= Triangular weights count genuine four-term progressions
{title2=$\text{weighted count}\geq\eta^4t^4-4\cdot6^{1/8}\alpha t^2$}
For $A\subseteq\mathbb Z_N$, $N\geq16$, put $\eta=|A|/N$, $M=\lfloor N/16\rfloor$, $J=\{0,\ldots,M-1\}$, and $w=1_J*1_J$ using <normalized convolution on a finite group>. The weights $w(x)w(d-M)$ are supported where $0\leq x\leq2M-2$ and $M\leq d\leq3M-2$, so $x,x+d,x+2d,x+3d$ are distinct integer points below $N$. If $t=M/N$, then $\mathbb Ew=t^2$ 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 $\eta^4t^4-4\cdot6^{1/8}\alpha t^2$. Since $t\geq1/32$, positivity follows from $\alpha<\eta^4/(4096\cdot6^{1/8})$.
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