Fermat's Last Theorem: A Physical Constructivist Proof

This paper presents a proof of Fermat's Last Theorem within the framework of physical constructivism and Minimal Physically Derivable Theories (MPDT), established by the Uniqueness Theorem (Burnstein 2026a). Under physical constructivism, mathematical objects exist only when their construction has been physically completed; mathematical notation is prescription rather than number. Fermat's Last Theorem is reframed not as a claim about the non-existence of integers in an infinite domain, but as a claim about the non-existence of a construction mode M capable of mapping the construction mode xⁿ + yⁿ into the construction mode zⁿ while preserving integer structure for fixed n > 2. The proof turns on the structural incompatibility of these two construction modes: for n > 2, the construction mode xⁿ + yⁿ introduces an additive step at the top level — after all multiplicative chaining is complete — that is absent from the construction mode zⁿ, whose additive steps appear only at the primitive base. Since for a fixed n each construction mode has exactly one available path, and the categorical irreversibility of dynamics in discrete quantum-geometrical space entails that the same output must arise from the same causal sequence, two construction modes with different unique primitive step sequences cannot produce the same completed integer construction. No valid M therefore exists for n > 2. The proof identifies n = 2 as the unique satisfiable case, where both construction modes operate entirely within the primitive addition layer and no structural incompatibility arises. The result is an instance of the broader principle running through the MPDT paper series: the distinction between what a formalism permits and what the physical universe instantiates.