Imagine arranging every number in a finite arithmetic system around a circle. There is one rule: add any two neighbours, and the answer must be a primitive element. A primitive element is a nonzero number whose successive powers visit every nonzero number in the system.

Sun and Hou asked whether such a circle always exists when the field has more than seven elements. Earlier work settled all sufficiently large fields, and all sufficiently large prime fields. This candidate closes the finite remainder.

There are seven hundred and twenty-eight field sizes to check. A mathematical bound settles six hundred and seventy-five. For the remaining fifty-three, the package gives actual circular orderings. A separate verifier checks that every field element appears exactly once and that every neighbouring sum has the required multiplicative order. It also checks the polynomial used to represent each field.

The smallest example on the page uses arithmetic modulo eleven. Its neighbouring sums are two, six, seven or eight: exactly the primitive elements in that field. This picture illustrates the rule; the downloadable certificates contain the full computational evidence.

The revised proof also gives a limited robustness result. Over the field with thirty-two elements, the bound still guarantees a circle after any six vertices are removed. This is a sufficient bound, not an optimal threshold.

The contribution needs perspective. Many small cases were already known, and the large-field theorem is inherited. This is a computer-assisted completion using established ideas. It remains an unrefereed candidate, without formal verification or independent external reproduction.

This is the Evidence Press briefing for the primitive-sum circle release, dated the twelfth of October, twenty twenty-six. The paper, code and certificates are linked on the page. This synthetic voice explanation is not additional evidence.