This paper studies how the geometry of a dynamical system is encoded in a spectral object called the diagonal landscape function, defined on a finite discrete torus equipped with the cat map. The main result gives an exact closed formula for the landscape in terms of the orbit structure of the map. The landscape is maximized uniquely at the origin, the unique fixed point of the system, whose existence follows from a single arithmetic identity that holds for every modulus simultaneously. This produces sharp spatial localization driven purely by arithmetic, with no disorder and no broken symmetry, a mechanism absent from classical localization theory. Further results include a trace formula linking the landscape to the dynamical zeta function, a complete spectral description of the operator, and a perturbation theorem showing how the localization degrades under an added Laplacian. All results are verified computationally to machine precision. This is the second paper in a three-part series. Paper I (doi:10.5281/zenodo.17866404) established the one-dimensional instance of the programme. Paper III provides the topological unification via Lefschetz numbers and cyclotomic fields.
Anup Chandra (Sun,) studied this question.