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

We define an integer valued recursion that uses divisor pair products on composite integers and a Goldbach type additive convolution on primes. We prove that its values admit a shape decomposition into a prime dependent factor and a universal coefficient depending only on the exponent pattern of the integer. These coefficients satisfy a recursion independent of the primes and admit a combinatorial interpretation as orbit counts of labelled unordered full binary trees under symmetric group actions, using the fact that tree automorphism groups are 2 groups. We also establish general bounds, supermultiplicativity under block union, exact closed forms for several infinite families, and related arithmetic corollaries.