Fixed-Threshold One-Bit Toeplitz Covariance Estimation under Sparse-Ruler Sampling

We study Toeplitz covariance estimation when fixed-threshold one-bit quantization is combined with deterministic sparse-ruler sampling, so that each observed bit is reused across many lag products. At a nonzero threshold the signs have nonzero mean, and this reuse gives raw sign products a coherent one-vertex variance component governed by weighted row sums; centering removes it and leaves a degenerate sparse-pair statistic. We prove a Gaussian variance contraction theorem for hollow quadratic forms of bounded coordinate transforms, including hard threshold signs: the variance is bounded by the squared correlation operator norm times the squared Frobenius norm of the edge weights, with constants independent of dimension, support size and maximum degree. For the oracle centered sparse-ruler estimator, the leading operator-norm term is \(\gamma_0L_1\kappa_{\rm obs}\sqrt{\varphi(\Omega)\log d/n}\), where \(\varphi(\Omega)=\sum_{s=1}^{d-1}q_s^{-1}\) is the coverage coefficient of the ruler; pooled marginal calibration from the \(n|\Omega|\) observed bits adds a plug-in term. A spectral-packing lower bound in a known-scale identity-neighborhood submodel shows that this dependence is intrinsic under balanced coverage geometry; in the non-saturated regime where the coverage term dominates, the oracle estimator is minimax rate optimal over this submodel.