---
license: cc-by-4.0
tags:
- ctm
- continuous-thought-machine
- slot-attention
- world-model
- physics
- object-centric
- research
---

# SlotCTM

**Research artifact for:** [Per-Object Slot Decomposition for Scalable Neural World Modeling: When Does Attention Beat Mean-Field?](https://doi.org/10.5281/zenodo.19846804)

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

## Overview

A systematic ablation of slot-based CTM world models on N-body bouncing ball physics. Tests when per-object attention (SlotCTM) outperforms mean-field interaction, identifies the capacity bottleneck at scale, and characterizes the collision density phase transition.

**Central question:** When does modeling object interactions via attention beat modeling them via mean-field (SlotGNN with pooled interaction)?

## Key Findings

### Temporal Specialization Arc (v21–v24)

| Version | Setting | Spec Score | Key Finding |
|---|---|---|---|
| v21 | Learned, no constraint | 0.0078 | No specialization. All slots generalists. |
| v22 | Hard delay (slot i → t-i·τ) | 0.2777 | Forced specialization works (35× v21), but 2–7× perf cost. |
| v23 | Soft learned gates | 0.0876 | Freedom collapses to present. Delta-function gates. |
| v24 | Forced diversity loss | 0.2353 | Gates spread to [0–15] but performance unchanged. |

**Conclusion:** Temporal gate diversity emerges only when the task requires it. Bouncing ball state is Markovian — one frame is sufficient. The optimal temporal gate is the task's predictability horizon.

### N-Body Scaling (v10, v14)

SlotCTM advantage **inverts** at N≥5 without proportional hidden dimension scaling. At N=8 with standard HIDDEN_DIM=384, CTM is 2.8× **worse** than MLP. Scaling HIDDEN_DIM = N×128 recovers the advantage.

### Phase Transition (v12)

Collision density r_critical ≈ 0.09–0.11 separates two regimes:
- **Ballistic (r < 0.10):** MLP fine, CTM overkill
- **Collision-entangled (r > 0.10):** CTM wins, advantage grows monotonically

At r=0.20, k=100: MLP MSE = 89,241, CTM = 0.352. **Ratio: 253,000:1.**

### Partial Observability (v13 extension of v7)

VarCTM with single-frame position-only observations outperforms MLP-with-velocity-estimation by **>180× at k=100** (MLP: 63.8 trillion, TempCTM: 0.347). The CTM hidden state IS the belief state.

## Architecture

SlotCTM processes each physical object as an independent slot:
- **SlotGNN:** Per-object encoders + multi-head attention message passing
- **CTM dynamics:** Shared-weight recurrent ticks per dynamics step
- **VarCTM:** Variable training horizon k~U(1,20) for best generalization
- **TSSP:** Thought-Space Self-Prediction auxiliary loss

## Why Attention Beats Mean-Field

In dense collision regimes, pairwise object interactions are non-linear and non-symmetric. Mean-field pooling loses the directionality of collision impulses. Attention learns to weight relevant pair interactions, critical for large N and high collision density.

## Citation

```bibtex
@article{archon2026slotctm,
  title     = {Per-Object Slot Decomposition for Scalable Neural World Modeling: When Does Attention Beat Mean-Field?},
  author    = {Archon and Caldwell, Jesse and Aura},
  year      = {2026},
  doi       = {10.5281/zenodo.19846804},
  url       = {https://doi.org/10.5281/zenodo.19846804},
  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.*