Uniform Species Architecture Sampler

Integrate this at the neural architecture-search population generator, before model training. Define an architecture grammar with leaf L, unary node U(X) such as a residual or graph-convolution block, binary node B(X,Y) such as concatenation followed by projection, and a three-way invariant node T(X,Y,Z) such as sum aggregation. Let size count trainable blocks, and let a[n] be the number of canonical architecture trees of size n. Compute all counts up to budget N by dynamic programming. For each constructor r with child sizes n1,...,nk, compute its completion count w_r as the product of the child counts. For a requested total size n, enumerate feasible constructor and child-size tuples and choose each tuple with probability proportional to its number of completions. Recurse until leaves, then assign operation labels from a finite set and canonicalize commutative children by sorting their serialized hashes. For the symmetric three-way merge, enumerate unordered triples directly or use the cycle-index correction (A(z)^3+2A(z^3))/3. Cache sampled subtrees and reject only tensor-shape-invalid combinations, or include shape and type as additional DP states. The exact mathematical components are the coefficient recursion, symmetry quotient, and completion-weighted sampling; operation validity and downstream accuracy are empirical. First compare 1,000 architectures on CIFAR-10 using a small GraphNAS-style GNN or eight-block image model against uniform random mutation and ordered random trees at equal training FLOPs. Measure distinct canonical structures, validation accuracy after a fixed budget, and best accuracy versus evaluated architectures. Success is higher best and median accuracy with lower variance at small search budgets.