A Variational Framework for Fermat-Type Equations

This paper establishes a continuous variational framework to resolve Fermat-type equations ($x^n + y^n = z^n$). By evaluating the continuous geometric volume against the absolute rigidity of the discrete integer lattice, we expose a fundamental structural incompatibility. For $n \ge 3$, asymptotic fractional decay forces the available continuous space strictly below the minimum volume required by the discrete grid. This geometric deficit causes the underlying arithmetic space to physically collapse, permanently sealing the solution set as empty.