Bicomplex Spatio-Temporal Residual Encoder

Integrate this module at the input of a distributed-sensor graph neural network or state-space model. Assume measurements y_i[n] from sensors at positions x_i = i Delta x, subtract a slowly varying baseline, and form a residual window of length T. Compute a differentiable approximation to the double transform: multiply each time sample by exp(-sigma_t n Delta t) and a temporal phase factor exp(-B_t omega_m n Delta t), apply an FFT over time, then apply a discrete Fourier transform over sensor index using spatial frequencies k_l = 2 pi l divided by N Delta x. Represent each coefficient by four real channels corresponding to the scalar, B_t, B_x, and B_t B_x components, and feed these channels to a small graph or temporal encoder. Add a fault-classification head and a location head. The location head can regress x_f directly, or estimate it from the spatial phase after learning the nuisance temporal factor. A practical training objective is class cross-entropy plus Huber location loss and an optional phase-consistency penalty that compares the observed spatial phase with the predicted temporal phase minus k_l times the predicted location. Pseudocode is: residual r equals y minus EMA(y); Z[m,l] equals the sum over n and i of r[n,i] times the temporal exponential times the spatial exponential; h equals Encoder of the real geometric components of Z; optimize classification loss plus location loss plus phase loss. The transform identities and Nyquist condition are paper-derived; damping values, frequency bins, phase unwrapping, and nuisance-factor estimation must be selected empirically. The first cheap experiment should use a one-dimensional synthetic wave line with 16 sensors, injected pulses at uniformly sampled locations, and additive Gaussian noise. Compare against a temporal CNN and a graph-temporal model. The falsifiable prediction is that localization error decreases approximately as 1 divided by spatial frequency magnitude times the square root of SNR until the spatial Nyquist boundary k = pi divided by Delta x, while deliberately injecting frequencies above that boundary produces an abrupt aliasing transition. A second prediction is that geometric temporal-spatial channels yield lower mutual confusion between perturbations with identical temporal waveforms but different locations than a single ordinary complex transform.