A mathematics‑only language model that thinks, learns, and replies exclusively in mathematical expressions. It starts with a pattern recognition engine (PRE) that maps any input (text, images, diagrams) to a mathematical structure. The core mathematics engine (CME) manipulates these structures. All extensions – interactive theorem proving, mathematical discovery, multi‑modal math, reinforcement learning from proofs, and self‑play – are integrated into a unified framework.
- Pattern Recognition Engine (PRE): Multi‑modal encoder (text, images, diagrams, sensor data) → outputs a mathematical object (tensor, graph, category, expression tree).
- Core Mathematics Engine (CME): A transformer/graph neural network that operates on mathematical objects. Its hidden states are computational graphs of mathematical operations. It is trained to predict the next mathematical step (e.g., simplification, equation solving, proof step).
- Output Renderer: Converts the resulting math object to LaTeX, MathML, or a formal proof language (Lean, Coq, Isabelle).
The model never generates natural language. It only outputs mathematics.
Goal: The model can produce formal proofs in a proof assistant language (e.g., Lean, Coq) and interact with a proof checker to verify correctness.
Integration:
- The CME is trained on a corpus of formal proofs (e.g., Mathlib for Lean) alongside their theorem statements.
- Output is a proof script (e.g.,
rw add_comm, simp, apply lemma). The model can also output tactic sequences. - The user (or an automated verifier) checks the proof; errors are fed back as a mathematical error object (e.g., “goal mismatch” → a set of remaining subgoals).
- The model then refines its output in a loop until the proof is accepted.
Training for ITP:
- Reinforcement learning from proof checker feedback (reward = 1 for a correct proof, 0 otherwise).
- Self‑play: The model generates conjectures and tries to prove them; if successful, the proof is added to the training set.
Example:
Input (as math): ∀ n : ℕ, n + 0 = n
Output: by simp or a full Lean proof.
Goal: The model generates new conjectures, definitions, or theorems that are likely to be true and interesting.
Mechanism:
- The CME is used as a generative model over mathematical expressions (e.g., sampling from its next‑symbol distribution).
- Generated expressions are filtered by a plausibility estimator (another neural network trained on known true statements).
- Plausible conjectures are then tested by the ITP module (attempted proof or counterexample search via symbolic computation).
- If a conjecture is proved, it becomes a new theorem. If disproved, it becomes a counterexample.
Discovery loop:
- Sample random expressions from the CME’s prior (or guided by novelty).
- Evaluate heuristic novelty (e.g., low embedding similarity to known theorems).
- Attempt proof using the ITP module (with time limit).
- If proved, add to knowledge base and reward the CME for generating it.
- If disproved, add as a counterexample and penalize.
Example discovery: The model might generate “The sum of the first n Fibonacci numbers is F_{n+2} – 1” (already known) but later discover “The sum of the first n Lucas numbers is L_{n+2} – 2” (less known).
Goal: The model can directly input mathematical diagrams (commutative diagrams, graphs, geometric figures, 3D shapes) and output their mathematical content.
Implementation:
- PRE is extended with a diagram encoder (CNN + graph neural network) that parses visual diagrams into mathematical structures.
- For commutative diagrams: extracts objects, morphisms, composition relations → outputs a category (or a diagram in a category).
- For geometric figures: extracts coordinates, lengths, angles, symmetries → outputs a set of constraints or a geometric construction.
- For graphs: nodes and edges → adjacency matrix, graph properties.
Training: Synthetic dataset of diagrams paired with their mathematical descriptions (e.g., a commutative diagram ↔ a set of equations between morphisms). Use supervised learning.
Example:
Input image of a triangle with side lengths a, b, c and an altitude h.
PRE → outputs: triangle with vertices (0,0), (c,0), (x,y) where x^2 + y^2 = b^2, (c-x)^2 + y^2 = a^2 and h = y.
CME can then compute area = (c * h)/2.
Goal: Improve the CME’s reasoning through direct feedback on its outputs (e.g., “your simplification is incorrect”, “your proof step is invalid”) without natural language.
Mechanism:
- The environment is a computer algebra system (CAS) or a proof assistant that can evaluate the correctness of a mathematical transformation.
- The model’s output (a math expression) is fed to the CAS; the reward is the inverse of the error (e.g., 1 if correct, 0 if wrong, or a continuous score based on similarity to ground truth).
- The model is trained with policy gradient (e.g., PPO) to maximize expected correctness.
Benefits: The model learns to avoid common mistakes (e.g., algebraic slips, incorrect integration) without needing human‑written explanations.
Integration with ITP: The same reward can be derived from proof checker feedback (number of remaining subgoals, success/failure).
Goal: The model improves by generating its own training problems and solving them.
Self‑play loop:
- Generator: The CME generates a mathematical problem (e.g., an equation to solve, a theorem to prove, a definition to unfold). This problem must be solvable (the model can check via symbolic computation or a simple heuristic).
- Solver: The same model (or a copy) attempts to solve the problem.
- Critic: A verifier (CAS or proof assistant) checks the solution.
- Reward: Both generator and solver are rewarded if the problem is well‑posed and correctly solved.
- Update: Both networks are updated via RL (or supervised learning on the successful pairs).
Curriculum: The model starts with simple problems (e.g., linear equations) and gradually increases difficulty based on its own success rate. This automatically creates a curriculum tailored to its current capability.
Example:
Generator outputs: Solve x^2 – 5x + 6 = 0. Solver outputs: x = 2 or x = 3. Critic verifies → reward.
All extensions share the same CME architecture. The only additional components are:
- ITP module: A wrapper that sends proof attempts to an external proof assistant and parses feedback into a mathematical error object.
- Discovery module: A sampler + novelty estimator + ITP loop.
- Diagram encoder: Part of the PRE.
- RLMF environment: A CAS or proof assistant with a reward function.
- Self‑play orchestrator: Manages the generator/solver loop and curriculum.
These components can be implemented as external tools that the model calls via API, or they can be partially learned (e.g., the novelty estimator can be a neural network trained on existing theorem databases).
- Pre‑train PRE on synthetic and real‑world data (text, images, diagrams) to map to math objects.
- Pre‑train CME on a large corpus of mathematical expressions (next‑token prediction on expression trees).
- Fine‑tune with RLMF using a CAS (e.g., SymPy) for algebraic tasks and a proof assistant (e.g., Lean) for theorem proving.
- Self‑play to generate curriculum and improve both generation and solving.
- Discovery runs continuously, adding new conjectures and proofs to the knowledge base.
User uploads an image of a right triangle with sides labelled (a, b, c) and an altitude (h) drawn.
PRE (diagram encoder):
Output: right_triangle(a, b, c) ∧ h = (a*b)/c ∧ a^2 + b^2 = c^2
CME (using geometric theorems):
Output: area = (a*b)/2 = (c*h)/2
User (text input): “Prove that (h = \frac{ab}{c}).”
PRE → prove h = (a*b)/c given right_triangle(a,b,c) ∧ a^2 + b^2 = c^2
CME (ITP mode) → outputs a Lean proof:
theorem altitude_of_right_triangle (a b c h : ℝ)
(h_def : h = (a*b)/c) (hyp : a^2 + b^2 = c^2) : h = (a*b)/c :=
by exact h_def(Actually, the model would output a more detailed proof, but the idea is that it outputs formal math.)
Discovery mode (autonomous): The model generates “The altitude of a right triangle is the geometric mean of the two segments of the hypotenuse.” It attempts to prove it using ITP, succeeds, and adds a new theorem to its internal knowledge base.
This LLM-Ω with all extensions is a self‑improving mathematical reasoner that:
- Thinks only in math.
- Learns only from math (no natural language supervision).
- Replies only with math (LaTeX, proofs, diagrams).
- Can prove theorems, discover new mathematics, understand diagrams, and learn from its own mistakes via RL and self‑play.
It represents a pure, formal, and autonomous mathematical intelligence.