= Four-term progressions in a quadratically uniform interval set
{title2=$\alpha\ll\delta^{32}\implies\Lambda_4(1_A)>\text{diagonal count}$}
Balance the indicator by its density on the interval, and extend it by zero to a cyclic group of order greater than twelve times the interval length and coprime to six. An eighth-power interval cube bound $\alpha$ bounds every four-term counting error by $\alpha^{1/8}$ through three Cauchy–Schwarz steps. The density contribution is bounded below by a constant times $\delta^4$, and the diagonal contribution tends to zero with interval length. A sufficiently small constant times $\delta^{32}$ therefore guarantees a nonconstant four-term progression. Stating the cube normalization distinguishes this parameter from the norm convention.
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