Data Entropment Labs https://entropment.com Author: Jack Kowalski Patent Pending Zero Divisors in Sedonion Algebra under Floating-Point Arithmetic ---------------------------------------------------------------- In exact Cayley–Dickson algebras over ℝ, zero divisors are well-defined, algebraically stable objects: nonzero elements whose product vanishes exactly. This property relies on exact equality between multiple coupled components across dimensions. When the same algebra is implemented using IEEE-754 floating-point arithmetic, this stability is lost. ### Reason Each multiplication step (sedonion → octonion → quaternion) introduces: - multiple floating-point multiplications, - multiple floating-point additions and subtractions, - independent rounding at each operation. As a result, algebraic equalities required for exact zero divisors are replaced by inequalities of the form: |component| < ε where ε is the machine epsilon at the working scale. Crucially, ε is not a neutral approximation error but a structural element of the numerical algebra. ### Consequence Zero divisors no longer behave as fixed algebraic points. Instead, they become dynamic numerical structures that: - drift toward zero under some rounding histories, - are repelled from zero under others, - may cross below epsilon (numerical annihilation), or remain finite depending on accumulated perturbations. Thus, in floating-point arithmetic: zero divisors are not static objects, but evolving orbits under projection. ### Practical Implication Repeated quaternion (or higher CD) multiplications do not converge to exact zero even when the corresponding real-algebra product would. This behavior is not a bug, noise, or instability. It is a direct consequence of treating floating-point arithmetic as a first-class algebra rather than an approximation of ℝ. The observed “non-zero residue” is an invariant of the projection history, not a violation of the algebraic construction.