{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace real nonnegative graph-edge weights in a directed GNN with unit-modulus complex gains and complex node states. Enforce node-wise U(1) gauge equivariance, so arbitrary phase choices at individual nodes cannot change predictions while gauge-invariant cycle phases remain available to encode directed relational structure.",
 "formulas_latex": [
  "$$A(\\Phi)_{ij}=\\begin{cases}\\varphi(v_i,v_j),\u0026(v_i,v_j)\\in\\mathcal A,\\\\0,\u0026\\text{otherwise},\\end{cases}\\qquad \\varphi(u,v)\\in\\mathbb T,\\ |\\varphi(u,v)|=1.$$",
  "$$\\varphi^{\\sigma}(u,v)=\\sigma(u)^{-1}\\varphi(u,v)\\sigma(v),\\qquad \\varphi(W)=\\prod_{s=0}^{k-1}\\varphi(v_{i_s},v_{i_{s+1}}).$$",
  "$$m_u=\\sum_{(u,v)\\in\\mathcal A}\\varphi(u,v)W h_v,\\qquad h_u^{\\sigma}=\\sigma(u)^{-1}h_u\\ \\Longrightarrow\\ m_u^{\\sigma}=\\sigma(u)^{-1}m_u.$$",
  "$$f(z)=z\\,g(|z|),\\qquad f(\\sigma^{-1}z)=\\sigma^{-1}f(z)\\quad (|\\sigma|=1).$$"
 ],
 "id": 105,
 "implementation": "1. Integration point: implement a directed message-passing layer for a graph batch. Store each node state as a complex tensor with shape [batch,nodes,channels], and assign every directed edge either a fixed gain φ_uv=e^{iθ_uv} derived from edge features or a learnable phase θ_uv. Do not use independent real and imaginary nonlinearities, since they break gauge equivariance. 2. Pseudocode: initialize complex states h; for each layer, compute q_v=complex_linear(h_v), messages m_u=sum over outgoing arcs (u,v) of exp(iθ_uv)q_v, normalize by degree, set z_u=W_self h_u+m_u, and update h_u=z_u*MLP_abs(|z_u|). Optionally concatenate invariant features |h_u| and sampled cycle gains. For a gauge test, sample random node phases σ_u, replace θ_uv by θ_uv−angle(σ_u)+angle(σ_v), and replace h_u by exp(−i angle(σ_u))h_u; outputs should transform by the same node phase. 3. The paper supplies the gain algebra and telescoping invariance; estimate phases from data or learn them, and use ordinary backpropagation through sin and cos. 4. First experiment: compare a 3-layer real directed GCN/GAT, a complex layer without gauge augmentation, and this equivariant layer on Cora or Citeseer with randomly oriented edges and a synthetic task where labels depend on directed cycle phase products. Measure accuracy, robustness to random node-wise phase reparameterization, and oversmoothing via pairwise feature cosine similarity. Success is unchanged predictions under gauge transformations and improved accuracy on cycle-dependent tasks at comparable parameter count.",
 "math_summary": "For a directed graph D with arc set A, each arc (u,v) receives a gain φ(u,v)∈T={z∈C:|z|=1}. The gain adjacency is A(Φ)_{ij}=φ(v_i,v_j) on arcs and zero otherwise. Under a node switching function σ:V→T, the switched gain is φ^σ(u,v)=σ(u)^{-1}φ(u,v)σ(v). A walk gain is φ(W)=∏_s φ(v_s,v_{s+1}), and cycle gains are invariant under switching because node phases telescope. Let node features transform as h_u^σ=σ(u)^{-1}h_u. Then the message m_u=∑_(u,v) φ(u,v)W h_v transforms as m_u^σ=σ(u)^{-1}m_u, provided W is complex-linear and shared across nodes. A nonlinear update must therefore be phase-equivariant, for example f(z)=z g(|z|), where g is a real MLP.",
 "math_tags": [
  "graph-theory",
  "representation-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "graph-nn",
  "embedding"
 ],
 "paper": {
  "arxiv_id": "2608.23655",
  "arxiv_url": "https://arxiv.org/abs/2608.23655",
  "summary_what_math_gives_to_ml": "The paper provides a principled way to attach unit-modulus complex phases to directed graph edges while preserving a useful gauge symmetry: node-wise phase reparameterizations change individual edge gains but leave cycle products invariant. This suggests graph-neural layers whose messages are complex and gauge-equivariant, allowing models to represent directional frustration, circulation, and phase consistency rather than only edge magnitudes. The spectral comparison theorem also gives a safety guarantee: phase-weighted propagation cannot have larger spectral radius than the nonnegative underlying adjacency, which can be used to build stable graph filters or constrain learned propagation operators.",
  "title": "On Spectra of $\\mathbb{T}$-Gain Digraphs",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "accuracy",
  "stability",
  "generalization"
 ],
 "title": "Gauge-Equivariant Complex Message Passing",
 "url": "https://synthcore.org/idea/105/gauge-equivariant-complex-message-passing",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}