# Zeno-Free Minkowski Routing

- ID: 2195
- Canonical URL: https://synthcore.org/idea/2195/zeno-free-minkowski-routing
- API JSON: https://synthcore.org/api/idea/2195.json
- API Markdown: https://synthcore.org/api/idea/2195.md
- Verification status: unverified
- Source: [arXiv:2608.12618](https://arxiv.org/abs/2608.12618)
- Category: dynamics
- Solves: stability, scalability, inference-speedup
- ML areas: moe-routing, mlp, inference-speedup, training-dynamics
- Math tags: dynamical-systems, geometry, convex-analysis, control-theory
- Ratings: usefulness 6/10; difficulty 7/10; novelty 8/10

## Idea description

Represent adaptive computation or MoE routing as a continuous latent trajectory that crosses convex mode walls, with a Minkowski norm defining computational speed. At a wall, choose the outgoing latent velocity by the same constrained variational rule as billiard reflection, and use the local path-length certificate to detect or prevent pathological accumulation of infinitely many routing events in finite depth.

## Mathematical statement

Let $X=\mathbb{R}^{d}$ have a smooth strictly convex norm $\|\cdot\|$, let $\Omega=X\setminus\bigcup_i K_i$ be the complement of convex routing obstacles $K_i$, and let $z(t)$ be a unit-speed latent trajectory with $\|\dot z(t)\|=1$ between events. At a smooth wall point $p$, the outgoing velocity $v_+$ is selected so that its Legendre covector $\mathcal{L}(v_+)$ lies on the affine line $\mathcal{L}(v_-)+\mathbb{R}f_0$, where $\ker f_0$ is the tangent hyperplane of the wall, and $\|\mathcal{L}(v_+)\|^*=1$. Strict convexity of the dual norm gives exactly two unit intersections, the incoming and outgoing covectors. The reflection is equivalently characterized by making $p$ a critical point of $x\mapsto\|x-q_1\|+\|q_2-x\|$ over the wall. If all walls visited on an interval contain a common pivot $p$, the paper proves $\|p-z(t_1)\|+\|z(t_2)-p\|\ge t_2-t_1$. Here $t$ is normalized computational depth, $z$ is the routing state, and the inequality is the measurable non-Zeno certificate.

## Key formulas

- $$p=\operatorname*{crit}_{x\in\partial\Omega}\left[\|x-q_1\|+\|q_2-x\|\right]$$
- $$\mathcal{L}(v_+)=\mathcal{L}(v_-)+\alpha f_0,\qquad \|\mathcal{L}(v_+)\|^*=1,\qquad \ker f_0=T_p(\partial\Omega)$$
- $$\|p-z(t_1)\|+\|z(t_2)-p\|\ge t_2-t_1$$
- $$C(t_1,t_2;p)=\|p-z(t_1)\|+\|z(t_2)-p\|-(t_2-t_1)\ge0$$

## Implementation notes

Use an adaptive residual or MoE block with latent routing state $z_k\in\mathbb{R}^d$. Associate each expert with a convex region or convex wall, such as an ellipsoid $K_i=\{z:(z-c_i)^TQ_i(z-c_i)\le1\}$ with $Q_i\succ0$, and integrate $z$ using small Euler steps. The router selects the active expert from the region containing $z_k$. When a wall is crossed, estimate the signed-distance crossing time, compute the collision point $p$, and update the velocity. For the Euclidean MVP, use $v_+=v_- -2(n^Tv_-)/(n^Tn)n$. For a general Minkowski norm, solve the scalar equation $\|\mathcal{L}(v_-)+\alpha f_0\|^*=1$ and select the nonzero root. Pseudocode is: predict $\tilde z=z+h v$; detect a crossing; interpolate $p$; solve for $\alpha$; set $v\leftarrow v_+$; execute the selected expert; and monitor $C=\|p-z_{k_1}\|+\|z_{k_2}-p\|-(t_{k_2}-t_{k_1})$. Convex wall geometry and the reflection equation come from the paper; discretization error, local certificate slack, and whether learned walls remain convex must be estimated. If $C<0$, reduce the next step by $h\leftarrow h\min(1,C_0/(C_0+|C|+\epsilon))$, or reject the update. First experiment: compare ordinary adaptive-depth residual routing, Euclidean reflected routing, and certificate-damped routing on CIFAR-10 using an 8-to-32-block MLP with identical parameters. Construct inputs that induce rapid switching and measure switch count, minimum inter-switch depth, and accuracy at fixed compute. The prediction is a sharp qualitative signature: for convex walls sharing a pivot and finite latent path length, switch counts should saturate as the integration step $h$ decreases, rather than grow proportionally to $1/h$. Deliberately use nonconvex walls or remove damping to test whether step-refinement-induced switching growth returns.

## Disclaimer

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