Two neighbouring polynomials can each have well-behaved roots without their roots fitting together in a useful order. This candidate proves something stronger for polynomials arising from the inverse of a quartic. Their roots strictly alternate, and the first belongs to the even-indexed polynomial. That means the two polynomials never share a boundary root. The proof uses classical Jacobi polynomials and finite free convolution. One comparison follows from signs. The harder comparison uses an explicit positive determinant to rule out root collisions during a continuous deformation. This is a uniform written argument, not an extrapolation from computer examples. There is also a geometric consequence. Combined with a separately cited rigidity theorem by liqsweep and earlier differential identities, the result describes every associated affine intersection. All its points are simple, two important quantities never vanish, and the number of geometric points is the integer part of d times d plus one, divided by six. A special boundary orbit explains the fractional correction. This does not say that every point has rational coordinates or provide their explicit locations. The dependency matters: the boundary ordering theorem does not use rigidity, but the affine count and simplicity do. The package includes the paper, exact symbolic checks, finite diagnostics, rejection controls and internal editorial records. Those checks are not formal verification or independent specialist approval. This is an unrefereed Evidence Press candidate dated September twelfth, twenty twenty-six. The full paper and evidence are linked on the release page. This is an AI-generated voice summary, not additional mathematical evidence.