Interval-Lifetime Feature Modules

Integrate this into a transformer, graph encoder, or mixture-of-experts router that must operate under several simultaneously varying conditions, such as token dropout, graph edge deletion, image resolution, and context length. Choose \(m\) binary factors and create one feature state \(z_S\in\mathbb{R}^d\) for each nonempty subset \(S\subseteq[m]\), or use a selected downward-closed subset when \(m>5\). Sample or hand-design \(J\) intervals \(I_j\) in the subset poset, assign each interval a latent vector \(h_j\in\mathbb{R}^{r_j}\), and assign embeddings \(E_{j,S}\in\mathbb{R}^{d\times r_j}\). For each minibatch, construct the active set \(J_S=\{j:S\in I_j\}\), compute \(z_S=\sum_{j\in J_S}E_{j,S}h_j\), and feed \(z_S\) to the corresponding layer or task head. If explicit transitions \(A_{S\to T}\) are used, add the displayed commutativity loss on sampled triples \(S\subseteq T\subseteq U\). Penalize total active rank with \(\lambda_0\sum_j|I_j|r_j\). The interval membership and ranks are design variables; the commutativity residual is measured directly from the learned matrices. Start with a small ViT or 3-layer GNN on CIFAR-100 or Cora, using two or three binary degradation factors, and compare against a dense per-condition adapter bank with the same baseline accuracy. Success means lower parameter count and peak activation memory at equal accuracy, graceful performance on unseen factor combinations, and a lower commutativity residual without a substantial training slowdown.