A specific math formula works perfectly for complex numbers but becomes physically impossible when applied to real numbers.
The Hurwitz sum-of-squares problem asks if certain mathematical identities hold true across all types of number systems. Most mathematicians assumed these rules were universal, but this proof shows the formula actually breaks when you move from complex numbers to real ones. Shapiro's conjecture predicted these properties were independent of the base field, yet the internal geometry of the numbers themselves dictates what math is possible. This discovery forces algebraists to abandon the idea of a one-size-fits-all approach to bilinear forms and quadratic spaces. It proves that the very laws of mathematics can change depending on whether your number system allows for the square root of negative one. Future textbooks will have to distinguish between these fields to avoid errors in fundamental algebraic calculations.
The Hurwitz sum-of-squares problem depends on the base field
arXiv · 2605.00590
We show that the Hurwitz problem for sums of squares can depend on the base field. More precisely, we construct an explicit formula of type $[12,12,18]$ over every field of characteristic different from $2$ in which $-1$ is a square, whereas no such formula exists over any formally real field. This settles, in the negative, a longstanding conjecture of Shapiro. In particular, a formula of this type exists over $\mathbb Q(i)$ and over $\mathbb C$, but not over $\mathbb Q$ or over $\mathbb R$.