How complicated can a space-filling arrangement of convex cells be? One proposed answer placed a particular average, called the harmonic degree, between three and four in three dimensions. This candidate supplies an exact counterexample.

The harmonic degree combines two counts: how many cells meet at a typical vertex, and how many vertices a typical cell has. In the constructed periodic mosaic, one repeating unit contributes three hundred and seventy-three vertices, four hundred and thirty-two cells, and three thousand two hundred and seventy-six incidences between them. The resulting harmonic degree is four hundred and sixty-eight divided by one hundred and fifteen: a little over four.

These are not just compatible numbers. The package supplies rational coordinates, supporting planes and a checked subdivision. Copies fit together along unchanged tetrahedral boundaries. A weighted construction also gives a genuine power diagram, often called a Laguerre diagram. It is not an ordinary unweighted Voronoi example.

The paper builds on established polytope constructions and an earlier method of inserting them into tilings. Its contribution is the explicit application to the proposed band, the exact certificate and further written refinements. Those refinements produce every harmonic degree between three and sixteen. They do not show that sixteen is the maximum.

The checks use exact arithmetic and deliberately damaged certificates. They are internal checks, not external peer review or formal verification of every infinite construction. Nor does an artificial counterexample tell us how frequently a structure appears in rocks.

The paper, evidence package and replay instructions are available on Evidence Press. This is synthetic speech.