Sparsity for Free: A Budget-Induced Equilibrium in Joint Topology–Parameter Search

Joint optimisation over a discrete topology $T$ and continuous parameters $\theta$ under a strict evaluation budget $B$ arises in neural architecture search, pruning, and variational quantum algorithms. We identify a budget-induced effect, dimensionality pressure: enlarging $T$ expands the search space of $\theta$, slows inner-loop optimisation at fixed $B$, and yields noisier fitness estimates that systematically bias outer-loop selection toward smaller structures, even without explicit sparsity regularisation. We formalise this as a Proposition composing standard CMA-ES dimensionality results with rank-based noisy-comparison selection, predicting a stationary distribution over $|T|$ concentrated below the dense baseline. To test the mechanism we introduce EvoluCMAES, a minimal two-loop framework whose only domain-specific component is the topology-mutation operator. Across two unrelated settings, the same fixed apparatus produces low-density solutions: in pruning of UCI tabular models, the equilibrium density settles at $\rho \approx 0.13$, roughly an order of magnitude below dense; in BB84 eavesdropping, discovered Eve circuits use 4–6 gates from an arbitrary-depth search space and approach the analytical Pauli-channel cloning bound under bit-flip noise. When a specific target sparsity is required for deployment, adapting only the mutation operator suffices: at $95\%$ sparsity on the UCI suite, mean accuracy reaches $0.972$, exceeding the strongest one-shot baseline at $0.946$. As a downstream consequence of the bound, the EvoluCMAES library is small enough to serve as a tractable action set for an off-the-shelf RL agent: a single Rainbow-DQN with feasibility masking and matched hyperparameters recovers $\approx 99\%$ of the dynamic-programming optimum on adaptive eavesdropping for both BB84 and E91/DIQKD.