Sharp detection and triangle likelihood limits for quadratic spherical kernel graphs

We study whether quadratic spherical geometry can be detected from a single observed graph. Latent positions are independent and uniform on ${\mathbb{S}}^{d-1}$; conditional on them, edges are independent with connection probability $K_{d,\beta }(t)=[1+\beta (d{t}^2-1)\mathrm{/}(d-1)]\mathrm{/}2$, where $0<\mathrm{\mid }\beta \mathrm{\mid }\le 1$. We test this model against $G(n,1\mathrm{/}2)$. Writing $N=(d-1)(d+2)\mathrm{/}2$ for the degree-two harmonic multiplicity, we prove that the total variation distance tends to one when $N\ll n^{3\mathrm{/}4}\mathrm{\mid }\beta {\mathrm{\mid }}^{3\mathrm{/}2}$ and to zero when $N\gg n^{3\mathrm{/}4}\mathrm{\mid }\beta {\mathrm{\mid }}^{3\mathrm{/}2}$. At criticality, if $N\to \mathrm{\infty }$ and ${\mathrm{\Delta }}_n=\sqrt{\left(\genfrac{}{}{0px}{}{n}{3}\right)}\mathrm{\mid }\beta {\mathrm{\mid }}^3\mathrm{/}{N}^2\to \mathrm{\Delta }\in (0,\mathrm{\infty })$, the signed triangle captures all asymptotic testing information and the total variation distance tends to $2\mathrm{\Phi }(\mathrm{\Delta }\mathrm{/}2)-1$, where $\mathrm{\Phi }$ is the standard normal distribution function. Polynomial finite-sample error bounds hold under polynomial signal scaling. The proof controls the likelihood of the dependent quadratic feature law through spectral recursions, covariance truncation and interpolation from a symmetric prior.