✗ Failed on benchmark 2026

Quotient-Fibre Mixing Network

Implementation & benchmark of arXiv:2607.29391 — Exponential mixing via invariant foliations and relatively Anosov homeomorphisms

Usefulness7/10

Difficulty5/10

Novelty7/10

Dynamics Stability Memory Accuracy Rnn Ssm World model Training dynamics Dynamical systems Statistics Control theory Geometry

Source paper: Exponential mixing via invariant foliations and relatively Anosov homeomorphisms arXiv:2607.29391 ⓘ · analyzed Aug 31, 2026

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 Not peer-reviewed.

Idea description

Split a recurrent or state-space model into a coarse quotient state $z_t$ and a leaf or fibre state $y_t$, where the quotient evolves autonomously and the fibre is driven conditionally by the quotient. Constrain the two transition operators to have independently measurable contraction or correlation rates, then allocate capacity and regularization to the slower branch. This is intended for sequence tasks containing both slowly evolving global variables and rapidly mixing local variables.

Formulas

$$\operatorname{Cor}_{\mu}(\psi,\varphi,f^{n})=\int_{M}\psi\,(\varphi\circ f^{n})\,d\mu-\left(\int_{M}\psi\,d\mu\right)\left(\int_{M}\varphi\,d\mu\right).$$

$$\rho_{\mathrm{full}}(n)\;\lesssim\;\max\{\rho_q(n),\rho_f(n)\},\qquad \rho_q(n)\approx\alpha_q^n,\quad \rho_f(n)\approx\alpha_f^n.$$

$$z_{t+1}=G_\theta(z_t,u_t),\qquad y_{t+1}=H_\theta(y_t,z_t,u_t),\qquad x_t=(z_t,y_t),\qquad \alpha_{\mathrm{pred}}=\max\{\alpha_q,\alpha_f\}.$$

Mathematical statement

The paper considers a measure-preserving map $f:M\to M$ with invariant foliation $\mathcal F$, quotient projection $\pi:M\to M/\mathcal F$, and induced quotient map $F:M/\mathcal F\to M/\mathcal F$. For observables $\psi,\varphi$, correlation is $\operatorname{Cor}_{\mu}(\psi,\varphi,f^n)=\int\psi(\varphi\circ f^n)d\mu-\int\psi d\mu\int\varphi d\mu$. The transferable dichotomy is that the full-system decay rate is bounded by the slower quotient or leafwise rate: $\rho_{\mathrm{full}}(n)\lesssim\max\{\rho_q(n),\rho_f(n)\}$. In a neural analogue, $z_t$ is the quotient state and $y_t$ the fibre state, with $z_{t+1}=G_\theta(z_t)$ and $y_{t+1}=H_\theta(y_t,z_t)$. If the branch contraction factors are $\alpha_q<1$ and $\alpha_f<1$, the predicted asymptotic memory factor is $\alpha_{\mathrm{full}}\approx\max(\alpha_q,\alpha_f)$, not their product.

Implementation notes

1. Integration point: replace the hidden-state transition of a GRU, vanilla RNN, or diagonal state-space model with a two-level transition. Let $z_t\in\mathbb R^{d_q}$ represent the quotient and $y_t\in\mathbb R^{d_f}$ represent the fibre. First compute $z_{t+1}=G_\theta(z_t,u_t)$, then compute $y_{t+1}=H_\theta(y_t,z_t,u_t)$, and predict from $[z_t,y_t]$. Apply spectral normalization separately to $G$ and to the derivative of $H$ with respect to $y$. 2. Pseudocode: initialize $z,y$; for each input $u$, update $z\leftarrow G(z,u)$, update $y\leftarrow H(y,z,u)$, and emit $R([z,y])$. During validation, create paired trajectories differing only in $z_0$, and separate pairs differing only in $y_0$. Estimate $\alpha_q$ and $\alpha_f$ from linear fits to $\log\|z_t-z'_t\|$ and $\log\|y_t-y'_t\|$, or from delayed autocorrelation slopes of hidden-state observables. 3. Computed versus estimated: the quotient/fibre construction and max-rate rule are transferred from the paper; finite-data rates, Jacobian norms, and effects of cross-coupling are estimated empirically. A stricter version enforces $\|J_G\|_2\leq\alpha_q$ and $\|\partial H/\partial y\|_2\leq\alpha_f$ using spectral normalization or projected power iteration. 4. First cheap experiment: train this model and a parameter-matched monolithic GRU on synthetic sequences from $z_{t+1}=0.98z_t+\epsilon_t^q$, $y_{t+1}=0.50y_t+Az_t+\epsilon_t^f$, with nonlinear mixed observations; then test on a small electricity-demand or Moving-MNIST task. Sweep spectral ceilings over $\alpha_q,\alpha_f\in\{0.3,0.5,0.7,0.9,0.98\}$. The falsifiable signature is that fitted full-system autocorrelation and perturbation decay satisfy $\alpha_{\mathrm{full}}\approx\max(\alpha_q,\alpha_f)$ within 20%. Changing the faster branch while holding the slower branch fixed should barely change the asymptotic tail. If either ceiling exceeds 1, perturbation norms and long-horizon loss should show a sharp stability transition.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Inconclusive

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a reproducible quotient-fibre triangular state model with separate quotient and fibre transitions, empirical rate fitting, stability testing, and a matched monolithic least-squares baseline. The max-rate prediction matched observed rates within 3.2% across the sweep (the equal-rate case has the expected finite-horizon Jordan transient), slow-branch invariance errors were at most 0.23%, and the fitted stability slope correctly changed from negative below rate 1 to zero at 1 and positive above 1. The split model did not beat the matched monolithic model on this simple identification task: rollout MSE was 0.00715811 versus 0.00715793.

Agent confidence: 9/10

Baseline: Monolithic one-step MSE 0.000896204; rollout MSE 0.00715793
Idea: Split quotient-fibre one-step MSE 0.000896222; rollout MSE 0.00715811; max-rate relative error <=3.17%; slow-tail invariance error <=0.23%; stability boundary correctly detected

Limitations: No nonlinear GRU, spectral normalization, real-world sequence dataset, or large-scale training comparison was tested. The verified result is for the explicitly triangular linear state model; cross-coupling and finite-sample neural optimization effects remain untested.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.29391) Aug 31, 2026

No significant difference vs. the tuned baseline (+14.9%, p=0.4041, wins 3 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0014
Idea mean: 0.0016
Effect (Δ): 0.0002 (+14.9%; negative = idea better)
Wins: 3 / 8 paired seeds
p-value: 0.4041 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±38.4%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

Only the registered dynamics track was tested. Explicit spectral normalization and contraction-ceiling sweeps were not implemented, and the mechanism signature used perturbation decay over the short eight-step benchmark window rather than long-horizon rollout stability.

How to run: python3 bench_experiment.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 9/10

The quotient-fibre GRU was evaluated on the registered built-in dynamics track with the canonical training loop, an equal-capacity recurrent baseline, a shared learning-rate search space, and 8 paired seeds. The trained-model perturbation signature confirmed the max-rate prediction within 3.47%, but the idea had higher test MSE and no significant win.

Baseline: Tuned rnn_small GRU, lr=0.01: mean test MSE 0.001394080012687482 over 8 seeds
Idea: Matched-capacity QuotientFibreGRU, lr=0.01: mean test MSE 0.001602079973963555 over 8 seeds; paired delta +0.000207999961276073, p=0.4041

Benchmark result