 Solution

Let $A\subseteq {\mathbb{F}}_5^n$ have density at least $\delta$ and put $f=1_A-\mathbb{E}{1}_A$. If $A$ has too few four-term progressions, part ii forces $\Vert f{\Vert }_{U^3}$ to be large. Part iii then produces a frequency graph with large additive energy. The Balog-Szemerédi-Gowers theorem extracts a large piece with small doubling, and a finite-field Freiman theorem makes the frequency selection approximately affine-linear there. Integrating these approximately linear derivative frequencies produces correlation of $f$ with a quadratic phase, as in the inverse theorem for the Gowers U3 norm over a finite field.

Restricting to a suitable level set of that quadratic phase produces a density increment on a structured affine subspace. Iterating these increments cannot continue indefinitely because density is at most one. Once $n$ is sufficiently large in terms of $\delta$, the iteration must instead terminate with the expected nontrivial four-term progression. This is the density-increment proof of the finite-field four-term progression theorem.