# Event-Driven Return-State Layer

- ID: 34
- Canonical URL: https://synthcore.org/idea/34/event-driven-return-state-layer
- API JSON: https://synthcore.org/api/idea/34.json
- API Markdown: https://synthcore.org/api/idea/34.md
- Verification status: unverified
- Source: [arXiv:2512.02291](https://arxiv.org/abs/2512.02291)
- Category: dynamics
- Solves: speedup, stability, scalability
- ML areas: ssm, rnn, inference-speedup, training-dynamics
- Math tags: dynamical-systems, linear-algebra, approximation-theory
- Ratings: usefulness 7/10; difficulty 6/10; novelty 8/10

## Idea 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.

## Mathematical statement

In the paper, a and b are coordinates along stable and unstable eigenvectors of a saddle point, with multipliers 0<lambda<1 and sigma>1. 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)|<C_2 epsilon^c, with epsilon measuring distance from the homoclinic parameter and c>0. For a neural layer, this motivates learning a return map while retaining exact exponential propagation inside each segment.

## Key formulas

- $$k=K(z,\sigma)=\left\lceil-\frac{\ln z}{\ln\sigma}\right\rceil.$$
- $$a\!\left(f_L^k(P)\right)=\lambda^k a(P),\qquad b\!\left(f_L^k(P)\right)=\sigma^k b(P).$$
- $$z'=h_k(z;\eta,\nu,\sigma)=\frac{\sigma-\sigma^kz}{\sigma-1}\nu+\frac{\sigma^kz-1}{\sigma-1}\eta.$$
- $$|z'-h(z;\eta,\nu,\sigma)|<C_2\varepsilon^c.$$

## Implementation notes

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)>1. 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.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.