# Cohomological Quotient RNN

- ID: 2660
- Canonical URL: https://synthcore.org/idea/2660/cohomological-quotient-rnn
- API JSON: https://synthcore.org/api/idea/2660.json
- API Markdown: https://synthcore.org/api/idea/2660.md
- Verification status: mechanism_failed
- Source: [arXiv:2608.21352](https://arxiv.org/abs/2608.21352)
- Category: dynamics
- Solves: stability, speedup, sample-efficiency
- ML areas: rnn, ssm, optimizer, training-dynamics
- Math tags: dynamical-systems, control-theory, optimization, statistics
- Ratings: usefulness 7/10; difficulty 5/10; novelty 8/10

## Idea description

Build a recurrent or state-space model with a base state carrying task-relevant dynamics and an explicitly contracting auxiliary state. If the training loss or energy depends on the auxiliary state, replace it by a quotient loss plus an analytically known telescoping correction; long-run optimization and invariant averages are then unchanged, while transient fiber effects decay geometrically.

## Mathematical statement

Let \(F(x,y)=(f(x),ay+\rho(x))\) with \(|a|<1\), where \(x\) is the quotient/base state, \(y\) is the contracting fiber state, and \(\rho\) is a learned or fixed forcing map. The invariant graph \(h\) satisfies \(h(f(x))=ah(x)+\rho(x)\). For the fiber-dependent potential \(\varphi(x,y)=g(x)+by\), define \(\psi(x)=g(x)+b\rho(x)/(1-a)\) on the invariant graph and \(u(x,y)=b(y-h(x))/(1-a)\). Then \(\varphi=\psi\circ\pi+u-u\circ F\), with \(\pi(x,y)=x\). Consequently, for every trajectory, \(\sum_{t=0}^{T-1}\varphi(F^t z)=\sum_{t=0}^{T-1}\psi(f^t x)+u(z)-u(F^Tz)\). Fiber differences obey \(|y_t-h(x_t)|=|a|^t|y_0-h(x_0)|\), and the time-average discrepancy is bounded by \(2\|u\|_\infty/T\) when \(u\) is bounded.

## Key formulas

- $$F(x,y)=(f(x),ay+\rho(x)),\qquad |a|<1,$$
- $$h(f(x))=ah(x)+\rho(x),\qquad \psi(x)=g(x)+\frac{b}{1-a}\rho(x),$$
- $$\varphi(x,y)=\psi(x)+u(x,y)-u(F(x,y)),\qquad u(x,y)=\frac{b}{1-a}\bigl(y-h(x)\bigr),$$
- $$\frac{1}{T}\sum_{t=0}^{T-1}\varphi(F^t z)-\frac{1}{T}\sum_{t=0}^{T-1}\psi(f^t x)=\frac{u(z)-u(F^Tz)}{T},\qquad |y_T-h(x_T)|=|a|^T|y_0-h(x_0)|.$$

## Implementation notes

1. Integration point: implement a two-state recurrent cell. The base state is \(x_{t+1}=f_\theta(x_t,\xi_t)\), while the auxiliary state is \(y_{t+1}=a y_t+\rho_\theta(x_t,\xi_t)\). Parameterize \(a=\tanh(\alpha)\) or use \(a=\sigma(\alpha)\) for one-sided contraction, and optionally clip \(|a|\leq a_{\max}<1\). Use a per-step objective \(\varphi_t=g_\theta(x_t,\xi_t)+b y_t\). 2. Pseudocode: first estimate or compute the invariant graph by running a detached burn-in trajectory, `h_x = fixed_point_or_long_burnin(f, rho, x)`; then compute `psi_t = g(x_t) + b*rho(x_t)/(1-a)` and train with `loss = mean(psi_t)` rather than `mean(phi_t)`. For exact finite-horizon equivalence, add the boundary correction `+(u(x_0,y_0)-u(x_T,y_T))/T`, where `u=b*(y-h_x)/(1-a)`. Backpropagate through the quotient loss, and use the full loss only for validation. 3. Computed versus estimated: \(a\), \(b\), and \(\rho\) are model quantities; \(h\) can be estimated by burn-in because contraction gives geometric convergence, or represented by a small auxiliary network trained to satisfy \(h(f(x))\approx ah(x)+\rho(x)\). Estimate the empirical contraction rate from \(\|y_{t+k}-\tilde y_{t+k}\|/\|y_t-\tilde y_t\|\), and estimate the coboundary residual from `phi - psi - u + u_next`. 4. First cheap experiment: compare a standard GRU or linear SSM against this split cell on sequential MNIST or Copy Memory with identical parameter counts and a synthetic objective containing \(b y_t\). Sweep \(a\in\{0.2,0.5,0.8,0.95,0.99\}\) and horizons \(T\in\{32,128,512\}\). The falsifiable signatures are: paired fiber trajectories must contract with slope \(\log|a|\) on a log-distance plot; the cumulative full-minus-quotient objective must remain bounded while its time-average decays as \(1/T\); and the residual of the cohomological identity should decrease with burn-in as \(|a|^t\). At \(a\to1\), convergence and boundary-error constants should deteriorate sharply, providing a measurable stability boundary rather than a benchmark-only comparison.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a readable scalar quotient-RNN verification with contraction, invariant-graph, finite-horizon, and formula sweeps. The predicted fiber ratios and log-slopes match a exactly for a in {0.2,0.5,0.8,0.95,0.99}, and the corrected telescoping discrepancy scales exactly as 1/T (T times discrepancy approaches 30.855). However, the authoritative psi=g+b*rho/(1-a) formula fails for general quotient dynamics: residuals range from 3.43 to 46.98, whereas replacing it with psi=g+b*h gives residuals below 1.4e-13. Thus the underlying contraction mechanism manifests, but the proposed method is not valid as stated.

### Mechanism check

- Verdict: Built a readable scalar quotient-RNN verification with contraction, invariant-graph, finite-horizon, and formula sweeps. The predicted fiber ratios and log-slopes match a exactly for a in {0.2,0.5,0.8,0.95,0.99}, and the corrected telescoping discrepancy scales exactly as 1/T (T times discrepancy approaches 30.855). However, the authoritative psi=g+b*rho/(1-a) formula fails for general quotient dynamics: residuals range from 3.43 to 46.98, whereas replacing it with psi=g+b*h gives residuals below 1.4e-13. Thus the underlying contraction mechanism manifests, but the proposed method is not valid as stated.
- Confidence: 9/10
- Limitations: No sequential-MNIST/Copy-Memory or GRU comparison was run, no learned invariant-graph estimator was tested, and the toy system is one-dimensional and linear. The corrected h-based formulation was verified, but it requires access to the invariant graph and is not the formula supplied in the idea.

## Artifacts

- [experiment.py](https://synthcore.org/code/1018/experiment.py)
- [report.md](https://synthcore.org/code/1018/report.md)
- [results.json](https://synthcore.org/code/1018/results.json)
- [results_v2.json](https://synthcore.org/code/1018/results_v2.json)
- [verify.py](https://synthcore.org/code/1018/verify.py)
- [Download all files as ZIP](https://synthcore.org/download/1018)

## Disclaimer

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