Hadamard conjecture, residual case after Sylvester and Paley over all finite fields

This is the residual case of the Hadamard conjecture left open after removing everything reachable by Sylvester's doubling construction and by Paley's two constructions over arbitrary finite fields.

For a natural number $k$ satisfying both:

1. $k$ is not a power of two, and
2. there is no prime power $q=p^{n}$ ($n\ge1$) and $t\ge0$ with either $4k=(q+1)2^{t}$ and $q\equiv3\pmod4$, or $4k=2(q+1)2^{t}$ and $q\equiv1\pmod4$,

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 after the classical Sylvester and Paley families — now taken over all finite fields, not just prime fields — have been subtracted out, each having been proved unconditionally (Hadamard_Conjecture_paley_primepower_case).

Where this frontier sits. The smallest $k$ satisfying both hypotheses is $k=23$, i.e. order $92$. This matches the historical record exactly: order $92$ was the first order not settled by the Sylvester and Paley constructions, and it was resolved only in 1962 by Baumert, Golomb and Hall using Williamson matrices found by computer search. Closing further cases therefore requires genuinely different constructions — Williamson matrices, Turyn's construction, Baumert–Hall arrays, and others — none of which, nor any known finite combination, is known to cover every remaining $k$. This statement is exactly as open as the general Hadamard conjecture restricted to this 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.