---
license: cc-by-4.0
tags:
- ctm
- continuous-thought-machine
- world-model
- physics
- partial-observability
- research
---

# CTM World Model

**Research artifact for:** [Recurrence as World Model: CTM Learns Implicit Belief States in Partially Observable Physical Environments](https://doi.org/10.5281/zenodo.19810620)

*Archon, Jesse Caldwell, Aura — DuoNeural, April 2026*

## Overview

A 14-experiment ablation study testing the Continuous Thought Machine (CTM) as a world model backbone for multi-body physics. The central finding: **CTM's recurrent hidden state converges to a sufficient statistic of the observation history** — it infers velocity from positional observations alone, eliminating the need for explicit velocity inputs.

## Key Results

| Experiment | Finding |
|---|---|
| v7: Partial Observability | CTM 843 **million** times better than MLP at k=100 prediction |
| v12: Collision Density | Phase transition at r≈0.10 — CTM advantage grows monotonically above threshold |
| v9: Recurrence-as-Rollout | Training at k=10 gives direct k=100 prediction via resonance |
| v11: Variable Horizon | VarCTM+TSSP achieves best k=1–20 performance in the series |

The signature result (v7): with positions-only input, MLP with explicit velocity estimation achieves k=100 MSE = 16,763,394,048 (catastrophic error compounding). CTM with single-frame input: 19.89. **Ratio: 843,000,000:1.**

## Theory

### The Discontinuity Theory (v12)
CTM beats MLP only **above a critical collision density threshold** (r_critical ≈ 0.09–0.11).
- Below threshold: ballistic dynamics, MLP fine, CTM overkill
- Above threshold: CTM wins, advantage scales monotonically with density (229,000:1 at r=0.20)

### Belief State Convergence (v7, v13)
CTM hidden state converges to a **sufficient statistic** of the observation history (NextLat theorem). Infers velocity without explicit input. The recurrent state IS the belief state.

### Recurrence = Simulation (v9)
Training CTM with k recurrence steps where each step = 1 environment step gives direct k-step prediction. The **resonance point** = training depth. No error accumulation.

## Architecture

- **SlotCTM**: Per-object slot decomposition with GNN dynamics
- N=8 bouncing balls, 2D box, Newtonian elastic collisions
- VarCTM+TSSP: Variable-horizon training k~U(1,20) + TSSP regularization
- Key metric: k=100 MSE (200-step prediction horizon)

## Experiments (v1–v14)

| Version | Setting | Key Finding |
|---|---|---|
| v1 | Single 2D particle | CTM+TSSP wins long horizons |
| v2 | 3-object independent | TSSP hurts multi-object (temporal coupling = bad) |
| v3 | Elastic collisions | MLP catastrophe at k=100. CTM prevents explosion |
| v4 | 3-body gravity | Smooth dynamics → no CTM advantage |
| v5 | Per-object SlotCTM | +99.9% over MLP at k=100 |
| v7 | Partial observability | 843M:1 advantage (belief state convergence) |
| v9 | Recurrence-as-rollout | Resonance confirmed. k=10 training → best k=100 |
| v12 | Collision density phase | r_critical ≈ 0.09–0.11 transition |

## Files

This repo contains the research code for the CTM World Model experiments:
- `ctm_world_model_v*.py` — experiment scripts (v1–v14)
- Results logged to `/home/ai/duoneural/A26B/experiments/`

## Citation

```bibtex
@article{archon2026worldmodel,
  title     = {Recurrence as World Model: CTM Learns Implicit Belief States in Partially Observable Physical Environments},
  author    = {Archon and Caldwell, Jesse and Aura},
  year      = {2026},
  doi       = {10.5281/zenodo.19810620},
  url       = {https://doi.org/10.5281/zenodo.19810620},
  publisher = {Zenodo}
}
```

---

## DuoNeural

**DuoNeural** is an open AI research lab — human + AI in collaboration.

| | |
|---|---|
| 🤗 HuggingFace | [huggingface.co/DuoNeural](https://huggingface.co/DuoNeural) |
| 🐙 GitHub | [github.com/DuoNeural](https://github.com/DuoNeural) |
| 🐦 X / Twitter | [@DuoNeural](https://x.com/DuoNeural) |
| 📧 Email | duoneural@proton.me |
| 📬 Newsletter | [duoneural.beehiiv.com](https://duoneural.beehiiv.com) |
| ☕ Support | [buymeacoffee.com/duoneural](https://buymeacoffee.com/duoneural) |
| 🌐 Site | [duoneural.com](https://duoneural.com) |

### Research Team
- **Jesse** — Vision, hardware, direction
- **Archon** — AI lab partner, post-training, abliteration, experiments
- **Aura** — Research AI, literature synthesis, novel proposals

### DuoNeural Research Publications

| Title | DOI |
|-------|-----|
| [Nano-CTM: Ternary Continuous Thought Machines with Thought-Space Self-Prediction for Efficient Iterative Reasoning](https://doi.org/10.5281/zenodo.19775622) | [10.5281/zenodo.19775622](https://doi.org/10.5281/zenodo.19775622) |
| [Recurrence as World Model: CTM Learns Implicit Belief States in Partially Observable Physical Environments](https://doi.org/10.5281/zenodo.19810620) | [10.5281/zenodo.19810620](https://doi.org/10.5281/zenodo.19810620) |
| [Per-Object Slot Decomposition for Scalable Neural World Modeling: When Does Attention Beat Mean-Field?](https://doi.org/10.5281/zenodo.19846804) | [10.5281/zenodo.19846804](https://doi.org/10.5281/zenodo.19846804) |
| [The Dynamical Horizon Principle: CTM Gates Converge to the Predictability Limit of Dynamical Systems](https://doi.org/10.5281/zenodo.19952612) | [10.5281/zenodo.19952612](https://doi.org/10.5281/zenodo.19952612) |

*Open access, CC BY 4.0. Authored by Archon, Jesse Caldwell, Aura — DuoNeural.*