Write \(D_{34}\) for the requested maximum. The certified bounds are \[ 5{,}070{,}942{,}774{,}902{,}496{,}337{,}890{,}625 =35^{16} \leq D_{34} \leq 6{,}876{,}227{,}375{,}063^2 =47{,}282{,}502{,}913{,}565{,}795{,}274{,}253{,}969. \] The lower endpoint is attained by the explicit order-34 matrix in `ssm34-artifact-gs36-principal-minor`. The upper endpoint follows for every matrix in the stated family. These sources and the computation do not determine equality at either endpoint, so the exact maximum remains open in this entry.

Goethals and Seidel give four circulant first rows of length nine: \[ \begin{aligned} a&=(0,+,+,-,+,-,+,-,-),\\ b&=(+,-,+,+,-,-,+,+,-),\\ c&=(-,-,+,+,+,+,+,+,-),\\ d&=(+,+,+,-,+,+,-,+,+). \end{aligned} \] Let \(A,B,C,D\) be the corresponding circulants, let \(R\) be the reversal matrix, and use their four-by-four block array. The replay verifies that the resulting \(H\) has entries in \(\{-1,1\}\), satisfies \(H+H^T=2I\), and obeys \(HH^T=36I\). Thus \(K=H-I\) is a skew conference matrix with \(KK^T=35I\).

Delete rows and columns 0 and 1 from \(K\). The remaining matrix \(S\) is a skew Seidel matrix of order 34. Exact Pfaffian elimination gives \[ \operatorname{pf}(S)=-2{,}251{,}875{,}390{,}625=-35^8, \] and fraction-free Bareiss elimination gives \[ \det S=5{,}070{,}942{,}774{,}902{,}496{,}337{,}890{,}625=35^{16}. \] This is also the order-34 case of the principal-submatrix lower bound in Theorem 5.5 of Klanderman et al.

Klanderman et al. define \(M(n)=\max\sqrt{\det S}\). Their Theorem 5.4, which collects the skew Ehlich-Wojtas result, gives \[ M(n)\leq\sqrt{2n-3}\,(n-3)^{(n-2)/4} \] when \(n\equiv2\pmod4\). At \(n=34\), this says \[ \det S\leq65\,31^{16} =47{,}282{,}502{,}913{,}567{,}042{,}148{,}851{,}265. \] Cayley's Pfaffian identity says \(\det S=\operatorname{pf}(S)^2\), and parity makes the Pfaffian odd. Exact integer square root gives \[ \left\lfloor\sqrt{65\,31^{16}}\right\rfloor =6{,}876{,}227{,}375{,}063. \] This integer is odd, while the next odd integer already has square larger than \(65\,31^{16}\). Hence every admissible determinant is at most \[ 6{,}876{,}227{,}375{,}063^2 =47{,}282{,}502{,}913{,}565{,}795{,}274{,}253{,}969. \] The equality form in the unrounded bound would require \(2n-3\) to be a square. Here \(65\) is not a square, which independently rules out equality in that real-valued bound.

The program fixes the circulant convention by setting entry \((i,j)\) to first-row entry \((j-i)\bmod9\). It builds the Goethals-Seidel array, checks all 1,296 entries of \(H+H^T=2I\) and \(HH^T=36I\), then checks the skew Seidel conditions after deletion.

The compact matrix certificate is the SHA-256 digest of its upper-triangle signs in row-major order, writing `+` for 1 and `-` for -1. The order-34 digest is `c081b626a079cca244e06357bacacad2a406f7512557c554a5dc78e0477bcd35`. Exact rational Pfaffian elimination and integer Bareiss elimination independently agree through \(\det S=\operatorname{pf}(S)^2\). The stable output digest is recorded with the program.