Hyperbolic Dynamics and Asymptotic Density in Families of Primitive Pythagorean Triples with Fixed Leg Difference

Abstract Primitive Pythagorean triples with prescribed absolute leg difference \(n=|b-a|>0\) are studied under the convention that \(b\) is the even leg. For admissible \(n>1\), a canonical reduced representative of \(x^2-2y^2=n\) is used in the transformation \((x+y\sqrt2)^2(\sqrt2-1)\) to construct an explicit representative of \(U^2-2Z^2=-n^2\) that minimizes \(|U|\) within its associated orbital class. The exceptional case \(n=1\) is treated separately. A common minimal key yields an indexing by family, signed branch, and orbital depth, together with simultaneous closed formulas for both legs and the hypotenuse. Sharp uniform exploration bounds, finite exact-counting procedures, and an explicit floor-function formula for the number of triples with bounded hypotenuse in a fixed stratum are established. The enumeration agrees with an independent Euclidean enumeration for \(c\le5000\). Finally, classical prime-ideal equidistribution in \(\mathbb Q(\sqrt2)\) is specialized to the quadratic cutoff \(x_p^2\le M\) for reduced representatives of admissible prime norms, yielding the limiting proportion $\frac{\sqrt2}{\log(3+2\sqrt2)}$. These results provide a canonical framework for nonredundant enumeration, exact counting, and asymptotic analysis of primitive Pythagorean triples with fixed leg difference.