address operator that performs dimensional transitions. Differentiation corresponds to multiplication by dx^{-1} (dimensional descent), and integration corresponds to multiplication by dx (dimensional ascent). We formalize value transport across dimensions via exchange laws, interpret factorial scaling as cumulative transport, and extend the framework to multi-dimensional, cyclic, and non-commutative address systems. This establishes calculus as a navigational algebra over structured space, compatible with the Generative Paradigm of Reset Dynamics. **Key contributions:**- dx as address operator (not infinitesimal)- Differentiation = dx^{-1}, Integration = dx- Value transport and factorial scaling- Cyclic and non-commutative extensions- Spectral interpretation as shift operators **Reset Dynamics Series Overview:**- Paper 1: Γ-Constrained Realizability and the Structural Inevitability of Non-Composability- Paper 2: Structural Closure and Generative Rupture- Paper 3: Division by Zero as Structural Rupture- Paper 4: Zero Is Not Atomic — Structured Zero and the Inevitability of Rupture- Paper 5: Welding Between Structured Zeros — From Rupture to Reconnection- Paper 6: Selection and Control in Rupture Dynamics- Paper 7: Structure Space and Addressable Memory- Paper 8 (this work): Address Calculus — Differential Operators as Address and Dimension Operators **Related Zenodo records:** - Paper 1: https://zenodo.org/records/19373151- Paper 2: https://zenodo.org/records/19609968- Paper 3: https://zenodo.org/records/19861236- Paper 4: https://zenodo.org/records/19923360- Paper 5: https://zenodo.org/records/19978542- Paper 6: https://zenodo.org/records/19979476- Paper 7: https://zenodo.org/records/19979922