A Pell-Orbit Certificate Framework for Prime-Power Coordinates in Quadratic-Exponential Diophantine Equations

This paper develops a general certificate framework for solving quadratic-exponential Diophantine equations that reduce to generalized Pell equations with restricted coordinates. The central problem is to determine all solutions of X^2-dY^2=N subject to a prime-power or finite-prime-support condition on the Pell coordinate YYY, such as Y=cq^k or Y=c∏{j=1}ˢp_je_j. The method decomposes the generalized Pell equation into finitely many unit orbits, proves that all orbit coordinates satisfy a common second-order binary recurrence, and converts the restricted-coordinate condition into a finite family of logarithmic approximation problems. Explicit lower bounds for linear forms in logarithms provide an initial effective bound, while a simultaneous continued-fraction reduction can sharply reduce all active Pell orbits using a common convergent. The remaining cases are resolved through an exact terminal recurrence audit. The paper also defines a finite Pell-orbit certificate containing the orbit seeds, recurrence data, logarithmic bounds, continued-fraction margins, reduction thresholds, and final exact checks. A certificate soundness theorem shows that successful verification of these data proves completeness of the reported solution set. The framework is extended to coordinates supported on a fixed finite set of primes and to broader quadratic-exponential equations that split into factorization, local-obstruction, and restricted Pell-coordinate branches. A previously resolved exponential Diophantine equation is included as a worked example demonstrating how the framework transforms an unbounded exponent problem into a finite, independently verifiable classification.