The complete cubic Walsh spectrum of a permutation-inverse Boolean family

Let $q=2^e$ with $e\ge 2$ even, put $d=(q^2+q+1)\mathrm{/}3$, and let $\sigma (X)=X+X^d+X^{dq}$ be the permutation of ${\mathbb{F}}_{q^2}$ introduced by Ding, Qu, Wang, Yuan, and Yuan. For $\alpha \in {\mathbb{F}}_q^{\ast }$, define the Boolean function $$f_{\alpha }(x)={\mathrm{Tr}}_{q^2}(\alpha ({\sigma }^{-1}(x){)}^3),x\in {\mathbb{F}}_{q^2}\mathrm{.}$$ In this paper, we determine the complete Walsh distribution of $f_{\alpha }$ in the remaining cubic case $\alpha \in ({\mathbb{F}}_q^{\ast }{)}^3$. More precisely, these functions are not bent but are $2$-plateaued: their Walsh values are precisely $0$ and $\pm 2q$, with exact multiplicities. The main new tool is a completion method for the outside Walsh coefficients: the punctured Fourier transform arising from the outside reduction is filled on the missing line, a modification invisible to outside frequencies, and the completed function is then identified with a Boolean component of a Kasami APN monomial. The APN property supplies a fourth-moment identity which, together with the known subfield spectrum and a Hasse divisibility congruence, forces the pointwise cubic spectrum.