The EML Operator as a Universal Analog Primitive: Translinear Circuits as its Natural Physical Substrate

In 2026, Odrzywołek demonstrated that the binary operator eml(x, y) = e^x − ln y, together with the constant 1, is sufficient to generate all elementary functions of mathematics in the expressive sense — a result analogous to the role of the NAND gate in Boolean logic. The immediate reaction was scepticism: implementing EML in digital hardware is costly, since every operation involves computing an exponential and a logarithm. This paper argues that such scepticism follows from a mistaken choice of substrate. Bipolar junction transistors (BJTs) obey an exponential current–voltage law by device physics, making exp and ln operations that emerge directly from the component’s behaviour without algorithmic approximation. We argue that EML finds its natural physical implementation in translinear circuits — a family of analogue circuits formalised by Barrie Gilbert in 1975 — and that this combination constitutes the continuous, analogue counterpart of the NAND–MOSFET system of digital computing. Hybrid digital–analogue architectures are discussed as the most practical path forward. This paper is theoretical and conceptual in nature; experimental verification constitutes a research programme that is proposed but not executed here.