Holonomy-Fixed State Filter

1. Integration point: use this module before a graph neural network's state decoder or immediately after its root-state logits. Inputs are a connected graph edge list, a permutation table g_e of shape [|O|] for every oriented edge, and root logits z of shape [batch, |O|]. The module can also provide a graph-level compatibility feature to the message-passing network. 2. Pseudocode: choose root r and build a spanning tree with DFS or BFS; set T_r to the identity permutation and propagate T_v=g_(u,v) composed with T_u along tree edges; initialize mask[o]=True; for every chord e=(u,v), compute h=T_v^{-1} composed with g_e composed with T_u, then set mask[o] = mask[o] and (h[o]==o); let F={o:mask[o]}; compute p=softmax(z), compatibility=sum(p[o] for o in F), and either train with L=L_task+lambda*(-log(compatibility+epsilon)) or set masked_logits[o]=-infinity for o not in F at inference. If F is empty, skip hard masking and return compatibility zero, or use a learned unconstrained fallback. 3. Computed exactly from the paper: tree transports, chord holonomies, F, and p(F). Estimated empirically: lambda, epsilon, and whether hard filtering or the soft loss is preferable; no subgroup enumeration is needed because generators suffice. Cache T_v and the mask when the graph is static. For a dynamic graph, recompute only the affected chord holonomy and rescan O(|O|) states. 4. First experiment: use a synthetic graph task with |O| in {8,16,32}, random permutation edge labels, and targets generated from a planted compatible root state. Compare a standard message-passing decoder, hard fixed-point masking, and the soft compatibility loss at equal parameter count. Measure exact assignment accuracy, invalid-edge rate, calibration, and preprocessing latency. The expected signal is lower invalid-assignment rate and faster convergence; on graphs with many inconsistent cycles, the hard mask should reduce decoder search from |O| candidates to |F| and provide near-zero compatibility violations whenever F is nonempty.