Hadamard conjecture, residual non-power-of-two case

This is the residual case of the Hadamard conjecture left open after removing the classical Sylvester family.

For a natural number $k$ that is not a power of two (there is no $m$ with $k = 2^m$), the claim is that there exists a $4k \times 4k$ matrix $M$ with every entry equal to $1$ or $-1$ and attaining Hadamard's determinant bound:

This is exactly the statement Hadamard_Conjecture for $k$ restricted to the case where $4k$ is not a power of two times $4$ in the classical sense covered by Sylvester's 1867 doubling construction ($k = 2^m$, giving order $2^{m+2}$). Sylvester's construction settles the conjecture unconditionally whenever $k$ is a power of two; for every other $k$ (e.g. $k = 3, 5, 6, 7, 9, \ldots$), existence of a Hadamard matrix of order $4k$ is known only case by case (Paley, Williamson, and many other constructions cover most small and many large orders, and no counterexample is known), and no proof covering every remaining $k$ is known. This restricted statement is therefore exactly as open as the general Hadamard conjecture, minus the powers-of-two case.

Formalization Note Matrix entries live in $\mathbb{R}$; membership in $\{1,-1\}$ is expressed as Finset membership. The determinant exponent $4k/2$ is a real number via Real.rpow; since $4k$ is always even this equals the natural number $2k$, but the statement keeps the same real-exponent form as Hadamard_Conjecture for direct compatibility.