Delay-Aware Frequency-Preserving Recurrent Coupling

Integrate this into a recurrent or state-space architecture whose hidden state is updated in several coupled branches, for example four diagonal SSM blocks or four small GRU-like modules connected by a graph. Use a continuous-time-inspired Euler update x_i[t+1] = x_i[t] + h*(A*x_i[t] - B*K*sum_j L[i,j]*x_j[t-d]), where \(h\) is the integration step and \(d\) is an integer delay. First choose \(A\) with a lightly damped complex pair at target frequency \(\omega_0\), and train only a readout on synthetic mixtures of sinusoids so the effect of coupling is isolated. Implement two variants: ordinary delayed coupling and phase-compensated coupling using a two-tap predictor \(\widehat{x}_j[t+d]=x_j[t]+d(x_j[t]-x_j[t-1])\), or a learnable FIR filter constrained to have frequency response near \(e^{\mathrm{i}\omega_0\tau}\). The paper contributes the modal delay characteristic equation and the distinction between delay-invariant zero-frequency consensus and delay-sensitive periodic modes. Estimate \(A,B,K\) from module weights and estimate dominant roots by scanning complex \(s=\alpha+\mathrm{i}\omega\) for minima of the characteristic determinant; measure the output spectrum with an FFT. Sweep \(\tau\), coupling gain \(\gamma\), and target frequency. The falsifiable prediction is a loss of phase locking or a dominant-frequency shift once \(\omega_0\tau\) becomes appreciable, with instability beginning when a characteristic root reaches \(\operatorname{Re}s=0\). The compensated model should preserve peak frequency within 5 percent up to delays for which the uncompensated model exhibits a measurable shift, while constant inputs should remain nearly delay-invariant.