Hadamard conjecture, residual case after Sylvester, Paley I, and Paley II

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

For a natural number $k$ satisfying all three of:

1. $k$ is not a power of two,
2. $4k$ is not of the form $(q+1)\cdot2^{t}$ for a prime $q\equiv3\pmod4$, and
3. $4k$ is not of the form $2(q+1)\cdot2^{t}$ for a prime $q\equiv1\pmod4$,

the claim is that there still exists a $4k\times4k$ 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 three classical, unconditionally-proved construction families (Hadamard_Conjecture_paley_sylvester_case from Paley's first construction, and Hadamard_Conjecture_paley2_sylvester_case from Paley's second construction, each combined with Sylvester's doubling construction via the Kronecker-product multiplicativity theorem) have been subtracted out. Further classical constructions exist in the literature (Williamson matrices, Turyn's construction, Baumert-Hall arrays, and others) that would shrink this residual further, but none of them, nor any known finite combination, is known to cover every remaining $k$. In particular the first genuinely difficult historical case, order $668=4\times167$ (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.