An Attempted Classical Proof of Fermat's Last Theorem via Diophantine Volume Bounds

We present a fully elementary proof of Fermat’s Last Theorem for all integers n > 2 using volume bounds on Diophantine identity sets. We define and analyze the set of positive integer solutions to the equation a^n + b^n = c^n with bounded variables, and show that its cardinality grows more slowly than the number of distinct exponentiated triples. By establishing a contradiction between the number of such solutions and the cost of expressing them under arithmetic constraints, we rule out the possibility of any nontrivial solution. The proof avoids the use of elliptic curves, modular forms, or analytic continuation, and instead relies on explicit counting,
additive structure analysis, and contradiction over encoding capacity. The argument is entirely classical and suitable for foundational number theory contexts.