Recursive Criticality of AI Self-Improvement

Computational notebook accompanying the paper “Recursive Criticality of AI Self-Improvement” by Mikhail Burtsev (London Institute for Mathematical Sciences, 2026).

This repository contains the publication-ready Jupyter notebook used for the model calculations, numerical scenarios, figures, and tables discussed in the paper.

The numerical scenarios are illustrative model experiments, not forecasts of AGI or ASI timelines.

• Recursive_Criticality_of_AI_Self_Improvement.ipynb — complete computational notebook, including the paper narrative, equations, simulations, figures, tables, and references.
• figures/ — created automatically when the figure-generation cells are run.

No external dataset is required. The numerical results are generated directly from the model and parameter choices defined in the notebook.

The notebook develops and explores a dynamical model of recursive AI self-improvement in which AI capability feeds back into the R&D process that produces successor systems.

It includes:

• the recursive-criticality framework and recursive reproduction number;
• a capability-dependent research frontier and frontier hardening;
• delayed recursive feedback through the AI development cycle;
• comparison of recursive and no-RSI trajectories;
• smooth-scaling, weak-supercritical, transient-takeoff, and rapid-transition reference scenarios;
• phase diagrams over recursive-gain and frontier-hardening parameters;
• threshold-crossing calculations for illustrative AGI and ASI capability levels;
• coupled multi-actor research systems and cross-actor transfer;
• closed-laboratory, open-ecosystem, and global-competition configurations;
• physical-compute/headroom constraints on deployable capability;
• numerical sweeps used to generate the publication figures and tables.

Clone the repository and create a Python environment:

git clone <REPOSITORY-URL>
cd <REPOSITORY-DIRECTORY>

python -m venv .venv
source .venv/bin/activate
python -m pip install --upgrade pip
python -m pip install jupyterlab numpy pandas scipy matplotlib networkx tqdm ipython numba

On Windows PowerShell, activate the environment with:

.venv\Scripts\Activate.ps1

Then start Jupyter:

jupyter lab

Open Recursive_Criticality_of_AI_Self_Improvement_text_synced.ipynb and run the cells from top to bottom.

The notebook creates a local figures/ directory and writes the publication figures there.

The committed notebook reports the following versions for its recorded execution:

The notebook additionally uses IPython, tqdm, and Numba. Numba is required for exact recomputation of the publication-resolution physical-headroom sweep.

Small rendering differences can occur across Matplotlib versions. The physical-headroom figure cell notes that the source publication PDF metadata reported Matplotlib 3.9.2, while the recorded notebook environment reports Matplotlib 3.10.0.

Running the relevant cells writes the following PDF files:

figures/
├── phase_diagrams.pdf
├── base_scenarios.pdf
├── strategic_network_topologies.pdf
├── compound_strategic_dynamics.pdf
└── physical_headroom_deployability.pdf

The last figure uses a 61 × 61 parameter sweep. Its publication-resolution recomputation requires Numba and is the most computationally intensive part of the notebook.

For a clean reproduction:

1. Start a fresh Python kernel.
2. Run all cells in notebook order.
3. Do not change the reference parameters unless intentionally exploring an alternative scenario.
4. Confirm that the figures/ directory contains the five generated PDFs listed above.
5. Compare the displayed threshold tables and figure outputs with the committed notebook outputs.

The notebook is self-contained: it does not download data or depend on external input files.

The model distinguishes recursive amplification from rapid progress caused by higher baseline research throughput. It also separates software-side recursive dynamics from constraints on physical deployability and extends the analysis from a single research actor to coupled research ecosystems.

The AGI and ASI thresholds used in the numerical scenarios are illustrative state-variable thresholds within the model. They should not be interpreted as empirical definitions or predicted dates.

If you use this notebook or its results, please cite the accompanying paper:

Mikhail Burtsev. Recursive Criticality of AI Self-Improvement. 2026. arXiv preprint - https://arxiv.org/abs/2609.00137v1

Mikhail Burtsev
London Institute for Mathematical Sciences