When More Search Reduces Coverage: Population-Aware Diffusion Search

More verifier-guided diffusion search need not improve the generated population. On class-conditional ImageNet, increasing branching beyond an initial gain raises Top-$K$ ADM precision while reducing recall and worsening FID. Retained paths become parents of later candidates, linking selection to future exploration. We investigate this feedback with retention-schedule interventions and introduce Moment-QUBO, a rule that balances verifier quality with matching the candidate population's first two Inception-feature moments. Applying it throughout search improves FID beyond replacing only the final selection, showing that early retention decisions matter beyond terminal reranking. Across two independently seeded SiT-XL/2 curves and a DiT-XL/2 transfer, its advantage over Top-$K$ widens with branching at matched denoiser evaluations. At the largest SiT budget, Moment-QUBO reduces FID by over 32% relative to Top-$K$ in both runs. Benefits depend on representation and operating point: CLIP-CMMD favors determinantal selection in the matched-precision SiT comparison, and ordinary DiT sampling offers lower FID and observed cost at lower ADM precision. These findings identify population retention as a design choice for mitigating adverse scaling in repeated diffusion search.