Hadamard conjecture, residual case after Sylvester and Paley

This is the residual case of the Hadamard conjecture left open after removing both the Sylvester family and the Paley-times-power-of-two family.

For a natural number $k$ satisfying both:

1. $k$ is not a power of two (there is no $m$ with $k = 2^m$), and
2. $4k$ is not of the form $(q+1) \cdot 2^{t}$ for any prime $q \equiv 3 \pmod 4$ and any natural number $t$,

the claim is that there still exists a $4k \times 4k$ matrix $M$ with every entry $1$ or $-1$ attaining Hadamard's determinant bound $|\det M| = (4k)^{4k/2}$.

This is the honest remainder of the conjecture after two classical, unconditionally-proved construction families (Hadamard_Conjecture_paley_sylvester_case, itself built from Sylvester's 1867 doubling construction and Paley's 1933 finite-field construction via the Kronecker-product multiplicativity theorem) have been subtracted out. Many further classical constructions exist in the literature (Paley's second construction for $q \equiv 1 \pmod 4$, Williamson matrices, Turyn's construction, and others), each covering additional orders not captured by hypothesis (2) above; none of them, nor any known finite combination, is known to cover every remaining $k$. In particular the first genuinely difficult historical case, order $668 = 4 \times 167$ (resolved only in 2004 by Kharaghani and Tayfeh-Rezaie via a dedicated non-generic construction), lies in this residual family. No proof of this statement for all such $k$ is known; it is exactly as open as the general Hadamard conjecture restricted to this shrinking family.

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.