Certified Boolean Quantization: Seven-Bit Counterexamples and Diffuse Gaussian Limits

Coordinate selection is a natural benchmark for compressing independent binary variables observed through binary noise. We construct two explicit fourteen-input, seven-bit quantizers that beat this benchmark near complete noise, including an exactly balanced quantizer. We then analyze a diffuse block winner-take-all family whose maximum coordinate influence vanishes and whose Gaussian limit crosses the coordinate benchmark between seven and eight output bits. Directed arithmetic certifies finite eight-bit witnesses and their noise windows. We also prove influence asymptotics, robustness to unequal blocks, sufficiency of noisy scores, and uniform convergence of the full information curve. MSC 2020: 94A29, 60F05, 65G30