What is proved

The main local result is Theorem thm:finite. It supplies:

1. one frozen integer, $N_0=12$;

2. a complete gate-list specification with no unspecified high-arity instruction;

3. exact gate and qubit counts;

4. a named resource polynomial;

5. an algebraic proof that every measured output has the complete correct prime list.

The proof actually covers all fourteen integers $2,\dots ,15$ with the same circuit. Thus the circuit description does not depend on which four-bit input is supplied.

This is an elementary finite benchmark. Its purpose is to meet the plan's certificate requirement without using the general Family 279 claim as a premise. Its construction is deliberately classical reversible computation inside the allowed quantum gate model. No quantum advantage is claimed.