Iteris: Agentic Research Loops for Computational Mathematics

Bin Dong; Leheng Chen; Wanyi He; Zihao Liu

arxiv: 2606.02484 · v1 · pith:V3UKFL3Wnew · submitted 2026-06-01 · 💻 cs.AI · cs.LG

Iteris: Agentic Research Loops for Computational Mathematics

Leheng Chen , Zihao Liu , Wanyi He , Bin Dong This is my paper

Pith reviewed 2026-06-28 14:46 UTC · model grok-4.3

classification 💻 cs.AI cs.LG

keywords agentic systemscomputational mathematicsnumerical linear algebraphase diagramcounterexampleconjugate gradientQR factorization

0 comments

The pith

An agentic research system generated numerical evidence, constructions, and proof drafts that contributed to verified results on two open problems in computational mathematics after human review and correction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents Iteris as a system built to handle open problems in computational mathematics, which often mix proofs with numerical experiments, adversarial constructions, and algorithm design. It describes applying the system to two problems from a recent workshop collection, where the system produced evidence and drafts that experts then reviewed, corrected, and used to reach verified outcomes. One outcome is a phase diagram comparing the asymptotic behavior of conjugate gradient and randomized coordinate descent on power-law spectra. The other is a counterexample showing that QR factorization with column pivoting can fail to pick well-conditioned submatrices even when coherence is low. A sympathetic reader would care because the work explores whether such systems can supply usable material in research loops that combine computation and theory, while keeping human validation central.

Core claim

Iteris generated numerical evidence, constructions, and proof drafts that led, after expert review and correction, to verified results: a phase diagram for the asymptotic comparison between conjugate gradient and randomized coordinate descent on power-law spectra, and a counterexample showing that QR factorization with column pivoting can fail to select well-conditioned submatrices even under low coherence.

What carries the argument

Iteris, the agentic research system that runs loops to produce numerical evidence, constructions, and proof drafts for open computational mathematics problems.

If this is right

Agentic systems can supply initial numerical evidence for asymptotic comparisons between iterative solvers on structured spectra.
Such systems can surface concrete counterexamples to conjectures about the behavior of matrix algorithms like column-pivoted QR under low coherence.
Research workflows in computational mathematics can incorporate AI-generated drafts that experts refine into publishable results.
Human review remains required to convert the system's outputs into verified theorems or diagrams.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the pattern holds on further problems, agentic systems could shorten the time spent on initial numerical exploration before humans focus on proofs.
The approach could be tested on other open questions that combine linear-algebraic analysis with algorithm design.
Over time, repeated use might reveal which kinds of computational mathematics tasks benefit most from this form of human-AI iteration.

Load-bearing premise

The material generated by the system supplied a non-trivial fraction of the insight or evidence, rather than the human experts performing the core discovery work during the review-and-correction step.

What would settle it

A controlled trial in which human experts solve the same two problems using only standard methods and without access to the system's generated evidence, constructions, or drafts would show whether the system's outputs were necessary for reaching the verified results.

read the original abstract

Recent advances in large language models and agentic AI systems have enabled significant progress in mathematical discovery, from solving competition problems to tackling research-level conjectures. However, open problems in computational mathematics have received comparatively less attention: research in this area often requires not only proofs but also numerical experimentation, adversarial constructions, and algorithm design. In this paper, we introduce an agentic research system, Iteris, designed for open problems in computational mathematics. We apply Iteris to two open problems from a recent Simons Workshop collection (arXiv:2602.05394). In these case studies, Iteris generated numerical evidence, constructions, and proof drafts that led, after expert review and correction, to verified results. The first result is a phase diagram for the asymptotic comparison between conjugate gradient and randomized coordinate descent on power-law spectra; the second is a counterexample showing that QR factorization with column pivoting can fail to select well-conditioned submatrices even under low coherence. These case studies suggest that agentic AI systems can participate meaningfully in research workflows for open problems in computational mathematics, while human validation remains essential.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Iteris shows an agentic loop producing starting material for two computational math results, but the paper gives no measure of how much the system contributed versus the human corrections that followed.

read the letter

The paper introduces Iteris and applies it to two open problems from the Simons Workshop collection. In one case it produced numerical evidence that helped build a phase diagram comparing conjugate gradient and randomized coordinate descent on power-law spectra. In the other it generated a construction that, after review, became a counterexample to a claim about QR factorization with column pivoting under low coherence.

What is new is the documented use of the system on these particular problems that mix numerical tests, adversarial constructions, and proof sketches. The work is honest about the need for human validation at the end.

The main limitation is the missing breakdown of labor. The abstract states that the results emerged after expert review and correction, yet there are no session logs, prompt traces, or before-and-after comparisons that would let a reader judge whether Iteris supplied a non-trivial fraction of the evidence or merely supplied raw drafts that experts then turned into the final claims. Without that, the claim of meaningful participation rests on an unverified assumption.

The methods section is also thin on the exact loop structure and prompts, which makes it hard to reproduce or extend the approach. The citation pattern is reasonable for an early system paper but does not engage deeply with prior agentic math work beyond the high-level motivation.

This is for readers already working on agentic systems for scientific computing or numerical linear algebra. Someone looking for a concrete example of AI-assisted counterexample search or phase-diagram construction will find the case studies useful as existence proofs, though they will still need to do their own implementation work.

It deserves peer review. The problems are real, the final results are externally verifiable, and the attribution question is fixable with additional documentation rather than a fatal flaw in the underlying idea.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Iteris, an agentic research system for open problems in computational mathematics. It applies Iteris to two problems from a recent Simons Workshop collection, claiming that the system generated numerical evidence, constructions, and proof drafts that, after expert review and correction, yielded verified results: a phase diagram for the asymptotic comparison between conjugate gradient and randomized coordinate descent on power-law spectra, and a counterexample showing that QR factorization with column pivoting can fail to select well-conditioned submatrices even under low coherence. The paper concludes that such systems can participate meaningfully in research workflows while emphasizing the essential role of human validation.

Significance. If the attribution of non-trivial contributions to Iteris can be substantiated, the work would provide concrete case studies of agentic AI assisting in generating verifiable numerical and constructive results in computational mathematics. This could highlight a pathway for AI participation in areas requiring both experimentation and proof elements. However, the absence of detailed evidence for the system's independent role limits the ability to evaluate novelty or impact beyond existing human-driven workflows.

major comments (2)

[Case studies] Case studies section: The central claim that Iteris supplied a non-trivial fraction of the numerical evidence, constructions, or proof drafts for the phase diagram and QR-pivoting counterexample is not supported by any session logs, prompt traces, before/after comparisons, or quantitative measures of the system's contribution versus human corrections. The text only states that results emerged 'after expert review and correction,' leaving the 'meaningful participation' assertion dependent on an unverified assumption about the division of labor.
[System description] System description: No details are provided on the exact prompts used, the loop structure of Iteris, or any metrics quantifying how much of the final verified evidence originated from the agent versus human intervention. This omission prevents independent assessment of whether the system performed core discovery work or primarily supplied raw material for human refinement.

minor comments (1)

[Abstract] The abstract and introduction could more explicitly distinguish between the system's raw outputs and the final verified results to avoid potential overstatement of autonomy.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important areas for improving transparency in both the case studies and system description. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

Referee: [Case studies] Case studies section: The central claim that Iteris supplied a non-trivial fraction of the numerical evidence, constructions, or proof drafts for the phase diagram and QR-pivoting counterexample is not supported by any session logs, prompt traces, before/after comparisons, or quantitative measures of the system's contribution versus human corrections. The text only states that results emerged 'after expert review and correction,' leaving the 'meaningful participation' assertion dependent on an unverified assumption about the division of labor.

Authors: We agree that the manuscript does not provide direct evidence such as session logs or quantitative contribution metrics to support the extent of Iteris's role. The presentation focused on the verified outcomes rather than process documentation. In the revised version, we will expand the case studies section to describe specific examples of numerical evidence and initial constructions generated by the system, along with the nature of subsequent human corrections, to better substantiate the claim of meaningful participation. revision: partial
Referee: [System description] System description: No details are provided on the exact prompts used, the loop structure of Iteris, or any metrics quantifying how much of the final verified evidence originated from the agent versus human intervention. This omission prevents independent assessment of whether the system performed core discovery work or primarily supplied raw material for human refinement.

Authors: We acknowledge the lack of implementation details in the current system description. We will revise this section to include a description of the agentic loop structure (hypothesis generation, tool use for computation, iterative refinement), representative prompts from the case studies, and a qualitative discussion of the division of labor between the system and human experts. Quantitative metrics were not collected during the work, so we will clarify this limitation explicitly. revision: yes

standing simulated objections not resolved

Complete session logs, full prompt traces, and before/after comparisons from the original experiments are unavailable, as they were not systematically archived during the exploratory process.

Circularity Check

0 steps flagged

No significant circularity: case studies rely on external workshop problems and post-hoc human validation rather than self-referential derivations or fitted predictions.

full rationale

The manuscript describes an agentic system applied to two external open problems (arXiv:2602.05394) and states that Iteris outputs were reviewed and corrected by experts to reach verified results. No equations, parameters, or uniqueness claims are defined in terms of the target results; the central narrative does not reduce any prediction or theorem to a fit or self-citation by construction. The attribution of insight is presented as a qualitative case study rather than a closed mathematical derivation, so the load-bearing steps remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The paper introduces a new named system (Iteris) whose internal design is not detailed in the abstract; no free parameters or mathematical axioms are mentioned.

invented entities (1)

Iteris no independent evidence
purpose: Agentic research loop system for generating numerical evidence and proof drafts in computational mathematics
The system is presented as the central new artifact of the paper.

pith-pipeline@v0.9.1-grok · 5723 in / 1114 out tokens · 25036 ms · 2026-06-28T14:46:08.544401+00:00 · methodology

discussion (0)

Reference graph

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DefineR β ∈R m×m by (Rβ)t,q =    dt, q=t, −βσtσqsdt, t < q, 0, t > q

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DefineR β ∈R m×m by (Rβ)t,q =    dt, q=t, −βσtσqsdt, t < q, 0, t > q

This constant controls the strictly upper-triangular couplings among the firstmselected columns. DefineR β ∈R m×m by (Rβ)t,q =    dt, q=t, −βσtσqsdt, t < q, 0, t > q. The firstmselected columns are aq = (Rβ):,q 0 ,1≤q≤m. In matrix form their firstmcoordinates are Rβ =   d1 −βσ1σ2sd1 · · · −βσ 1σmsd1 0d 2 · · · −βσ 2σmsd2 ... ... ... ... 0 0· · ·...