 Triangular weights count genuine four-term progressions

For $A\subseteq {\mathbb{Z}}_N$, $N\ge 16$, put $\eta =|A|/N$, $M=\lfloor N/16\rfloor$, $J=\{0,\dots ,M-1\}$, and $w=1_J\ast 1_J$ using normalized convolution on a finite group. The weights $w(x)w(d-M)$ are supported where $0\le x\le 2M-2$ and $M\le d\le 3M-2$, so $x,x+d,x+2d,x+3d$ are distinct integer points below $N$. If $t=M/N$, then $\mathbb{E}w=t^2$ and the sum of absolute values of its Fourier coefficients is $t$, by Parseval identity on a finite group. Expanding the four indicators around their subset density leaves four terms with one balanced factor. Expand both weights in Fourier series and absorb each character $e(rx+sd)$ into the factors at $x$ and $x+d$. Character multiplication preserves the Gowers U3 norm, and the cyclic four-term progression bound with torsion factors bounds every error by $6^{1/8}\alpha$. Thus the weighted count is at least ${\eta }^4{t}^4-4\cdot 6^{1/8}\alpha t^2$. Since $t\ge 1/32$, positivity follows from $\alpha <{\eta }^4/(4096\cdot 6^{1/8})$.