# Cosymplectic Reeb-Hamiltonian Layer

- ID: 427
- Canonical URL: https://synthcore.org/idea/427/cosymplectic-reeb-hamiltonian-layer
- API JSON: https://synthcore.org/api/idea/427.json
- API Markdown: https://synthcore.org/api/idea/427.md
- Verification status: mechanism_failed
- Source: [arXiv:2607.05231](https://arxiv.org/abs/2607.05231)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: ssm, world-model, training-dynamics
- Math tags: symplectic, differential-geometry, dynamical-systems
- Ratings: usefulness 6/10; difficulty 6/10; novelty 6/10

## Idea description

Replace an unconstrained latent transition by a layer with a distinguished scalar coordinate \(t\) and a symplectic leaf state \(x=(q,p)\). The layer advances \(t\) through a Reeb drift while updating \(x\) with a symplectic Hamiltonian step, preventing arbitrary mixing between progression and content coordinates and potentially improving long-horizon stability.

## Mathematical statement

A cosymplectic manifold is a \((2n+1)\)-dimensional space with closed forms \(\alpha\) and \(\beta\) satisfying \(\alpha\wedge\beta^n\neq0\), where \(\alpha\) defines the transverse direction and \(\beta\) is symplectic on its leaves. The Reeb vector field \(R\) is uniquely defined by \(\iota_R\alpha=1\) and \(\iota_R\beta=0\). Use coordinates \(z=(x,t)\), \(x=(q,p)\in\mathbb{R}^{2n}\), \(\alpha=dt\), and \(\beta=\sum_i dq_i\wedge dp_i\). The paper's thickening constructs the symplectic form \(\Omega=\beta+d\theta\wedge\alpha\) on \(M\times\mathbb{S}^1\); computationally this means treating \((x,t)\) as the leaf-plus-transverse state and preserving the canonical symplectic form on \(x\). For a Hamiltonian \(H(x,t)\), the cosymplectic moment condition \(\iota_{X_H}\beta=-dH+\phi_H\alpha\) means that the leaf component obeys \(\iota_{X_H}\beta=-d_xH\), while \(\phi_H=\partial_tH\) is the coefficient of the transverse 1-form. In canonical coordinates this gives \(\dot q=\partial H/\partial p\), \(\dot p=-\partial H/\partial q\), together with an independently chosen Reeb speed \(\dot t=v_H\). The exact geometric guarantees apply when \(H\) and the forms satisfy the stated closure and nondegeneracy assumptions; the neural adaptation should test whether approximate preservation of these structures improves learned dynamics.

## Key formulas

- $$\alpha\wedge\beta^{n}\neq 0,\qquad \iota_R\alpha=1,\qquad \iota_R\beta=0.$$
- $$\Omega=\beta+\mathrm{d}\theta\wedge\alpha\quad\text{on }M\times\mathbb{S}^{1}.$$
- $$\iota_{X_H}\beta=-\mathrm{d}H+\phi_H\alpha,\qquad \dot q=\nabla_p H(q,p,t),\qquad \dot p=-\nabla_q H(q,p,t).$$
- $$\begin{aligned}p_{k+1/2}&=p_k-\frac{h}{2}\nabla_qH(q_k,p_{k+1/2},t_k),\\ q_{k+1}&=q_k+h\nabla_pH(q_k,p_{k+1/2},t_k),\\ p_{k+1}&=p_{k+1/2}-\frac{h}{2}\nabla_qH(q_{k+1},p_{k+1/2},t_k),\\ t_{k+1}&=t_k+h\,v_\psi(q_k,p_k,t_k,u_k).\end{aligned}$$

## Implementation notes

1. Integration point: use this as the transition block in a recurrent state-space model, latent world model, or sequence encoder. Split each hidden vector into \(q,p,t\), with \(q,p\in\mathbb{R}^{d/2}\) and one scalar \(t\); project the original hidden state into this representation once at the input and project back after the transition stack. Let an MLP \(H_\psi(q,p,t,u)\) produce the leaf Hamiltonian, where \(u\) is the current observation, action, or conditioning vector. Let a second small MLP \(v_\psi(q,p,t,u)\) produce Reeb speed, initialized near one for physical time or near a learned positive constant for abstract depth. 2. Pseudocode: compute \(g_q=\nabla_qH_\psi\) and \(g_p=\nabla_pH_\psi\) with automatic differentiation; perform the three implicit/semi-implicit updates in the displayed leapfrog formula, using one fixed-point iteration or two Newton iterations for the half-step \(p_{k+1/2}\); then set \(t_{k+1}=t_k+h\,\mathrm{softplus}(v_\psi)\) if monotonic time is required. Feed \((q_{k+1},p_{k+1},t_{k+1})\) to the next block. Add an optional residual observation update after the geometric transition, but do not let that update alter \(t\). 3. Computed from the mathematics: the canonical leaf split, Hamiltonian gradients, symplectic leapfrog map, and Reeb/transverse separation. Estimated empirically: the Hamiltonian MLP parameters, step size \(h\), the best parameterization of \(v_\psi\), and whether the approximate learned \(\beta\)-preservation survives the observation residual. Monitor the Jacobian determinant of the \((q,p)\) transition and the symplectic error \(\|D\Phi^TJ D\Phi-J\|_F\), where \(J=\begin{bmatrix}0&I\\-I&0\end{bmatrix}\). 4. First experiment: train a small latent ODE/RNN or action-conditioned world model on a synthetic harmonic oscillator and a chaotic double pendulum, comparing an equal-parameter GRU, an unconstrained MLP residual transition, and the proposed layer at equal rollout FLOPs. Evaluate one-step prediction, 100- to 1000-step rollout error, energy drift, and gradient norm stability. The target signal is lower long-horizon error and bounded energy drift at similar one-step accuracy; on a real test, use a small MuJoCo trajectory dataset and compare validation rollout error at horizons 10, 50, and 200.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a PyTorch cosymplectic-inspired Hamiltonian transition with two fixed-point leapfrog half-steps, plus a residual MLP baseline and oscillator rollout experiment. The mathematical sanity check passed: leapfrog had determinant 1, near-zero symplectic error, and bounded energy after 1000 steps, while Euler energy exploded. However, after matched one-step training, the Hamiltonian idea was worse at horizons 10, 50, and 200 and had slightly higher energy drift, so no empirical win was observed.

### Mechanism check

- Verdict: Built a PyTorch cosymplectic-inspired Hamiltonian transition with two fixed-point leapfrog half-steps, plus a residual MLP baseline and oscillator rollout experiment. The mathematical sanity check passed: leapfrog had determinant 1, near-zero symplectic error, and bounded energy after 1000 steps, while Euler energy exploded. However, after matched one-step training, the Hamiltonian idea was worse at horizons 10, 50, and 200 and had slightly higher energy drift, so no empirical win was observed.
- Confidence: 9/10
- Limitations: Only a small synthetic harmonic oscillator was tested; no chaotic double pendulum, observations/actions, learned Reeb coordinate, GRU comparison, FLOP normalization, multiple seeds, or real-world dataset evaluation was performed. The learned Hamiltonian used a small MLP and two fixed-point iterations, so the negative learning result is not a broad proof against the architecture.

## Artifacts

- [experiment.py](https://synthcore.org/code/1297/experiment.py)
- [report.md](https://synthcore.org/code/1297/report.md)
- [results.json](https://synthcore.org/code/1297/results.json)
- [Download all files as ZIP](https://synthcore.org/download/1297)

## Disclaimer

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