A Shape Decomposition for a Divisor–Goldbach Recursion and Closed Forms for Shape Coefficients

We define an integer recursion that uses divisor-pair products on composites and a Goldbach-type additive convolution on primes. We prove that the values admit a shape decomposition into a prime-dependent part and a universal coefficient depending only on the exponent pattern of the integer. These coefficients satisfy a prime-free recursion and have a combinatorial interpretation as an orbit count of labelled unordered full binary trees under symmetric-group actions, using the fact that tree automorphism groups are 2-groups. We obtain bounds, supermultiplicativity under block union, and exact closed forms for several infinite families, together with arithmetic corollaries.