Cryptic constructions of Hadamard matrices

On August 12, 2026, Levent Alpöge of Anthropic posted on X constructions for Hadamard matrices of orders $n \in \{668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964\}$, resolving all previously unknown orders below 2000 (but leaving the full conjecture open). A matrix $H \in \mathcal{M}^{n \times n}(\left\{ \pm 1\right\})$ is Hadamard of order $n$ if $$ H H^\intercal = n I_n .$$It is conjectured that such matrices exist for all orders $4n$, where $n$ is any positive integer. This is an open problem that has long intrigued me, and it’s exciting to see progress! However, the post was, in the unfortunate style du jour, a “vaguepost”: an intentionally cryptic post to bait users into engagement. This is a term that I hadn’t come across before but that I think is apropos (especially in light of the similar Jacobian conjecture resolution). To me, this was immediately reminiscent of the anonymous maths Stack Exchange user, “Cleo”, who posted closed-form solutions to intricate…