Arrow Logic Algebra Boolean Gates Using the Complex Numbers, Quaternions and Octonions

Most of quantum computing today is written in the mathematics of quantum mechanics: complex vectors and the matrices that act on them. This mathematics has served physics superbly, yet it is not a Euclidean number system. Its imaginary unit came in as an afterthought, attached to the time derivative of Schrödinger's equation; two states of one system cannot be multiplied and divided back; and a turn about a general axis is a matrix, not a number. The arrow numbers, the complex numbers, quaternions and octonions, are such a system: lengths multiply, every nonzero arrow divides, and the Euler relation turns an arrow about any axis. What if we build a Boolean algebra on this Euclidean mathematics? With a bit written as a sign, ±1, the basis units multiply as the XOR of their labels; a half turn is NOT, and a quarter turn is its square root. Through zero, the sign of a sum of three inputs gives the majority gate, hence AND, OR and NAND. Reversibly, the quotient of the two orders of a quaternion product is the AND of two bits, AB(BA)⁻¹ = (−1)^(ab), which builds the Toffoli gate; the two groupings of an octonion product give the AND of three. Thus each doubling gives up one rule of arithmetic, and the quotient that measures it carries one more degree of AND. These identities are known; we read them as gates. In standard CMOS, NOT and every product by a fixed unit are wiring, the Toffoli gate takes fourteen transistors, and the product and the zero crossing make a full adder; SPICE simulations confirm every truth table. Submitted to the Journal of Applied Mathematics and Physics. The checks archive contains the Julia script of the algebra (59 tests), the switch-level circuit checks and the SPICE netlists and driver, each check with a control that must fail, and the logs of the runs.