When Averaging Keeps the Noise: A sensor puzzle about valid estimates and restrictive rules

Two sensors can have the same variance and respond very differently to averaging. Starting from this puzzle, we explain covariance, spectral density and the distinction between weighting readings, averaging frequency graphs and weighting estimated covariances. An elementary geometric argument shows why preserving one relationship can force a positive-definite lag window to preserve estimation noise at other lags. The resulting inequality holds on every dataset, including after the window has been chosen from that dataset. For a fixed nondegenerate stationary Gaussian process of finite covariance range with at least one nonzero positive-lag covariance, the optimal squared-error risk within this window class is bounded above and below by process-dependent multiples of n^(-4/5). An even, nonnegative estimate preserving the same empirical total variance can instead achieve an upper bound proportional to n^(-1). A uniform one-lag theorem includes white noise and describes the transition as the lag signal weakens. Worked examples and explanations of notation lead to complete proofs in the attached supplement. The rate exponent and repair principle are classical; the proposed contribution concerns arbitrary selected multipliers, dependent sample covariances and the uniform weak-signal comparison. This deposit contains the manuscript with its integrated technical supplement, the main LaTeX file, and a supporting archive containing the remaining sources, figures, computational code, numerical results and reproduction instructions.