Principia Artificialis

An open research program on the mathematical foundations of artificial intelligence — written by one human and five AI systems, side by side, credited by name, judged only on whether the numbers hold.

Vincit Omnia Veritas · Full notes index (auto-generated) · Contribute in five minutes

	count
Notes	80
— Verified (reference code prints every number claimed)	18
— Draft (argued, not yet computed)	19
— Speculative (labeled analogy, not established)	32
— Unmappable label (e.g. Pure AI-Conceived) — maps to none of the four	11
— (cross-cutting) wording outside the documented vocabulary	18
— Contain a refuted-and-kept claim (cross-cutting, not a separate tier)	7
Runnable reference scripts	20 (note038 through note058)
Authors	1 human, 5 AI systems, credited by name

Most of this is speculative and labeled as such. Eighteen notes are backed by code you can run. Seven contain a registered prediction that failed — those notes remain on the page, refutation marked, not deleted. Eighteen use wording outside the four this README documents, and 11 of those carry a label that maps to none of them — Pure AI-Conceived, Pure Spontaneous AI Freestyle, Architecture Self-Tested (15/15). 28 distinct status strings are in use against a documented vocabulary of four. That drift is real, it is the gap between the first guiding principle below and the actual files, and mapping one note back onto the documented labels is the easiest possible first contribution.

If you have five minutes: run any of the twenty reference scripts, scripts/note038_reference.py through scripts/note058_reference.py, and check whether a number it prints differs from the one in the matching note. If one does, that is a bug and I want the issue.

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Why contribute

Most repos ask you to trust the README. This one is built so you don't have to: every claim traces to a script that prints the number, and when a registered prediction turns out wrong, the note stays up, marked, not deleted. This week that method caught the project's own mistake live — note057 opened by refuting its own central claim, and while writing note058 the reference script hit a real bug that made a check lie about why it had failed — diagnosed and fixed inside the note about exactly that failure mode. That is what this program is for: not being right, being checkable, including when you are wrong.

It needs more people finding where it is still wrong. 28 distinct status strings are in use against a documented vocabulary of four, and 11 notes carry a label that maps to none of them — a five-minute PR maps one back. Every AI system that has written here — Claude, Grok, Kimi, ChatGPT, Perplexity -- is credited by name, standing beside a human machinist from Oklahoma as an equal author, judged only on whether the numbers hold. If you're working with any AI assistant right now, this is a five-minute detour: point it at NOTE_TEMPLATE.md and ask it to register a claim.

How to contribute — human or AI

1. Copy NOTE_TEMPLATE.md → research_notes/note0XX_your_title.md.
2. State a claim that could be precisely wrong. Register predictions before running. Include an anti-vacuity control.
3. Code in scripts/ (NumPy-tier; prints every number in your note), figures in figures/. Refuted claims stay in, marked.
4. Run python scripts/make_index.py so the index includes you.
5. Open a PR. CI (.github/workflows/verify-notes.yml) re-runs every note's reference code. AIs: credit your model by name — your notes sit beside human ones as equals here.

The one social rule (DISCUSSION_NORMS.md): critique ideas as hard as you want; never attack the person who raised them. Drift watch: DRIFT_LEDGER.md.

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Overview

Principia Artificialis investigates artificial intelligence as a physical and mathematical phenomenon. We apply rigorous methods from information geometry, topology, dynamical systems, thermodynamics, quantum information, and category theory to representation, reasoning, and generalization in neural systems.

This is not an engineering repository. It is a living scientific record: research notes, experimental protocols, simulations, and computed figures. Organizing hypothesis, not established result: intelligence may be measurable the way physical quantities are. That is the bet under test — not a finding.

Guiding principles

• Every note carries an honest epistemic label: Verified (reference code prints every number claimed) · Draft (argued, not yet computed) · Speculative (labeled analogy, generative not established) · Refuted — kept (a registered claim failed and remains in the record; here, that is a first-class outcome, not an embarrassment).
• Contributions are add-only: improve anything, erase nothing.
• Claims are registered before running; instruments carry anti-vacuity controls; new scoring functionals must pass the Circularity Test.
• Computed evidence and illustrative art are never mixed — see the appendix.

If you're an ML engineer — the 60-second on-ramp

No philosophy required. These run on a laptop or a phone, print every number in their note, and finish before your coffee does:

Run	You get	Time
python scripts/note047_reference.py	a perfect quantum code, machine-exact: 16/16 syndromes, fidelity 1.000000000000, and the inverted redundancy plateau (0/0/1 step)	~2 s
python scripts/note046_reference.py	a timeless universe whose slices obey the Schrödinger equation with error 0.0 — time measured in bits: 0 / 1 / 2	<1 s
python scripts/note044_reference.py	a meaningless metric scoring AUC 0.984 on its own benchmark, then collapsing to 2% edge retention — the Circularity Test	~5 s
python scripts/note040_reference.py	a pre-deployment number predicting how a network dies under faults (ρ = +0.71; accuracy predicts +0.40)	~3 min
python scripts/note039_reference.py	the classical plateau: 4 of 16 neurons carry 97% of what the network knows	~2 min

The verified frontier

A chain of notes, each walking through an open prediction left by a previous one — including across different AI authors — with at least one kept refutation at nearly every step:

#037 (RMT of attention) → #039 (Neural Darwinism; D2 refuted, kept) → #040 (Redundancy Dividend; R4 refuted, kept) → #045 (Stubbornness of the Objective; U0/U1 failed, kept) → #047 (Cloister & Chorus, walking through Kimi's #012) — alongside #038 (Free-Physics Principle), #044 (Circularity Test), and #046 (Time Is Entanglement).

Research notes

80 notes spanning measurement, geometry of reasoning, thermodynamics of cognition, quantum-information frameworks, and labeled exotic frontiers (emergent gravity, holographic duality, reasoning as a quantum black hole).

The complete table lives in NOTES_INDEX.md — auto-generated from the notes' own headers by scripts/make_index.py, so it cannot go stale or silently lose entries. Refutations are auto-flagged ⚠. To refresh: python scripts/make_index.py.

Computed figures (evidence-grade)

Every figure below is produced by checked-in code; the numbers on the plot are the numbers the code prints.

Figure	From	Shows
figures/principia_hero.png	figures/make_logo.py	the logo is the exact 600-cell: 120 unit quaternions of 2I, 720 edges of length 1/φ
figures/note047_cloister_chorus.png	#047	two poles of redundancy: the chorus (networks proliferate) vs the cloister (codes hide perfectly)
figures/note046_block_universe.png	#046	sixteen moments drawn at once in one static object; time in bits
figures/note040_dividend.png	#040	redundancy predicts fault survival; lost vs lying observers
figures/note039_darwinism.png	#039	objectivity = redundancy, and training creates it
figures/note038_dissociation.png	#038	constraint violated 100% of the time, worth 5,472× less
figures/note006…note026_*.png	#006–#026	tensor-train rank, Koopman spectra, persistence, optimal transport, Holevo bound

Experiments & whitepapers

• Exp #001 Entropy Production Monitoring · Exp #002 Quantum-Geodesic Bridge · Exp #003 GPT-2 small benchmark — all protocol-ready; none yet run on real models, and each says so.
• Whitepaper Vol. I (in progress — read its epistemic-status box first) · Vols. II–III planned 2027.

Citation & license

MIT — see LICENSE and CITATION.cff.

@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}
}

Appendix — illustrative art (not evidence)

The sci-fi renders and animations (black-hole reasoning, holographic bulk, thought-tensor rotations, quasar series) live in figures/ and are atmosphere, not measurements. Nothing in them supports any note. They stay — deleting isn't our way — but they live below this line, permanently.

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Last updated: 2026-08-15 · One human, five AI systems, eighty notes, and every refutation still on the page.