Local Hard-Sphere Poisson-Nernst-Planck Models for Ionic Channels with Permanent Charges

The paper states and justifies Theorem 4.1: (i) at Q=0 the first-order zero-current flux J11 equals M00, giving the sign of J11 from r^2−l^2; and (ii) for small nonzero Q it derives ∂J11/∂V<0, a geometry-dependent critical potential V^c<0 where the sign of ∂J11/∂Q flips, and the one-sided sign conclusions in (b)–(c). These are presented as direct consequences of equations (4.2)–(4.4) assembled from the matching system, with J11 linear in Q to first order, J11=(M00+M01 Q)/H(a) and M00 explicitly given; see the displayed Theorem 4.1 and the lead-in formulas (4.2)–(4.4) in the PDF . The candidate solution follows the same matching-and-expansion framework and uses an equivalent affine-in-(V,Q) representation J11(V,Q)=M00+κ(V^c−V)Q+O(Q^2) to read off all signs, which aligns with the paper’s conclusions. Two minor issues are worth noting: (1) the paper’s Theorem states J11=M00 at Q=0, while (4.2) shows J11=M00/H(a); H(a)>0 so signs agree but the normalization is inconsistent in notation (the text later restores the 1/H(a) factor) ; (2) the statement “for non-zero small Q, ∂J11/∂V<0” implicitly presumes Q has fixed positive sign, because the linearized dependence implies ∂J11/∂V is proportional to Q (the candidate flags this). These do not affect the main sign conclusions. Overall, both the paper and model are consistent and essentially use the same proof architecture.