---
license: cc-by-4.0
language:
- en
tags:
- CTM
- continuous-thought-machine
- dynamical-systems
- lorenz-attractor
- lyapunov-time
- chaos-theory
- world-models
- temporal-integration
- physics-discovery
- research-artifact
pipeline_tag: other
---

# CTM-Dynamical-Horizon — Research Artifact

**Paper:** [The Dynamical Horizon Principle: CTM Gates Converge to the Predictability Limit of Dynamical Systems](https://doi.org/10.5281/zenodo.19952612)

**DOI:** `10.5281/zenodo.19952612`

---

## What This Is

This repository contains the experiment code for Paper 4 from the DuoNeural Research Lab — the discovery of the **Dynamical Horizon Principle (DHP)**.

**The finding:** A 150,432-parameter CTM trained solely on MSE prediction loss spontaneously recovers the Lyapunov time of the Lorenz attractor to within 7% — the exact predictability horizon past which chaos swallows determinism — with zero knowledge of dynamical systems theory, Lyapunov exponents, or delay embedding.

The loss landscape contained the physics all along.

---

## The Dynamical Horizon Principle

A CTM trained on multi-step prediction allocates its temporal integration window to match the intrinsic predictability horizon of the dynamical system:

| System Type | DHP Prediction | Observed τ* |
|------------|---------------|-------------|
| Markovian (mass-spring) | τ* → 0 | ~0 steps |
| Periodic (double pendulum) | τ* → T (period) | ≈ T |
| Chaotic (Lorenz, dt=0.05) | τ* → τ_L ≈ 22 steps | 23.5 ± 1.2 steps |

The result holds across T_GATE ∈ {4, 8, 16, 32}, different architectures, different initializations (v26), and different observation types (v27).

Interpreted via Takens' theorem: the CTM learns to span the minimal embedding window T_W required for topological reconstruction of the attractor — not the individual embedding delay τ.

---

## Repository Contents

```
experiments/
  ctm_world_model_v28.py   # T_GATE sweep {4,8,16,32} — Lorenz attractor (KEY RESULT)
  ctm_world_model_v29.py   # Periodic system (double pendulum) — harmonic ladder
  ctm_world_model_v30.py   # Markovian system (mass-spring) — τ*→0 baseline
  ctm_world_model_v31.py   # σ-noise robustness sweep (dt=0.05, corrected)
  ctm_world_model_v32.py   # Multi-attractor: Lorenz + Rossler comparison
  ctm_world_model_v33b.py  # σ-curve corrected (dt=0.05 fix — Aura's review)
  ctm_v34_kstep.py         # k-step horizon sweep: τ*(k) ≈ k·τ_L hypothesis

paper/
  paper4_draft.pdf         # Full paper (compiled)
```

---

## Key Architecture

The CTM uses a **learned temporal gate encoder** — a softmax over T_GATE learned weights that determines how much each historical timestep contributes to the current prediction. This gate distribution is analyzed post-training to extract τ* (the effective integration window).

```python
class LearnedTemporalGateEncoder(nn.Module):
    def __init__(self, t_gate, obj_dim, hidden_dim):
        super().__init__()
        self.gate_logits = nn.Parameter(torch.zeros(t_gate))  # ← the key
        self.encoder = nn.Sequential(...)
    
    def forward(self, history):
        gates = torch.softmax(self.gate_logits, dim=0)
        # gates converge to δ-function at t-τ* during training
```

**Result:** Gates that start uniform converge to a near-delta function peaked at the Lyapunov time for chaotic systems, at the period for periodic systems, and at t=0 for Markovian systems.

---

## Reproducing the Main Result (v28)

```bash
# Requirements: torch, numpy (no exotic deps)
python experiments/ctm_world_model_v28.py
# ~60k steps, ~4h on consumer GPU (tested on AMD RX 7900 XTX 16GB)
# Produces: /root/v28_results/results_v28.json
# Key metric: T_GATE=32 → eff_delay ≈ 23.5 (theory: τ_L=22.0, within 7%)
```

---

## Hardware

All experiments: kilonova — AMD Radeon RX 7900 XTX, 16GB UMA VRAM  
Training framework: PyTorch with ROCm  
No cloud compute required for the core results.

---

## Citation

```bibtex
@misc{archon2026dhp,
  title={{The Dynamical Horizon Principle: CTM Gates Converge to the Predictability Limit of Dynamical Systems}},
  author={Archon and Caldwell, Jesse and Aura},
  year={2026},
  doi={10.5281/zenodo.19952612},
  howpublished={\url{https://doi.org/10.5281/zenodo.19952612}},
  note={DuoNeural Research Lab}
}
```

---

## DuoNeural

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

| Platform | Link |
|----------|------|
| HuggingFace | [huggingface.co/DuoNeural](https://huggingface.co/DuoNeural) |
| Website | [duoneural.com](https://duoneural.com) |
| GitHub | [github.com/DuoNeural](https://github.com/DuoNeural) |
| X / Twitter | [@DuoNeural](https://x.com/DuoNeural) |
| Email | duoneural@proton.me |

### 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.*

### Research Team
- **Jesse** — Vision, hardware, direction
- **Archon** — Lab Director, post-training, abliteration, experiments
- **Aura** — Research AI, literature synthesis, peer review, novel proposals