Let \[ C_{31}=\sup_{f\in\mathbb R^{C_{31}},\,\sum_x f(x)=0,\,f\ne0}\frac{\|Hf\|_4}{\|f\|_4}. \] The certified interval in this fixture is \[ 1.5693<C_{31}<1.6453. \] The lower inequality comes from the explicit rational vector in `ch31-artifact-interval-certificate`. Its interval-evaluated ratio is approximately 1.5693754353.

For the upper inequality, the convolution kernel is \[ h(j)=\frac{2}{31}\sum_{k=1}^{15}\sin\frac{2\pi kj}{31},\qquad h(0)=0. \] Its row \(\ell^1\)-norm is enclosed near 2.70676032010662 and is strictly below 2.707. The multiplier has modulus one away from the zero mode, so \(\|H\|_{2\to2}=1\). The convolution estimate gives \(\|H\|_{\infty\to\infty}\leq\|h\|_1<2.707\). Riesz-Thorin interpolation therefore yields \[ C_{31}\leq\sqrt{\|h\|_1}<\sqrt{2.707}<1.6453. \] The endpoints remain separated. This entry supplies a certified bracket rather than the requested ten-place sharp value.

For odd \(N=2m+1\), extend the transform to every complex function by sending the zero mode to zero. Its multiplier has absolute value zero or one, hence Parseval gives \[ \|H\|_{2\to2}=1. \] Fourier inversion writes it as convolution with \[ h_N(j)=\frac{2}{N}\sum_{k=1}^{m}\sin\frac{2\pi kj}{N}. \] Thus \(|Hf(x)|\leq\|h_N\|_1\|f\|_\infty\). Interpolating the whole-space bounds at exponents two and infinity gives \[ \|H\|_{4\to4}\leq\|h_N\|_1^{1/2}. \] Restriction to real zero-mean functions can only decrease the operator norm. At \(N=31\), interval evaluation gives \(\|h_{31}\|_1<2.707\), producing the upper endpoint in the bracket.

The candidate-generation run used one hundred seeded projected-gradient starts on the zero-mean subspace and reported the best ratio \[ 1.5693754353475171. \] The interval artifact records a rounded rational representative of that incumbent and certifies the weaker lower endpoint 1.5693.

The objective is a nonconvex homogeneous quartic ratio. Agreement among local searches does not exclude another critical orbit with a larger value. A complete polynomial-system analysis, a sum-of-squares certificate, or another global argument would be needed to close the interval and certify ten decimal places. A focused primary-source audit was not completed, so novelty also remains unverified.

The first thirty coordinates are terminating decimals and therefore exact rationals. The final coordinate is defined as minus their exact sum. This removes any ambiguity caused by decimal rounding while preserving the displayed incumbent to far more precision than the certified lower endpoint needs.

Sage's `RealIntervalField(200)` encloses pi, every sine, the kernel entries, the transformed vector, and both fourth-power sums with directed rounding. The assertions certify a ratio above 1.5693, a kernel row norm below 2.707, and the final interpolated upper bound below 1.6453. The calculation certifies this witness and the analytic bound. It does not search all stationary points.