Tagged-Nullity Arithmetic: A Non-Contradictory System for Division by Zero with Provenance Tracking

Division by zero is handled in mainstream computing by absorbing error values (NaN in IEEE 754) that destroy information about the error's origin. We propose tagged-nullity arithmetic: x/0 for x ≠ 0 produces a tagged object ⊥(x) that retains the dividend (the "tag") and its provenance history, while 0/0 = 0 by convention (following meadows). The original intuition — that the tag can be "returned" by multiplying by zero, ⊥(x)×0 = x — is shown to collapse arithmetic (it implies 2 = 1) unless associativity is abandoned, which destroys expression equivalence and testability (Richardson's theorem). We resolve this by making tag recovery a dedicated operation extract(⊥(x)) = x rather than multiplication. The resulting system is consistent, preserves ordinary algebra, and provides error traceability useful for debugging numerical software. A reference Python implementation with tests is included. This upload includes both the theoretical paper (PDF) and the reference implementation (Python script) as supplementary material