First Fall Is Not Solving: An Exact Degree Classification of the Quadratic Semaev–Weil Family

Does the first fall degree really predict the solving degree of the Weil-restricted Semaev systems used in elliptic-curve index calculus? In P1.3, we give an exact degree classification for the quadratic Semaev–Weil family, prove an unconditional solving-degree bound, and construct an infinite family showing that field-equation nonredundancy is not automatic. The honest conclusion: first fall is not solving. The gap between them can grow without bound—but the classification currently applies only to the n=m=2 family in odd characteristic. First Fall Is Not Solving: An Exact Degree Classification of the Quadratic Semaev–Weil Family