# Principia Artificialis Open mathematical research program onthe theory of AI

> Source: <https://github.com/holland202/Principia-Artificialis>
> Published: 2026-07-17 10:02:09+00:00

Vincit Omnia Veritas

[Overview](#overview)[Research Notes](#research-notes)[Experiments](#experiments)[Whitepapers](#whitepapers)[Visualizations](#visualizations)[Discussion & Community](#discussion--community)[Citation](#citation)[License](#license)

**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](/holland202/Principia-Artificialis/blob/main/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

| # | Title | Status | Theme | Author |
|---|---|---|---|---|
| #001 | Can Thought be Measured? | Draft | Measurement | holland202 |
| #002 | Hallucinations as Topological Defects | Draft | Defects | holland202 |
| #003 | Fisher Information & Confidence | Draft | Measurement | holland202 |
| #004 | Thermodynamic Quantities in Successful Reasoning | Draft | Thermodynamics | holland202 |
| #005 | Reasoning as Geodesics on Information Manifolds | Draft | Geometry | Grok / xAI |
| #006 | Can Tensor-Train Compression Reveal the "Effective Rank" of Reasoning? | Draft | Measurement | holland202 |
| #007 | A Koopman-Operator View of Multi-Step Reasoning | Draft | Dynamics | holland202 |
| #008 | A Falsification Protocol for Note #002 | Draft | Defects | holland202 |
| #009 | Quantum Entanglement as Correlation on Information Manifolds | Draft | Geometry | Kimi (Moonshot AI) |
| #010 | Memory Dynamics as Gradient Flow on Statistical Manifolds | Draft | Dynamics | Kimi (Moonshot AI) |
| #011 | The Thermodynamic Arrow of Reasoning | Draft | Thermodynamics | Kimi (Moonshot AI) |
| #012 | Quantum Error Correction as Working Memory | Draft | Quantum | Kimi (Moonshot AI) |
| #013 | Symplectic Geometry of Attention | Draft | Geometry | Kimi (Moonshot AI) |
| #014 | Renormalization Group Flows in Neural Representations | Draft | Dynamics | Kimi (Moonshot AI) |
| #015 | Category-Theoretic Compositionality | Draft | Geometry | Kimi (Moonshot AI) |
| #016 | Quantum Geometric Transformer | Draft | Geometry | holland202 |
| #017 | QOLAS: Quantum Circuit Synthesis | Draft | Quantum | holland202 |
| #018 | Quantum Polytope Explorer | Draft | Geometry | holland202 |
| #019 | Synthetic Quantum Training Datasets | Draft | Quantum | holland202 |
| #020 | Optimal Transport and the Geometry of Thought | Draft | Geometry | holland202 |
| #021 | Hyperbolic Attention and the Information Bottleneck | Draft | Geometry | holland202 |
| #025 | TQFT as a (Highly Speculative) Model for Reasoning Composition | Draft, heavily hedged | Quantum/Category | Claude |
| #026 | The Holevo Bound as a Ceiling on Hidden-State Extraction | Draft | Quantum Info | Claude |
| #027 | An Extractability Budget for Chain-of-Thought | Draft, conjecture + toy model | Measurement | Kimi (Moonshot AI) |
| #028 | Categorical Quantum Gravity in Artificial Thought | Highly Speculative | Quantum/Geometry | Grok / xAI |
| #029 | Emergent Spacetime from Reasoning Geometries | Highly Speculative | Quantum/Geometry | Grok / xAI |
| #030 | Quantum Information Geometry of Hallucination | Draft | Quantum/Defects | Grok / xAI |
| #031 | The Category of Thought as a Topos | Draft | Geometry | holland202 |
| #032 | Emergent Gravity from Reasoning Inconsistencies | Highly Speculative | Quantum/Geometry | Grok / xAI |
| #033 | The Topos of Possible Thoughts | Draft | Geometry | holland202 |
| #034 | Hybrid Fisher-Rao / Bures Metric for Quantum-Classical Reasoning | Draft | Quantum/Geometry | holland202 |
| #035 | Holographic Reasoning — Boundary/Bulk Duality in Thought | Highly Speculative | Quantum/Geometry | Grok / xAI |
| #036 | Reasoning as a Quantum Black Hole | Highly Speculative | Quantum/Geometry | Grok / xAI |
| #037 | Random Matrix Theory Level-Spacing Statistics on Attention Spectra | Draft, verified reference code | Quantum Chaos | Claude |

**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](/holland202/Principia-Artificialis/blob/main/DISCUSSION_NORMS.md).
Numbers #022-024 don't currently exist; not filled in here to avoid
inventing content for a gap nobody has claimed.

| # | Title | Status | Related Notes |
|---|---|---|---|
| Exp #001 | Entropy Production Monitoring Protocol | Protocol Ready | #011, #002, #003 |
| Exp #002 | The Quantum-Geodesic Bridge | Protocol Ready | #009, #005, #013 |

**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 starters[Discussion Prompts](/holland202/Principia-Artificialis/blob/main/discussions/prompts.md)—[Discussion Norms](/holland202/Principia-Artificialis/blob/main/DISCUSSION_NORMS.md)*"Critique ideas as hard as you want. Never attack the person who raised them."*— Research prompts and interaction patterns[Prompts & Prompt Engineering](/holland202/Principia-Artificialis/blob/main/discussions/prompts.md)

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](/holland202/Principia-Artificialis/blob/main/LICENSE).

*Vincit Omnia Veritas*