Unverified 2026

Reciprocal-Gain Loxodromic Memory

Implementation & benchmark of arXiv:2608.19393 — Outer Contact Billiards

Usefulness6/10

Difficulty5/10

Novelty5/10

Dynamics Stability Memory Speedup Rnn Ssm World model Initialization Dynamical systems Control theory Spectral theory Linear algebra Geometry

Source paper: Outer Contact Billiards arXiv:2608.19393 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Use paired recurrent channels with exactly reciprocal gains while applying a common phase rotation. One channel carries a controlled expanding mode and the other a matching contracting mode, creating a tunable hyperbolic memory spectrum without the optimization fragility of an unconstrained recurrent matrix.

Formulas

$$z_1'=\rho e^{i\varphi}z_1,\qquad z_2'=\rho^{-1}e^{i\varphi}z_2,\qquad \rho=e^r.$$

$$\begin{pmatrix}x'\\y'\\u'\\v'\end{pmatrix}=\begin{pmatrix}\rho\cos\varphi&0&-\rho\sin\varphi&0\\0&\rho^{-1}\cos\varphi&0&-\rho^{-1}\sin\varphi\\\rho\sin\varphi&0&\rho\cos\varphi&0\\0&\rho^{-1}\sin\varphi&0&\rho^{-1}\cos\varphi\end{pmatrix}\begin{pmatrix}x\\y\\u\\v\end{pmatrix}.$$

$$|z_{1,k}|=e^{rk}|z_{1,0}|,\qquad |z_{2,k}|=e^{-rk}|z_{2,0}|,\qquad Q_k=|z_{1,k}|^2|z_{2,k}|^2=Q_0.$$

Mathematical statement

Let z_1=x+iu and z_2=y+iv be two complex state channels. The paper's loxodromic map is z_{1,k+1}=rho exp(i phi) z_{1,k} and z_{2,k+1}=rho^{-1} exp(i phi) z_{2,k}, where rho>0 and phi are constant on an invariant leaf. The real eigenvalue moduli are rho and rho^{-1}; hence the logarithmic growth rates are r and -r when rho=exp(r). The product quadratic invariant Q(z_1,z_2)=|z_1|^2|z_2|^2 is preserved because |z_1'|^2=rho^2|z_1|^2 and |z_2'|^2=rho^{-2}|z_2|^2. Euclidean norm stability occurs only at r=0, but for any bounded r the exact amplification and forgetting rates are known in advance.

Implementation notes

1. Integration point: implement this as a structured recurrent block for long-context sequence models, latent world models, or diagonal state-space layers. Pair every expanding channel with a contracting channel and use the pair product as a diagnostic or optional regularizer. 2. Pseudocode: for every pair store real coordinates (x,u) and (y,v), compute r=r_max*tanh(r_hat) and phi=pi*tanh(phi_hat), set a=exp(r), c=cos(phi), s=sin(phi), then update x'=a*(c*x-s*u), u'=a*(s*x+c*u), y'=a^(-1)*(c*y-s*v), and v'=a^(-1)*(s*y+c*v). Add a separate input term Bx_t after the structured transition. 3. Computed exactly: reciprocal gains, common phase, and the autonomous transition. Estimated empirically: useful r ranges, effects of input injection, and whether the downstream readout exploits the expanding channel. Track g1=log(|z1,t+1|+epsilon)-log(|z1,t|+epsilon), g2 analogously, and Q_t; epsilon prevents numerical singularities. 4. First cheap experiment: compare width-128 versions with GRU and a diagonal SSM on adding, delayed copy, and permuted sequential MNIST at lengths 1,000 and 10,000. Sweep r_max in {0,0.01,0.05,0.1}. The predicted signature is fitted log-amplitude slopes of +r and -r in the two channels, with pair-product drift below 1% over 10,000 autonomous steps. At r=0 the transition is norm-preserving; increasing |r| should produce a measurable crossover to exponential amplification and forgetting, with observed slopes agreeing with the specified values within 10%.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

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