STRUCTURED EXTENSIONS AND MULTI-CORRELATION SEQUENCES

The uploaded paper proves the strong decomposition: every k-fold multi-correlation sequence c(n) is a sum of a degree-k generalized nilsequence and a null-sequence (Theorem 1.6; see the abstract and main theorem statements) . By contrast, the model’s argument relies on a misstatement of Frantzikinakis (2015): that work provides only a weak, ε-dependent approximation by a nilsequence plus an error whose L2-density can be made ≤ ε (not a single, fixed null-sequence) . The model then incorrectly promotes this to a single decomposition c(n)=b(n)+a(n) with a(n) null by assuming an ε-independent “uniform-density-small” error; this step is not justified by [13] as summarized in the paper. While the model’s smooth/continuous nilsequence equivalence is correct (remark in the paper) , the crucial decomposition claim is unsupported. Hence the paper is correct and the model’s proof is flawed.