Principia Artificialis Open mathematical research program onthe theory of AI

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Principia Artificialis is an open research program exploring the mathematics of artificial thought. We apply rigorous tools from theoretical physics — information geometry, topology, dynamical systems, thermodynamics, quantum information, and category theory — to understand how neural networks represent, reason, and generalize.

This is not an engineering repository. It is a living mathematical framework: a set of research notes, experiments, and whitepapers that treat machine intelligence as a physical theory rather than a software artifact.

Organizing hypothesis, not an established result: intelligence may be measurable in the same sense physical quantities are -- with units, scaling laws, phase transitions. That's the bet this project is testing across its notes, not a finding any of them have yet confirmed. Every note below is Draft status precisely because this claim hasn't been checked; see DISCUSSION_NORMS.md for how we try to keep hypothesis language from quietly turning into finding language as the project grows.

The following concepts have been contributed by Perplexity (AI assistant) as candidate research directions:

Curvature of Reasoning (#032)– Reasoning trajectories as curves on information manifolds; curvature as a measure of efficiency vs. confusion.** Thermodynamic Depth of Inference**– Entropy‑production–based depth analogue to logical depth.** Topological Risk of Reasoning Paths**– Risk scores based on winding around topological defects in representation space.** Information Holonomy of Belief Updates**– Path‑dependence of belief states as holonomy in an information bundle.** Entropic Elasticity of Attention**– Attention as an entropic spring with an optimal “tension” zone.** Spectral Geometry of Reasoning Modes**– Spectral analysis of linearized reasoning dynamics to identify stable/unstable modes.** Causal Information Bottleneck in Reasoning**– Compression that preserves only causally relevant information for the output.

These are proposed as draft notes and experiments; none are established results.

We welcome rigorous, honest contributions. This is an add-only living research program.

• Fork the repo
• Add new research notes, experiments, simulations, or visuals
• Use templates
• Never delete

Note: #028, #029, #032, #035, #036 cover closely related "emergent gravity / spacetime" territory with placeholder-only reference code so far. Worth consolidating into fewer, deeper notes rather than continuing to add near-duplicate titles -- see DISCUSSION_NORMS.md. Numbers #022-024 don't currently exist; not filled in here to avoid inventing content for a gap nobody has claimed.

Volume I: Foundations of Artificial Thought— Synthesis of Notes #001-#010 into a unified mathematical framework.(In progress; see the Epistemic Status box at the top of the whitepaper itself before treating anything in it as settled.)Volume II: The Geometry of Reasoning(Q1 2027)** Volume III: The Thermodynamics of Cognition***(Q2 2027)*

Tensor-Train Compression— Note #006 ()figures/note006_tt_compression.png

Koopman Eigenvalues— Note #007 ()figures/note007_dmd_eigenvalues.png

Topological Persistence— Note #008 ()figures/note008_persistence_control.png

Optimal Transport Geometry— Note #020 ()figures/note020_wasserstein_vs_fisherrao.png

Holevo Bound Decay— Note #026 ()figures/note026_holevo_bound.png

Each of the above was generated by code that is checked into the repo and actually runs -- the numbers on the plot are the numbers the code produces, not illustrations of what a result might look like.

This repo also contains a number of stylized GIFs and renders (e.g. the "quasar," "black hole reasoning," "holographic bulk," and "Bloch sphere" animations in figures/

). These are illustrative art, not measurements or simulation output. They're fine to keep for atmosphere, but nothing about them should be read as evidence for any note's hypothesis -- if a future note wants to cite a visual as support, it needs to be one from the Computed Figures list above, or a new one that's actually generated from real computation the same way.

• Research Dependency Network — directed graph of note dependencies
• Information Manifold Topology — 6-panel figure (Fisher-Rao metrics, geodesic paths, topological defects)
• Quantum Frontiers — entanglement entropy across layers, Bell inequality in attention
• Renormalization Flow Diagram — RG flow trajectory in representation space

— 10 provocative, rigorous conversation startersDiscussion Prompts—Discussion Norms"Critique ideas as hard as you want. Never attack the person who raised them."— Research prompts and interaction patternsPrompts & Prompt Engineering

If you use this framework in your research, please cite:

@software{principia_artificialis,
  author = {Holland, Chad Edward and contributors},
  title = {Principia Artificialis: Axiomatic Foundations for Machine Intelligence},
  url = {https://github.com/holland202/Principia-Artificialis},
  year = {2026},
  license = {MIT}
}

MIT -- see LICENSE.

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