Imagine two junctions connected by three separate routes. That is a theta graph. A proper colouring assigns colours to the vertices so that neighbours never share a colour. Mathematicians can collect every possible colouring into one algebraic object, called the chromatic symmetric function.

A graph is called e-positive when this object can be written using a particular family of elementary symmetric functions without any negative coefficients. The question is not whether colourings exist. It asks for a deeper positivity hidden inside all those colourings together.

Earlier work settled theta graphs whose shortest route has one edge or two. This candidate proves the next case: one route has three edges, while the other two may be arbitrarily long, provided neither is shorter.

The proof starts from an expansion already established by Chen, He and Wang. Some individual terms are negative. Rearranging their parts leaves the underlying symmetric function unchanged, so the negative contributions can be grouped with positive ones. But several negative terms may arrive at the same target. The proof must cover their combined cost.

For one small example, a positive contribution of sixty absorbs two contributions of minus eight, leaving forty-four. A gap calculation controls every group in the infinite family. The revision also proves exactly when one class of groups cancels to zero.

Exact computations check more than nine hundred and seventy thousand compositions through twenty-four vertices. These support the written argument; they do not replace its universal proof. The general theta-graph conjecture remains unresolved here.

This is an unrefereed Evidence Press candidate dated the twelfth of October, twenty twenty-six. The manuscript and replay code are linked on the page. This synthetic voice briefing is not additional scientific evidence.