Zero Divisors in Sedonion Algebra under Floating-Point Arithmetic

Author: Jack Kowalski <jack@entropment.com>

Zero Divisors in Sedonion Algebra under Floating-Point Arithmetic

In exact Cayley–Dickson algebras over \mathbb 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:

| \text{component} | < \varepsilon

where \varepsilon is the machine epsilon at the working scale.

Crucially, \varepsilon 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 \mathbb R.

The observed "non-zero residue" is an invariant of the projection history, not a violation of the algebraic construction.