Hadamard conjecture: Paley over prime-power fields, times a power of two

This is the case of the Hadamard conjecture reachable by Paley's constructions over an arbitrary finite field (not merely a prime field), combined with Sylvester's doubling construction.

Let $q = p^{n}$ be a prime power with $n \ge 1$. Suppose $k$ satisfies either

for some $t \ge 0$. Then there exists a $4k \times 4k$ matrix $M$ with every entry $1$ or $-1$ attaining Hadamard's determinant bound $|\det M| = (4k)^{4k/2}$.

Role. Paley's constructions (Paley_construction_I and Paley_construction_II) are stated and proved for an arbitrary finite field of odd characteristic, but the earlier combination theorems on this platform instantiated them only at the prime fields $\mathbb{Z}/q\mathbb{Z}$. Instantiating instead at the Galois field $\mathrm{GF}(p^{n})$ strictly extends the covered orders. Since $n=1$ recovers the prime case, this theorem subsumes both Hadamard_Conjecture_paley_sylvester_case and Hadamard_Conjecture_paley2_sylvester_case.

The genuinely new orders come from proper prime powers. For example $q=25$ gives order $2(25+1)=52$, i.e. $k=13$ — and $k=13$ is not reachable from any prime via either Paley construction, since none of $51, 25, 12$ (Paley I) nor $25, 12$ (Paley II) is prime. Likewise $q=49$ gives $k=25$, $q=121$ gives $k=61$, $q=169$ gives $k=85$, and $q=343$ gives $k=86$.

Formalization Note The field of order $p^{n}$ is Mathlib's GaloisField p n; its Fintype instance is obtained from the derived Finite instance, and its cardinality from GaloisField.card (stated with Nat.card). The determinant bound follows internally from the orthogonality identity $MM^{\mathsf T}=4kI_{4k}$.