Light from a distant source can reach us along different paths when gravity bends it through layers of matter. Each path can appear as a separate image. How many images can a collection of point masses produce when those masses lie in several lens planes?

This candidate supplies a tighter upper bound. For two planes containing two masses each, the previous general ceiling was forty-one images. The new bound lowers it to thirty-three. A published construction already produces twenty-five. The remaining gap, between twenty-five and thirty-three, is still unresolved.

The proof keeps track of where each ray crosses every plane. This preserves one fact: in the equations, a plane couples directly only to its immediate neighbours. Turning the equations into polynomials then leads to a counting problem along a path. A short recurrence evaluates the bound for any number of planes and any positive number of masses in each plane.

The improvement matters particularly as the number of planes grows. It does not turn the bound into a linear function of the mass counts, and it does not prove that any lens reaches the new ceiling.

The package contains a written proof and exact, reproducible checks. It also certifies twenty-five separate regular images in the earlier published example. That certificate establishes a lower example, not a complete catalogue of its images.

The assumptions exclude external shear and continuous matter, and only regular, unobstructed images are counted. The work remains an unrefereed candidate. Supplied ChatGPT checks are not human peer review or unaffiliated reproduction.

This is Evidence Press, the multiplane lensing sparse bound, dated the first of October twenty twenty-six. The paper and evidence are linked on the release page. This synthetic AI voice is explanatory, not additional scientific evidence.