A partition breaks a whole number into an unordered sum. Three, for example, can be three, two plus one, or one plus one plus one. An overpartition adds a choice: the first occurrence of each distinct part may be marked. That gives eight overpartitions of three.

This collection studies two-colour versions. Think of two boxes of marked parts. The first box accepts any positive size. The second accepts only multiples of a chosen number, called the parameter. We count pairs whose combined total is the number we want.

The first paper gives exact certificates for four divisibility patterns. One count is always divisible by thirty-two, two by sixty-four, and one by one hundred and twenty-eight, along their specified arithmetic progressions. These statements come from earlier work. The contribution is a separate complete certificate, with careful attribution.

The second paper explains which parameters make every count at four n plus three divisible by eight. There is an exact answer: the parameter must leave remainder zero or one on division by four. The other two classes fail, and the proof supplies a counterexample for every excluded parameter.

There is also a larger family modulo sixteen. Remove all factors of two from the parameter. If what remains leaves remainder one on division by eight, then the required dyadic progression always works. This is a sufficient condition, not a complete classification. A related theorem connects divisibility with numbers that cannot be represented by either of two simple quadratic forms.

Why can a computer help prove an infinite statement? In the certificate paper, modular-form theory supplies a rigorous finite checking bound. The parameter results instead use written parity and support arguments. Exact computations check the recorded certificates and many examples, but examples alone are not the proof.

These are specialised number-theory results, with no practical application demonstrated. The two papers, code, assumptions and limitations are linked on Evidence Press. This is an unrefereed candidate collection dated the twelfth of October, twenty twenty-six. The synthetic briefing is not additional scientific evidence.