Reset Dynamics, Paper 18: Residuum Algebra and the Logic of Structural Interference

We introduce residuum algebra as an algebraic framework for handling division-by-zero structures in non-invertible systems, building directly on Paper 16's residual memory fields and Paper 17's localization into discrete residua. Core contributions:- Formalization of division-by-zero as projection onto residual subspaces (always defined, never an error).- Residuum space as a quotient structure, incorporating Paper 17's empirical findings (short correlation length, hierarchy, lateral inhibition).- Algebraic operations: addition ⊕ (coexistence), multiplication ⊗ (sequential composition), division ⊘ (projection).- Composite decomposition theorem connecting to Extra 1's zero spectrum (prime residua correspond to prime Reset lengths).- Interference energy shift ΔE(π₁,π₂) quantifying emergence (whole ≠ sum of parts), with constructive (ΔE>0), additive (ΔE=0), and destructive (ΔE