Wavelet-Based Parity Detection Revisited: Representation Dependence, Generalization, and Mechanistic Analysis

Parity is exactly determined by the least significant bit (LSB), so machine learning is unnecessary for solving the task. This work instead uses parity as a controlled diagnostic for studying how a classical signal-processing pipeline preserves, amplifies, or suppresses access to symbolic information embedded in a numerical representation. Integers from 0 to 10,000 are encoded as fixed-width 32-bit binary signals, decomposed with a level-3 Daubechies-2 (db2) discrete wavelet transform, summarized by mean absolute coefficient magnitude, and clustered independently per subband with k-means. Clustering is unsupervised, while cluster-to-parity calibration uses training labels only under a stratified 60/20/20 train/validation/test protocol. The frozen configuration reaches 84.26% accuracy on the held-out test set (95% Wilson CI: 82.60-85.79%), and 84.20% +/- 0.57% across 20 stratified resplits. Raw binary summary statistics reach 61.30%, while masking the LSB reduces validation accuracy to 48.15%, consistent with chance. The A3 approximation band alone retains 83.20%, whereas detail bands remain near chance. Performance is strongly representation-dependent: moving the parity bit changes validation accuracy up to 98.60%, and changing the wavelet boundary mode ranges from 54.45% to 83.20%. Cross-magnitude extrapolation degrades from 79.98% on 10,001-20,000 to 59.69% on 100,001-1,000,000. These results do not show that wavelets discover the arithmetic rule of parity. Rather, they show that information already present in the binary encoding becomes more or less recoverable depending on multiscale filtering, spatial alignment, and boundary handling.