Exactly Unbiased Single-Stage Randomized Hadamard Companders: Uniform High-Rate Theory and Gaussian Tradeoffs

For every continuous positive probability density f and every B = 2^b >= 4, we construct a B-ary companded scalar quantizer whose decoder uses a reconstruction map satisfying an exact moving-average identity. The resulting reconstruction is unbiased at every deterministic scalar input. We place this quantizer after one randomized Hadamard transform with affine pairwise-independent coordinate dithers. For unit-norm signals, exact finite-rate identities reduce vector reconstruction and the mean-squared error of one fixed inner-product query to an unweighted scalar Rademacher risk and a scalar Rademacher risk with a correlated quadratic weight. For the symmetric Hadamard-admissible density class, a central-comparison and tail argument yields a high-rate approximation that is uniform over all finite dimensions and all signal and query directions. For the Gaussian point density with variance v > 2, the resulting functionals are C_rec(v) = (pi/6) v^(3/2)/(v - 2)^(1/2) and C_IP(v) = (pi/6) v^(5/2)/(v - 2)^(3/2). Both suprema over dimension are approached by flat aligned Rademacher sequences. Within the Gaussian point-density family, reconstruction is uniquely minimized at v = 3, fixed-query inner-product estimation is uniquely minimized at v = 5, and the Pareto-efficient set is [3, 5]. Moving from v = 3 to v = 5 lowers the sharp inner-product leading constant by approximately 30.99% and raises the reconstruction leading constant by approximately 24.23%. Exact unbiasedness is a finite-rate statement, whereas the risk comparison is a high-rate result for the fixed exactly unbiased single-stage architecture; the optimization varies only v within the Gaussian point-density family.