# Finite-State Extinction Monitor and Threshold Scheduler

- ID: 1630
- Canonical URL: https://synthcore.org/idea/1630/finite-state-extinction-monitor-and-threshold-scheduler
- API JSON: https://synthcore.org/api/idea/1630.json
- API Markdown: https://synthcore.org/api/idea/1630.md
- Verification status: unverified
- Source: [arXiv:2608.00340](https://arxiv.org/abs/2608.00340)
- Category: dynamics
- Solves: stability, speedup
- ML areas: rnn, graph-nn, training-dynamics, scheduler, inference-speedup
- Math tags: graph-theory, dynamical-systems, combinatorics, bifurcations
- Ratings: usefulness 5/10; difficulty 4/10; novelty 8/10

## Idea description

Treat a quantized recurrent network as a finite deterministic state-transition system and distinguish absorption from latent periodic behavior during inference or training. Use the observed extinction threshold to adapt the activation threshold or recurrent gain, stopping once all tested trajectories reach the zero state and increasing the threshold when trajectories enter nontrivial cycles.

## Mathematical statement

Let F_R: \Omega_0(G)    o\Omega_0(G) be the deterministic update map at replacement threshold R, where \Omega_0(G) is the set of admissible three-state configurations. The transition digraph has one outgoing edge f    o F_R(f) per state. Let 0_G be the all-healthy absorbing state and D_H its weakly connected component. A trajectory from f_0 becomes extinct exactly when f_0\in D_H; otherwise, finiteness of \Omega_0(G) implies eventual entry into a directed cycle. Define the empirical extinction fraction q_R=(1/B)\sum_{b=1}^B 1\{\exists t\le T:F_R^t(f_0^{(b)})=0_G\}. Use q_R and detected cycle lengths to control R or the recurrent gain. The paper's global bounds transfer as a sanity check for maximum degree \Delta: extinction is guaranteed by R=\Delta+1, while persistence can occur for lower R when a cross-degree core exists.

## Key formulas

- $$D_H=\text{the weakly connected component of }D_R(G)\text{ containing }0_G,\qquad f_0\text{ extinct}\Longleftrightarrow f_0\in D_H.$$
- $$\exists\,0\leq i<j\leq |\Omega_0(G)|:\ F_R^i(f_0)=F_R^j(f_0),\quad L=j-i,$$
- $$q_R=\frac{1}{B}\sum_{b=1}^{B}\mathbf{1}\left[\min_{0\leq t\leq T}\|h_t^{(b)}\|_0=0\right],\qquad R\geq \Delta(G)+1\Rightarrow q_R=1\text{ for every initial state}.$$

## Implementation notes

Use this in inference-time stabilization of a quantized RNN or recurrent GNN whose hidden state h_t is represented by ternary values {-1,0,+1}, or whose activations are hard-thresholded into healthy, active, and inactive states. Define F_R by one recurrent forward pass with threshold R. For B random hidden-state probes, store a hash of each state for at most T steps; declare extinction when the all-zero state is reached and declare a cycle when a repeated hash appears, recording L=j-i. Pseudocode: initialize R; for each probe, iterate h=F_R(h), hash h, stop at zero or repeated hash; compute q_R and the fraction p_cycle; if p_cycle exceeds a target alpha, increase R by one or shrink recurrent gain, while if q_R is already one for K consecutive batches, reduce R to save computation or preserve responsiveness. The paper supplies the finite-state absorption and cycle logic and the conservative degree bound; trajectory statistics and the appropriate neural threshold are empirical. First cheap experiment: use a ternary GRU-like recurrent classifier on MNIST-permuted sequences and a binary GNN on synthetic temporal node labels, comparing fixed thresholds with the scheduler. Sweep R around Delta+1 and the estimated cross-degree threshold. The prediction is a monotone rise in q_R, with a transition from nonzero cycle rate to near-zero cycle rate near the empirical extinction threshold; for unweighted graphs, R=Delta+1 must yield extinction from every probe.

## Disclaimer

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