{
 "artifacts": null,
 "category": "dynamics",
 "description": "Build a recurrent or state-space layer whose latent state has a stable coordinate and an unstable coordinate, and analytically jump across long stretches of repeated dynamics until an event boundary is reached. Instead of applying the same transition k times, compute the integer return time from a logarithm and apply the closed-form powers directly.",
 "formulas_latex": [
  "$$k=K(z,\\sigma)=\\left\\lceil-\\frac{\\ln z}{\\ln\\sigma}\\right\\rceil.$$",
  "$$a\\!\\left(f_L^k(P)\\right)=\\lambda^k a(P),\\qquad b\\!\\left(f_L^k(P)\\right)=\\sigma^k b(P).$$",
  "$$z'=h_k(z;\\eta,\\nu,\\sigma)=\\frac{\\sigma-\\sigma^kz}{\\sigma-1}\\nu+\\frac{\\sigma^kz-1}{\\sigma-1}\\eta.$$",
  "$$|z'-h(z;\\eta,\\nu,\\sigma)|\u003cC_2\\varepsilon^c.$$"
 ],
 "id": 34,
 "implementation": "Integrate this as an adaptive-step recurrent cell or diagonal state-space block. Maintain two latent vectors A_t and B_t with learned positive rates represented as lambda=exp(-softplus(r_lambda)) in (0,1) and sigma=1+softplus(r_sigma)\u003e1. Let z=norm(B_t)_2+epsilon be the event coordinate and choose k=clip(ceil(-log(z)/log(sigma)),0,K_max). Perform the exact skipped propagation A_next=lambda^k A_t and B_bar=sigma^k B_t. Project B_bar to a scalar u=norm(B_bar)_2 and apply the return map u_next=((sigma-u)/(sigma-1))*nu+((u-1)/(sigma-1))*eta, where u=sigma^k z; rescale the direction of B_t and set B_next=u_next B_t/(z+epsilon). Add a learned return injection and residual connection to preserve representational capacity. During training, either use a straight-through gradient for the ceiling operation or replace it with a soft mixture over k-1, k, and k+1; at inference use the hard integer jump. The exact mathematics supplies the powers and event index; lambda, sigma, eta, nu, and the return injection are learned empirically. First compare a 64-dimensional cell on sequential MNIST or a copy-memory task against a standard GRU and a diagonal SSM with the same hidden size. Log the average skipped steps, wall-clock time, gradient norms, and task loss. Success requires matching baseline accuracy while reducing recurrent transition evaluations by at least 2x and avoiding exploding gradients; ablate the return map and the exact exponential jump separately.",
 "math_summary": "In the paper, a and b are coordinates along stable and unstable eigenvectors of a saddle point, with multipliers 0\u003clambda\u003c1 and sigma\u003e1. Repeated application of the left affine branch obeys a(f_L^k(P))=lambda^k a(P) and b(f_L^k(P))=sigma^k b(P). The exit or return index is k=K(z,sigma)=ceil(-ln(z)/ln(sigma)), where z=b(P), so the layer can skip k repeated transitions. After the jump, the paper's return coordinate is h_k(z;eta,nu,sigma)=((sigma-sigma^k z)/(sigma-1)) nu + ((sigma^k z-1)/(sigma-1)) eta, where nu and eta are the return values from the two endpoints of the unstable fundamental segment. The approximation error in the original two-dimensional construction is bounded by |z'-h(z)|\u003cC_2 epsilon^c, with epsilon measuring distance from the homoclinic parameter and c\u003e0. For a neural layer, this motivates learning a return map while retaining exact exponential propagation inside each segment.",
 "math_tags": [
  "dynamical-systems",
  "linear-algebra",
  "approximation-theory"
 ],
 "ml_areas": [
  "ssm",
  "rnn",
  "inference-speedup",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2512.02291",
  "arxiv_url": "https://arxiv.org/abs/2512.02291",
  "summary_what_math_gives_to_ml": "The paper constructs an explicit first-return reduction from a two-dimensional piecewise-affine saddle dynamics to a discontinuous one-dimensional map with infinitely many geometrically scaled linear pieces. The transferable asset is not the specific border-collision application, but the combination of logarithmic state partitioning, exact event-driven iteration counts, self-similar parameter scaling, and a controlled approximation error. This suggests neural modules that route or transform activations according to their logarithmic magnitude while sharing parameters across scales. A second opportunity is an event-driven recurrent layer that analytically skips long runs of nearly linear dynamics instead of evaluating every intermediate step.",
  "title": "From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps",
  "year": "2025"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "speedup",
  "stability",
  "scalability"
 ],
 "title": "Event-Driven Return-State Layer",
 "url": "https://synthcore.org/idea/34/event-driven-return-state-layer",
 "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
   }
  }
 }
}