On modular invariants of the truncated polynomial ring in rank four

We prove the rank-4 case of the conjecture of Ha-Hai-Nghia for the invariant subspace of the truncated polynomial ring ${\mathcal{Q}}_m(n)={\mathbb{F}}_q[x_1,\dots ,x_n]\mathrm{/}(x_1^{q^m},\dots ,x_n^{q^m}),$ under a new, explicit technical hypothesis. Our argument extends the determinant calculus for the delta operator by deriving crucial rank-4 identities governing its interaction with the Dickson algebra. We show that the proof of the conjecture reduces to a specific vanishing property, for which we introduce a sufficient condition, the “matching hypothesis” (${\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$), relating the degree structures of Dickson invariants. This condition is justified by theoretical arguments and verified computationally in many cases. Combining this approach with the normalized derivation approach from our prior work, we establish the conjecture. As a result, the Lewis-Reiner-Stanton Conjecture is also confirmed for rank four under the given hypothesis.