The circulant Hadamard conjecture: real circulant Hadamard matrices exist only in orders 1 and 4

Theorem 1.1: a real circulant Hadamard matrix of order $n\ge1$ exists if and only if $n\in\{1,4\}$. Section 2 reproves Turyn's restriction $n=4u^2$ with $u$ odd, and the main argument excludes every odd $u>1$ through an alternating product of character values over the primes dividing $u$. Corollary 1.2 deduces the Barker-sequence classification. It says nothing about the (non-circulant) Hadamard conjecture on existence in every order divisible by 4, nor about complex or group-developed variants.

Theorem 1.1: a real circulant Hadamard matrix of order $n\ge1$ exists if and only if $n\in\{1,4\}$. Section 2 reproves Turyn's restriction $n=4u^2$ with $u$ odd, and the main argument excludes every odd $u>1$ through an alternating product of character values over the primes dividing $u$. Corollary 1.2 deduces the Barker-sequence classification. It says nothing about the (non-circulant) Hadamard conjecture on existence in every order divisible by 4, nor about complex or group-developed variants.