Strong Hankel Asymptotics for the Franel Numbers with an Interior Logarithmic Spectral Singularity

For the Franel numbers fm = ∑mk=0(mk)3, we prove the conjec-tured limit H1/n2n → 9/4 for Hn = det(fi+j )0≤i,j≤n, with explicitfinite-order bounds, using their triangular-lattice spectral measureon [−1, 8]. The density has an interior logarithmic van Hove singu-larity; its angular logarithm belongs to H1/2(T) but to no strictlyhigher Sobolev space. Ordinary Szegő theory also determines thelinear term and fixed-shift asymptotics. We establish the full finerexpansionlog Hn = n(n + 1) log 94 + (n + 1) log A− 14 log n + Cunif + 12∑k≥1k|Vk |2 + o(1),with explicit A, Cunif , and Fourier coefficients Vk . An obliqueLegendre-projection comparison and a uniform dyadic H1/2 esti-mate control the integrated two-point variance at all spatial scales,yielding the strong limit without relying on the reported gap inGeronimo’s 1979 proof. We also prove sub-spacing bounds andfixed-weight sine-kernel universality. An exact finite-torus honey-comb adjacency-trace model links all Franel moments and minorsto the carbon π-electron tight-binding spectrum.