This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical k-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-R symmetric signal tensor of order~k 3 and dimension~d 3 from a single Gaussian observation at signal-to-noise ratio~λ, through the profile maximum likelihood estimator, the MLE restricted to normalized rank-R tensors of coherence at least~κ. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite- (k, d) error bound recovers the asymptotically optimal rate~d k of Perry, Wein and Bandeira, with explicit dependence on the rank~R and the coherence~κ. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical k-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.
Azäis et al. (Tue,) studied this question.