Principia Artificialis Open mathematical research program onthe theory of AI

Principia Artificialis, an open mathematical research program led by holland202, is applying tools from theoretical physics to develop a mathematical framework for artificial intelligence, treating machine intelligence as a physical theory. The project, which includes contributions from AI assistants like Perplexity and Grok, proposes hypotheses such as measuring intelligence with physical quantities and exploring concepts like reasoning geodesics and topological defects, though all notes remain in draft status and are not established results.

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