There is a difference between an AI that can answer a mathematical question and an AI that can participate in mathematical discovery.
The first retrieves patterns.
The second has to search.
It has to conjecture.
It has to fail.
It has to construct counterexamples.
It has to revise its assumptions.
And, eventually, it has to produce something stronger than a convincing paragraph:
a proof that a formal system accepts.
That is the experiment I want to explore with Astra.
Not: “Can Astra solve physics?”
A much more interesting question is:
Can a scientist-developer build a loop in which an AI explores scientific ideas, while mathematics itself acts as the verification layer?
The proposed architecture is simple:
Scientist → Question → Astra → Conjecture → Search → Lean 4 → Proof/Failure → Scientist
The scientist chooses the territory.
Astra explores it.
Lean becomes the referee.
And failure becomes data.
To make this concrete, I would start with seven problems.
1. Can Astra turn numerical physics into formal mathematics?
Numerical simulations are extraordinarily powerful.
But a simulation that appears to converge is not the same thing as a machine-checkable guarantee of convergence or error.
So the first challenge is:
Can Astra derive formal error bounds for a carefully chosen numerical method used in physics?
For example, rather than attacking the full Einstein equations, begin with a constrained numerical formulation and ask Astra to transform:
numerical scheme → assumptions → stability analysis → explicit error bound → Lean certificate
The goal isn't another simulation.
It's a certificate for the simulation.
2. Can Astra mine hidden quantitative information from proofs?
Many mathematical proofs establish that something exists without telling us exactly how fast it converges or how much error remains.
That hidden quantitative information may already be encoded in the proof.
So the second challenge becomes:
Can Astra perform proof mining and extract explicit convergence rates from existing existence arguments?
The pipeline:
Classical proof → constructive interpretation → quantitative bound → formal verification
If successful, Astra would not merely reproduce proofs.
It would extract information that was implicit inside them.
3. Can Astra discover new structure in integrable systems?
This is where things become genuinely exploratory.
Give Astra a parametrized family of nonlinear differential equations.
Don't tell it the answer.
Let it search.
The objective:
Can Astra discover a previously unknown Lax pair for a nontrivial family of nonlinear PDEs?
Conceptually:
PDE family → symbolic ansatz → coefficient search → candidate Lax pair → compatibility check → formal proof
This is not theorem retrieval.
It is mathematical exploration.
And if the candidate fails, Astra should explain why.
4. Can Astra discover sharper entanglement bounds?
Tensor networks provide a remarkable meeting point between mathematics, quantum information and theoretical physics.
The question:
Can Astra discover stronger bounds on entanglement or coding capacity for generalized tensor-network constructions?
Here the search space is combinatorial, algebraic and physical at the same time.
Astra would have to move between:
network topology → tensor constraints → entropy inequalities → conjecture → proof/counterexample
The interesting output may not be a theorem.
It might be a new conjecture worth proving.
5. Can Astra map the boundary between quantum and classical simulation?
Quantum many-body systems often sit on a complicated boundary:
Some structures are efficiently simulatable.
Others appear computationally intractable.
But where exactly is the frontier?
The fifth challenge:
Can Astra identify and formally characterize a new family of Hamiltonians whose quantum behavior admits efficient classical simulation?
The experiment becomes:
Hamiltonian family → complexity analysis → candidate classical algorithm → complexity bound → formal verification
The real target is not merely a faster algorithm.
It is a map of the boundary between tractable and intractable physics.
6. Can Astra construct new Ricci-flat geometries?
Geometry offers another extreme test.
Instead of asking an AI to generate aesthetically interesting manifolds, ask something much harder:
Can Astra discover a mathematically valid family of geometries satisfying the conditions required for Ricci-flatness?
The system would need to navigate:
geometric assumptions → manifold construction → metric candidate → curvature calculation → Ricci-flat condition → proof
The important distinction:
A neural network can generate an approximation.
A mathematical system has to explain why the object exists.
7. Can Astra turn scientific literature into a map of unknown mathematics?
Perhaps the most ambitious problem isn't solving one theorem.
It's discovering what hasn't been solved yet.
Imagine feeding Astra a corpus of mathematical-physics papers and asking it to extract:
claim → assumptions → missing step → precise theorem → conjecture → proof attempt
The objective:
Can Astra convert ambiguous scientific claims into a machine-checkable map of open mathematical problems?
Some claims might be proved.
Some disproved.
Others might become precisely formulated open problems.
That changes the role of AI from:
literature assistant
to:
research-question generator.
The experiment is bigger than the seven problems
These seven problems are deliberately heterogeneous.
They test different capabilities:
| Problem | What Astra must demonstrate |
|---|---|
| Numerical error bounds | Formalization |
| Proof mining | Extraction |
| Lax pairs | Discovery |
| Entanglement | Conjecture generation |
| Classical simulation | Complexity reasoning |
| Ricci-flat geometry | Mathematical construction |
| Literature mining | Research intelligence |
Together they create something more interesting than a benchmark.
They create a scientific loop.
And the developer-scientist becomes a new kind of researcher.
Not someone who simply asks an AI questions.
Someone who builds an environment in which the AI can search, fail, learn, formalize and try again.
The rule: no fake breakthroughs
There is one rule I would make non-negotiable.
We should never confuse:
“Astra generated a plausible solution”
with
“Astra solved the problem.”
A genuine result should survive independent verification.
Ideally:
Astra proposes → Lean checks → human scientist reviews → independent reproduction
Until then, it is a hypothesis.
That distinction matters enormously.
Because the real scientific breakthrough may not be:
“Astra solved seven problems.”
It may be:
“We built a reproducible system in which an AI can participate in mathematical discovery without asking humans to trust the AI.”
That is a much more interesting experiment.
The First Prototype
I wouldn't start with all seven.
I'd pick one.
Build the smallest possible loop:
Problem → Astra → Candidate → Lean → Feedback → Astra
Then measure:
- How many conjectures are generated?
- How many are mathematically meaningful?
- How many survive symbolic checking?
- How many reach Lean?
- How many fail?
- Can Astra learn from those failures?
- Does human intervention decrease over time?
If the answer is yes, we have something worth scaling.
Seven problems then become seven laboratories.
And Astra becomes something different from another chatbot.
A research instrument.
The question I'd put at the top of the repository:
What happens when you give an AI not an answer, but a mathematical universe to explore—and give mathematics the final vote?
That's the experiment.
And perhaps the most interesting result won't be a theorem.
It will be discovering what kind of scientist an AI can become.
created by Seyed Alireza Alhosseini Almodarresieh
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