Critical Points of Line Restrictions of Signed Point-Charge Potentials

Abstract

Let \(α>0\) and restrict a finite inverse-power potential with arbitrary real coefficients to a line. Combine terms having the same projected centre and squared height \((a,b^2)\), delete classes whose coefficient sum is zero, and let \(m\) be the number of remaining classes. We prove the following dichotomy. If \(m=0\), exact cancellation occurs if and only if every class sum vanishes, and every ordinary point of the original domain is critical. If \(m\geq1\), there are at most \(2m-1\) critical points: when all effective heights are positive the zeros are counted with analytic multiplicity, while in the presence of effective sources on the line the assertion is one global distinct-point bound. This proves the signed line conjecture of Gabrielov--Novikov--Shapiro, valid throughout their range and in fact for every \(α>0\). The structural input is a projective paired Haar theorem: for finite \(β>1\), the full \(2m\)-dimensional space \(\sum L_j/Q_j^β\), with pairwise nonproportional positive-definite binary quadratics and arbitrary real linear numerators, has at most \(2m-1\) projective zeros counted with multiplicity. For positive charges, the substitution \(p=2α\) proves Conjecture~3 of Edelsbrunner--Fillmore--Oliveira throughout its stated range \(p\geq1\) and extends the same conclusion to every \(p>0\). For every \(n\), an explicit positive \(n\)-charge configuration attains \(2n-1\) simple critical points.