Two-Channel Fractal Renormalization Network

Use this block as a replacement for a 3-to-1 pooling or message-passing stage in a hierarchical image, graph, or token model. Partition the input into three child groups recursively. Each child emits two vectors: a neutral content vector a and a defect or disagreement vector b. First align the three child vectors with shared linear projections. Then compute the neutral parent using the paper-inspired rule a_parent = lambda^3 times the elementwise product of the three a vectors plus 2 times x^3 times the elementwise product of the three b vectors. Compute b_parent with a small gated MLP G_theta that receives all six child vectors. Apply RMS normalization after the cubic operation and use softplus parameterizations for x and lambda. At each level, estimate the largest local Jacobian eigenvalue with two power-iteration steps or Hutchinson-vector products. Add a penalty max(0, rho_hat - 1 + epsilon)^2 when contractive propagation is desired; alternatively target rho_hat near one for maximal signal persistence. The paper supplies the ternary recursion, the two boundary-state interpretation, and the fixed-point stability criterion. The learned defect map, Jacobian estimate, and task loss are empirical additions. First test a six-level model on CIFAR-10 using a small hierarchical CNN, comparing an ordinary pooling baseline and the fractal block at matched parameter count. Sweep q = x/lambda and measure loss, gradient amplification, and maximum stable depth. The predicted signature is a sharp change when the dominant Jacobian eigenvalue crosses one: below the boundary, perturbations decay approximately as rho^n; above it, perturbations grow or remain persistent. Near the deconfinement-like setting, active routing paths should scale approximately as L^1.3 with hierarchy depth L, rather than as the full number of sites.