Zero Domain Algebra (ZDA) proposes a formal algebraic framework for handling degenerate arithmetic operations — division and multiplication by zero — as first-class mathematical objects rather than undefined errors. Instead of propagating exceptions, ZDA assigns labeled elements in a parallel domain 𝒵, preserving full provenance of zero-related events through four core operations: fusion, erasure, projection, and partial resurrection. The framework establishes axiomatic foundations (associativity, commutativity, idempotence, partial monotonicity), constructs a canonical model with existence proof, and characterizes resurrection policies and their monotonicity properties. Key structural results include a non-homomorphism theorem, an emergence theorem for atomic composites, and a universal property of resurrection. Formal comparison with related approaches — partial algebras, error monads, annotated semirings, and wheel theory — demonstrates distinct advantages in information preservation. A complete Python prototype with property-based unit testing accompanies the theory, ensuring reproducibility and alignment between formalism and implementation.
François Gauthier (Wed,) studied this question.