Reset Dynamics, Paper 3: Division by Zero as Structural Rupture: Irreducibility, Branching, and Phase Transition

This is Paper 3 in the Reset Dynamics series (3/3). **Title:** Division by Zero as Structural Rupture: Irreducibility, Branching, and Phase Transition **Abstract:**We propose a structural reinterpretation of division by zero as a rupture operator rather than an undefined arithmetic expression. Instead of assigning a scalar value, division by zero generates a branching space of structures: x ÷ 0 := Gen(R, Γ, x), where Gen enumerates all reachable structures under rule set R and resource bound Γ. We prove an Irreducibility Theorem, showing that no operator within a single algebraic universe (including extensions such as wheels, meadows, or compactifications) can faithfully reproduce rupture behavior. The obstruction is structural: rupture produces multiple non-embeddable worlds, not multiple values within a shared space. We further establish a Branch Growth Law and a Phase Transition Theorem, demonstrating that rupture emerges beyond a critical resource threshold. Classical arithmetic is recovered as a single-branch projection of this broader framework. Apparent paradoxes (e.g., division by zero leading to 1=2) are reinterpreted as illegal cross-branch projections rather than intrinsic inconsistencies. **Division by zero does not fail because a value is missing, but because the result no longer belongs to the same ontological category as its inputs.** --- **Reset Dynamics Series Overview:** - **Paper 1:** Γ-Constrained Realizability and the Structural Inevitability of Non-Composability Introduced the core concept of resource-constrained realizability, established the necessity of a reset operator, and identified "0", "1", and "∞" as prototypical structural phase boundaries. - **Paper 2:** Structural Closure and Generative Rupture Formalized the dynamics of structural closure and generative rupture, distinguishing well-founded state spaces from rupture-triggered expansion. Established the core duality: collapse (x·0) reduces to a closure unit, while rupture (x÷0) expands into structured possibility spaces. - **Paper 3 (this work):** Division by Zero as Structural Rupture Takes the rupture operator as its central object, demonstrating that classical division by zero—the prototypical "undefined" operation—is a powerful instance of structural rupture. Proves irreducibility, formulates branching growth laws, identifies phase transitions, and resolves classical paradoxes. --- **Keywords:** division by zero, structural rupture, generative systems, non-invertibility, branching dynamics, phase transition, Reset Dynamics **License:** CC BY 4.0 (recommended for maximum reuse) **Related Zenodo records:** - Paper 1: https://zenodo.org/records/19373151 - Paper 2: https://zenodo.org/records/19609968