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

For the Franel numbers fm = ∑m
k=0
(m
k
)3, we prove the conjec-
tured limit H1/n2
n → 9/4 for Hn = det(fi+j )0≤i,j≤n, with explicit
finite-order bounds, using their triangular-lattice spectral measure
on [−1, 8]. The density has an interior logarithmic van Hove singu-
larity; its angular logarithm belongs to H1/2(T) but to no strictly
higher Sobolev space. Ordinary Szegő theory also determines the
linear term and fixed-shift asymptotics. We establish the full finer
expansion
log Hn = n(n + 1) log 9
4 + (n + 1) log A
− 1
4 log n + Cunif + 1
2
∑
k≥1
k|Vk |2 + o(1),
with explicit A, Cunif , and Fourier coefficients Vk . An oblique
Legendre-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 in
Geronimo’s 1979 proof. We also prove sub-spacing bounds and
fixed-weight sine-kernel universality. An exact finite-torus honey-
comb adjacency-trace model links all Franel moments and minors
to the carbon π-electron tight-binding spectrum.