Spectral Message Basis

Integrate this into the communication layer of a multi-agent transformer or mixture-of-experts router. Each agent first maps its local hidden state $h_i^t\in\mathbb{R}^{p}$ into a shared observable vector $\phi_i^t\in\mathbb{R}^{d}$ using a fixed linear projection plus disagreement and confidence features. During a calibration phase, log consecutive global observable vectors and fit $\widehat K=YX^{\mathsf T}(XX^{\mathsf T}+\alpha I)^{-1}$. Compute left and right eigenvectors, sort modes by $|\lambda_j|$, and select the smallest $k$ whose held-out reconstruction error is below a chosen threshold: $\|\phi-\sum_{j\le k}v_ju_j^{\mathsf T}\phi\|_2/\|\phi\|_2\le\eta$. At runtime, transmit only $c_{1:k}=U_k^{\mathsf T}\phi_i^t$ between agents or across devices; reconstruct the approximate message as $V_kc_{1:k}$ before the receiving attention or router block. Include the mode eigenvalues and reconstruction residual in the message metadata so the receiver can reject compression when the residual is high. Use $|\lambda_2|>0.9$ as the paper's metastability/attribution validity condition, but measure its predictive value on the target task rather than hard-coding it as a theorem. Compare full hidden-state communication against random projection, PCA, and the spectral basis on an 8-agent synthetic consensus benchmark and a small QA debate model. Report bandwidth per round, wall-clock communication time, answer accuracy, and reconstruction error. A successful result is at least 4x lower transmitted dimension with less than 0.5 percentage-point accuracy loss and stable performance under changing initial agent states.