Iteris: Agentic Research Loops for Computational Mathematics

Iteris: An Agentic Research Loop for Solving Open Problems in Computational Mathematics

Researchers have developed an AI system called Iteris. It does not just solve math problems. It acts like a scientist. It can run experiments and try to build proofs. It can even realize when a mathematical direction is failing. This helps humans solve complex mathematical mysteries.

While frontier reasoning models solve competition-level math, they struggle with open-ended research. Real discovery in computational fields is messy. It requires a loop of numerical experimentation, adversarial construction, and rigorous proof development. Iteris bridges this gap. It treats research as a long-horizon coordination problem rather than a single prompting task.

The Problem

Current AI approaches to mathematics are optimized for "isolated exercises." These are problems with known solutions and clear objectives. This works for benchmarks. It fails in computational mathematics. Progress here requires navigating between different modes of thought. You might run a numerical simulation to find a pattern. Then you might switch to designing a counterexample. Finally, you attempt a formal proof.

Existing agentic frameworks often suffer from "local inertia." The system gets stuck refining a single, suboptimal path. It lacks a mechanism to probe alternative directions. Current models also produce "plausible-sounding" but unsound arguments. They might identify a correct intuition. However, they often fail to provide the auditable obligations required for peer-reviewed results.

How It Works

The core architectural choice in Iteris is the decoupling of exploration from planning. The system operates on an "explore–plan–execute" loop to prevent over-committing to dead-ends .

1. Explore: An exploration agent performs light, repository-level probing. It reads recent files and drafts "scratch reasoning." This tests if a research direction is becoming locally stagnant. It does not write the official plan. It only provides an advisory signal to the planner.
2. Plan: The plan agent maintains a global view of the project state. It consumes the exploration advisory. It then writes a TASK_POOL.json file. This file acts as a structured contract. It specifies the goal, scope, and verification requirements for the next tasks.
3. Execute: Specialized execution agents carry out discrete tasks. These are categorized by research mode: Foundation agents (auditing definitions), Experiment agents (running numerical diagnostics), Proof agents (constructing lemma artifacts), and Review agents (evaluating evidence and blocking invalid routes).

To maintain stability over hundreds of iterations, Iteris uses files as long-term memory and structured messaging. This keeps research facts separate and checkable. It prevents the "hallucination drift" common in long-context conversations.

Results

The authors applied Iteris to two open problems from a Simons Workshop collection. The system generated valid mathematical artifacts rather than just text.

Case 1: CG vs. RCD Phase Diagram

The first problem compared Conjugate Gradient (CG) and Randomized Coordinate Descent (RCD) on power-law spectra (eigenvalues that decay according to a power law).

• Result (Theorem 1): Iteris established a fixed-parameter phase diagram. This defines the asymptotic cost ratio between the two methods.
• Mechanism: The system moved from source auditing to identifying a "limiting CG floor," $F(p)$. This floor is the minimum error level reachable by fixed-degree CG. It serves as the primary divider in the phase diagram .

Case 2: QRCP Orthonormal Selection

The second problem concerned whether QR factorization with column pivoting (QRCP) reliably selects well-conditioned submatrices. This is measured by coherence (a metric describing how spread out a matrix's information is).

• Result (Theorem 2): Iteris constructed a low-coherence counterexample. This proves that QRCP can fail even when coherence is low.
• Mechanism: The system transitioned from failed proof attempts to "reverse construction" mode. It treated failed proofs as diagnostic tools. This helped it identify a structural obstruction .

In a direct comparison, the authors tested a standalone high-reasoning model (GPT-Pro). GPT-Pro produced a meaningful high-level sketch. However, it failed to provide the paper-quality closure or the necessary boundary checks that the Iteris loop achieved.

What's Missing

The paper is honest about the necessity of the "human-in-the-loop." Iteris is a tool for human-verified research.

First, the system struggles with "semantic alignment" and exposition. In the QRCP case, the generated proof trajectory was correct. However, it was "circuitous and required substantial reorganization" to be readable. The system might find the discovery, but it will not write a clean narrative.

Second, the system can produce overstrong, unsupported claims. In the CG analysis, the agent initially relied on an unjustified assumption. While the Review agent and human oversight caught the error, verification is not yet fully autonomous.

Finally, the mathematical results are presented as conservative stochastic upper bounds. The authors explicitly state these bounds may not be sharp. They do not address complexities like finite-precision effects or restarted solvers.

Should You Prototype This

Depends on your access to domain experts.

If you want a "set and forget" solver for conjectures, this is not it. The system requires significant human steering. It also requires massive manual effort to audit the outputs.

However, the Iteris framework offers a strong blueprint for engineering. The separation of exploration from planning is a vital pattern. Using structured file-based memory for long-horizon agents is also a proven strategy. If you are building agents for highly technical R&D, these architectural choices are worth adopting.

Code is reportedly available; see the paper for the canonical link to the QRCP obstruction repository.