The Institute for Applied Ontological Mathematics
One mathematical foundation.Endless possibilities.
You already trust mathematics every time you cross a bridge. We put that same certainty into the machines, models and instruments now making decisions for us: proved before they run, measured when they do, and published for you to check.
Curious why it works? The Principia: from a single distinction to machines that can be trusted →
Why we exist
Intelligence is accelerating. Mathematics is how we stay in command of it.
- The pace
Machines now improve faster than we can test them.
Artificial intelligence, and the superintelligence coming after it, already designs chips, balances power grids, reads medical signals and writes software, and gets better every month. Testing a few cases can no longer keep up.
- The language
Mathematics is the one language they cannot talk their way around.
Code, models, circuits and matter itself all obey it. A proof does not sample cases or trust a demonstration: it covers every case, and a computer checks it line by line.
- What we do
We put that language in charge.
We prove what an intelligent system may do before it acts, measure what it actually did, and publish the evidence, so engineers, leaders, regulators and you can check it for yourselves.
Understand the mathematics and you understand the machine. Govern the mathematics and you govern the machine, and the matter it moves.
Start with the question you came with
Each door opens the page written to answer it, and every page leads to the next.
Engineers & technologists
How does mathematics become hardware?
Category theory made executable: proved rewrite rules that run the same on every substrate, and a compiler that realises a problem’s own mathematics where it runs best, with proof of what it guarantees.
Mathematics that runs →Executives & partners
Does it work in a real system?
Four research records: a proof-governed solar supervisor, certified AI on Akida silicon, hive sensor data that carries its own proof, and distributed wind whose every decision is checked.
See the research →Mission & safety leaders
Can autonomy be bounded?
Learned systems may advise; only what is established may act. How we separate belief, evidence and permission, and test it by making the model wrong.
Believe, establish, permit →Press & storytellers
What is the Institute, in plain words?
Who we are, why mathematics helps hives, farms, power systems and health, and the fields we work in, told without jargon.
Our foundations →Donors & foundations
What does a gift make possible?
Equipment, compute, field sites and bench time, each becoming a named capability in a running program, published with its outcome.
Ways to give →Researchers & reviewers
How do I check your claims?
A step-by-step walk from everyday evidence to formal verdicts, and live labs that recompute our results in your own browser.
Check the evidence →Why it matters
Intelligent machines are taking on real work. We make that work trustworthy.
The trick is to keep three things apart that a machine naturally blurs together: what it believes, what is actually established, and what it is permitted to do. Blur them and a confident guess can move a turbine, a grid or a patient’s dose.
Mathematics provides it. A proof shows that a rule holds in every case, and a computer checks that proof line by line. We build systems in which learned intelligence advises freely and established knowledge decides what acts, and we publish the evidence so you can see exactly where that line sits.
Kept apart, these three turn autonomy from a leap of faith into something your engineers can build against, your leadership can stand behind and your auditors can check. How it works →
The corpus
We are learning the language the universe is written in.
Every project here serves one larger aim: a body of mathematics complete enough to explain physical reality, to help us understand it, and to let us steer it, and to reach beyond it where physics runs out. Each theorem is a word in that language, checked by a machine before anything is built on it. This is how much of it we have learned, written and gathered so far.
The aim
Physical reality, and beyond
A language complete enough to explain it, to understand it, and to steer it. Today 700,182 formal theorems are within our reach.
What we read
9,616
papers and reference works, searchable across 1,554,557 indexed passages, and 287 papers of our own.
What we built
208,375
theorems in our own formal library, every one with its proof written out in full.
- 7.6 million lines of Lean and Agda
- 2,111 proof plans
- 5,777 accepted generations of our proof store
What works with it
683
agent tools that let AI search, prove, translate and verify across the whole corpus, every step on the record.
- 226 skills
- 253 verification gates
- 3,291 automation programs
What we stand on
491,807
theorems from the open libraries of formal mathematics: 234,964 in Lean (Mathlib and 35 more libraries) and 256,843 in Isabelle’s Archive of Formal Proofs (824 entries), with 115,728 Agda files catalogued for search.
Counted 7 November 2026, from our own stores. The commons beneath is credited to its authors, never claimed as ours.
Already in a machine
Featured research · 001 · with Morningstar
Let the sun set the pace. Let proof hold the line.
A solar supervisor whose rules are proved before they run: a current budget that composes, a reserve floor no admitted task can break, and evidence that expires. The proved core and every recorded run are open now; commissioning on a real controller is next.
- 19
- Safe Agda core modules
- 72
- Recorded simulation runs
- 36
- Paired policy comparisons
- 0 s
- Hardware monitoring so far
Featured research · 002 · with BrainChip
Every model, with its paper trail. Proof-carrying AI on Akida.
A certificate chain around BrainChip’s model conversion: the exact model in, how it was quantized, converted and placed on the chip, and what came out. Three model families certified, and a model reproduced bit for bit on a real AKD1500.
- 25 / 25
- Outputs reproduced exactly on a real Akida chip
- 126
- Model layers bound to their exact bytes
- 3
- Model families certified end to end
- 0
- Findings in the independent hostile audit
Also in the recordResearch 003 · What makes a sensor reading worth keeping? Verifiable sensor data, at the hive →Research 004 · Wind power that proves its decisions: assurance for distributed wind →
With the support of
You would be in good company
Research and equipment partners, sponsors and technology platforms already stand behind this work, each with a record of what their support made possible. Meet them all →
One foundation, many fields
The same mathematics, carried into every field we work in
One method (state the question precisely, build the construction, check it with a proof, test it against the world, publish the evidence) carried into each field. What is learned in one transfers to the next, because the foundation underneath is the same.
- active · live labsQuantum systemsExact amplitudes, machine-checked circuits, certificates anyone can re-check.
- active · research 002Intelligent systemsAssurance for AI/SI: what a model claims, checked against a proof kernel.
- activeTrusted hardwareProofs compiled toward circuits; receipts from FPGAs and accelerators.
- activeFundamental physicsA machine-checked ledger scoring one generative idea against fifty measured effects.
- active · research 003 · live apiaryAgriculture & pollinatorsAn instrumented research apiary: live hive cameras and continuous colony data.
- seeking partnersNeurotechnologyInterpretable neural-signal models and verified acquisition pipelines.
- buildingDistributed & swarm systemsMany agents, one verifiable account of what the group did.
- active · research 001Energy & infrastructureSolar and wind power governed by proof: admit the work, keep the reserve, check every decision.
Active research · owned and operated
We don’t just theorise. We operate.
The Institute builds, owns and runs its own research in the world: living systems, real markets and real hardware, each one a place where the mathematics has to hold under load. Two of them are open to you now.
an Institute project · research apiary · live cameras
Brumble Bee Farms
Instrumented hives with live cameras and continuous colony data, open findings for researchers and educators, bee rescue, and raw Michigan honey.
brumblebeefarms.com ↗an Institute project · farm-to-door marketplace
Freshly Farmed
Shop nearby farms, delivered to your door: a farmers’ market online, where farmers list their harvest free and sell directly to neighbours.
freshlyfarmed.org ↗Separate companies whose software the Institute uses
Give
From a donated machine to a result anyone can check
Picture the instrument you fund a year from now: running in a named program, its results public, and your support acknowledged beside it, or kept anonymous if you prefer. IAOM is a 501(c)(3) private operating foundation, organized as a Focused Research Organization (FRO). Every gift becomes a named capability in a running program, and everything it produces is published with its evidence.
- 1
Your contribution
Equipment, data, funding or a research partnership.
- 2
Infrastructure
It becomes compute, instruments or field sensors in a running program.
- 3
Mathematical model
The question is stated precisely, as mathematics a machine can read.
- 4
Machine-checked
The result is proved or refuted by a proof kernel, not by reputation.
- 5
Public result
Published with its evidence, so anyone can re-check it without asking us.
Computing & storage
Servers, GPUs, workstations, SSDs, networking.
Scientific instruments
Sensors, cameras, test equipment, FPGAs and accelerator boards.
Agricultural research
Apiary equipment, hive sensors, cameras, orchard monitoring systems.
Energy & infrastructure
Solar, batteries, power systems, connectivity.
Also welcome: research grants, datasets, specialised equipment and scientific partnerships. Where the chain is already complete end to end, in quantum systems, you can walk it yourself →
How we know
You never have to ask how sure we are. Every claim says.
Every claim we make tells you exactly how strong it is. It answers four questions separately, and the first answer comes from a proof checker, a program that accepts a proof only when every step is correct. That precision is what lets partners build on our results the day they appear.
A proof tells you what a statement guarantees; an experiment tells you how well it fits the world. We bring both, each recorded in its own right.
Four questions every claim answers
- 1Is it proved?by a proof checker, under assumptions written down
- 2Does it run?as real code, with a receipt for what ran
- 3Does it match the world?measured, with its conditions and uncertainty
- 4Who else has checked it?from rechecked in-house to independently reproduced
Certificate of standing · closed 5 Nov 2026
Our solar supervisor sends a load only when the reserve holds and its evidence is fresh.
- ✓1 · Proved3 theorems, safe Agda
- ✓2 · Runs72 runs, rebuilt from source
- ○3 · Matches the worldnot yet: the real controller is next
- ◐4 · Checked byus, in-house; independent review planned
kernel sha256 e43df565…5d3f · ledger head 4aab4264…9000Verify it →
For an engineer: a specification you can build against, with its assumptions written down.
For a leader: a risk you can size, because every claim says which questions it has answered.
For a donor: progress you can watch: every answer your gift helps settle is published.
Check it yourself
Everything here can be checked, by you, today
A proof-governed solar supervisor
Inspect all 72 recorded runs, the theorems behind them, and the full comparison.
Open the record →Proof-carrying quantum labs
Build a circuit and watch the verified core recompute it in your browser.
Open the labs →The full record
Every platform, project and proved result across all the work, kept by Apoth3osis, which licenses its software to the Institute.
Open the record ↗Papers and preprints
The public research record behind this work, every result published in full.
Read the research →Nullius in verba
Take nobody’s word for it — ours included.
Whatever brought you here, there is a next step that fits, and whichever you take, what comes back carries its evidence.
- 01∃ExploreSee the work running in a real machine: solar, neuromorphic AI, a research apiary, distributed wind.The four studies →
- 02⊢CheckRecompute a result yourself, in your browser, exactly. Nothing to install and nothing to take on trust.Walk the evidence →
- 03⊗PartnerBring a question, an instrument or a dataset, and we will build the proof around it together.Ways to partner →
- 04ΣGiveFund the next experiment. Every gift is reported back with the evidence it made possible.Donate →
New here? Meet the Institute first.
Our Technology is Ontological. Our Engineering is Constructive.