Épi: Turning compute into verifiable improvement

Today, we’re introducing Épi, a self-evolving framework for autonomous research. Working in verifiable environments on our sandbox infrastructure, Épi’s agents produced 120 record-setting results in just a few days, including new constructions and stronger bounds. For example, in polyomino enumeration, Épi lowered the upper bound on Klarner’s constant from 4.5238 to 4.29569. It also proved that the Cohn–Elkies method cannot reach the exact sphere-packing density in dimensions 10 and 11, even at its theoretical limit. For Tuza’s conjecture, Épi produced an independently verified, computer-assisted proof of a new case, covering an infinite family of graphs.

These findings show that frontier models already have capabilities that can advance research. We built Épi to turn those capabilities into a sustained research process that carries verified results, useful methods, and unfinished questions forward. Our goal is to make discovery cumulative, with each advance expanding the scope and depth of future research.

Epi: Turning compute into progress

Épi brings together environments that evolve with research, agents that develop and test their own approaches, and infrastructure that keeps experiments running at scale.

Environments that evolve

Progress starts with a robust verifier: a reliable way to check whether a candidate is valid and improves on the reference result. Within each investigation, Épi freezes the environment’s configuration and verifier, so every candidate faces the same test as the research changes direction.

Across investigations, the frontier moves as verified advances create new starting points and new research demands shape the environment itself. This is the core of Épi’s self-evolution: agents that make progress within environments, and environments that evolve with their discoveries.

Agents that develop their own methods

Because a difficult problem rarely reveals the right method at the outset, Épi’s agents develop their approaches as evidence accumulates, following a promising direction further or finding a different way into the problem. In the process, the investigation can change the method itself.

Research history remains available across sessions, allowing that development to continue, while independent review brings another perspective. An idea set aside earlier can become useful when the problem is understood differently.

Infrastructure for discovery at scale

Breadboard, the execution infrastructure powering Épi, combines high concurrency in isolated sandboxes with the stability that extended research requires. It sustains many investigations in parallel, so research can keep moving while its longest experiments are still running.

As experiments continue across sessions and across different timescales of computation, one investigation can follow a difficult question further while others open new directions.

Mathematical advances with Epi

Épi’s mathematical advances include new constructions, stronger bounds and a new computer-assisted proof. They also establish limits of the Cohn–Elkies method, showing where a stronger mathematical approach is needed.

A problem family groups related instances. Each final result for a distinct instance or parameter setting counts once. A certified interval counts once, and repeated improvements to the same instance contribute one record.

New constructions and stronger bounds

MAX-4-CUT: Épi reformulated the problem and lowered the NP-hardness ratio from 0.9870848895 to 84931/86101 ≈ 0.986411307651.¹³

Ramsey multiplicity c₄,₅: A construction on 1,024 vertices set a new upper bound of 4427841832181/2⁴⁹ ≈ 0.007865427122378.¹²

Spencer discrepancy: The advance fits inside an 8 × 8 matrix. All 256 sign assignments were evaluated exactly, giving an exhaustive certificate that raises the record lower bound from 7/√17 to 5/√8.⁹

Certified numerical results

N-queens constant: The constant α describes how quickly the number of non-attacking arrangements of n queens on an n × n chessboard grows. Nobel, Agrawal, and Boyd bounded α using large convex optimization problems.¹¹ Épi extended their calculation and narrowed the enclosing interval to a width of 3.9367 × 10⁻⁹, approximately 84× narrower than the published interval. A separate verification checked all 67,108,864 projected coordinates of the construction and all 57,340 constraints. Feasibility was verified exactly, and directed rounding kept the objective bounds valid despite numerical error.

Klarner’s constant: This constant describes the growth in the number of shapes formed by joining squares edge to edge on a grid as more squares are added. Mathematicians have studied bounds on this growth for more than half a century. Building on Bui’s recurrence method, Épi lowered the upper bound from 4.5238 to 4.29569 by extending the calculation to 1,186 neighborhood types. An exact certificate covers the full recurrence system.¹⁴

Quantum capacity: Épi found 40-qubit input states that support positive capacity at noise parameters 0.1198803 for the independent X–Z channel and 0.1193415 for the Two-Pauli channel. Both exceed the published thresholds. A separate interval calculation confirms that coherent information remains strictly positive after accounting for numerical error.¹⁷

Feigenbaum dimension: For the quartic attractor, Épi obtained the certified dimension interval [0.6425750638035370, 0.6425750659339700], more than 3.6 million times narrower than the rigorous interval proved by Burbanks, Osbaldestin, and Thurlby. This comparison concerns the width of the proved enclosure; it does not measure a gain over every prior numerical estimate.¹⁶

Bₕ[g] coefficients: Épi improved all five coefficients for h = 3 through 7. These coefficients bound the size of sets with a limited number of representations of an integer as a sum. The search optimized cosine kernels, and interval bounds certified their minima across the entire domain.¹⁵

Water-network design: Épi raised the global lower bound for the waterund36 network-design problem to 662.80702839, within approximately 0.00001001 of the known feasible value. The release includes an exact certificate for the bound.²⁰

Limits of the Cohn–Elkies method

The sphere-packing results cross a different kind of threshold: they establish where a leading mathematical method must fall short. The problem asks how densely identical spheres can fill a space. The Cohn–Elkies method puts an upper bound on that density and gives the exact answer in dimensions 8 and 24. Elsewhere, there can be a gap between what the method allows and what any packing can achieve.

Épi used Li’s discrete-reduction method to bound the best answer the Cohn–Elkies program could ever produce.⁸ It improved these lower bounds in dimensions 9, 10, 11 and 13. In dimensions 10 and 11, they crossed a decisive threshold: they exceeded known upper bounds on the actual packing density.¹⁸

This establishes a limitation of the method itself. Even if the Cohn–Elkies program were solved perfectly, its answer would remain above the true packing density by at least approximately 0.35% in dimension 10 and 1.92% in dimension 11. Closing those gaps requires a stronger method.

A new case of Tuza’s conjecture

Épi’s research extends from improving numerical bounds to proving new cases of mathematical conjectures. For Tuza’s conjecture, it produced a computer-assisted proof for graphs with a fully connected eight-vertex core and an independent set of other vertices. Among the outside vertices that form triangles, it allows up to three distinct ways of connecting to the core, extending Zeng’s two-type result.¹⁹ Each pattern can repeat arbitrarily many times, so the theorem covers an infinite family of graphs. It guarantees that if at most k triangles can be chosen without sharing an edge, at most 2k edges suffice to meet every triangle. The proof passed independent verification.

Twenty numerical bound records across twelve problem families

The table lists all 20 numerical bound records. They cover 12 problem families: Cohn–Elkies contributes four dimensions, Bₕ[g] contributes five parameter values, quantum capacity contributes two channels, and the other nine families contribute one result each.

The entries measure different kinds of progress: a tighter upper or lower bound, a narrower enclosing interval, or a higher noise threshold at which quantum communication remains possible. The direction of improvement is marked for each row.

Sustained research in practice

The evaluations below examine how Épi improves existing results, from successive gains within a single investigation to outcomes across a broader set of problems.

Seventeen improvements in 56 minutes

Épi improved a published construction for packing 34 regular pentagons into an equilateral triangle. Starting from Hyra’s reference arrangement, credited to Haowei Lin on Friedman’s pages in July 2026, it reduced the triangle’s side length from 12.98141 to 12.97564 through 17 successive improvements in a 56-minute session.⁶

All 17 candidates passed the same fixed verifier, keeping the standard of validity and improvement consistent throughout the search. The total reduction reached 19 times the minimum improvement required for this case.

The first two candidates accounted for 82.4% of the improvement within 9.4 minutes. Épi continued searching after these early gains, producing another 15 verified improvements that reduced the side length from 12.97665 to 12.97564. Together, those later results contributed the remaining 17.6% of the total reduction.

One hundred construction records across eight problem families

Épi set 100 construction records across eight problem families, covering geometric packing, vehicle routing, and analytic inequalities. These include 46 circle packings in a quadrant, 36 ball packings in four dimensions, nine pentagon packings, five circle packings in a semicircle, and one record in each of the remaining four families.

We compared the results against published tables and registries, including Packomania, Erich Friedman’s packing pages, CVRPLIB, EinsteinArena, and the mathematical optimization registry. The constructions, mathematical certificates, and complete result tables are available in Épi Results.

Building on Hyra’s constructions

Épi also built on recent AI-generated constructions, improving nine pentagon packings released by Tencent’s Hyra. In addition, Épi reduced routing overhead across Hyra’s collection of 61 quantum circuits by 4,140 CNOT gates, or 1.60%.⁵ The constructions are available in Épi Results.

Comparison of released constructions. Routing totals are recomputed from the circuit files, with three CNOTs per SWAP.

When the search finds no improvement

In this failure case, Épi investigated flat polynomials for seven hours, exploring 31 approaches, abandoning 27, and submitting 20 candidates for evaluation. None improved on the published construction. Every candidate had a higher objective value than the reference, on a task where lower values are better.

The tree below traces how these approaches developed, showing where the search branched and which directions were abandoned. Four directions remained open when the run ended.

Even when a search finds no improvement, Épi preserves what was tried and which approaches were abandoned. Later investigations can review this history when deciding what to revisit and what to try next.

Research that builds on itself

These 120 records and the Tuza theorem extension are a starting point. Mathematics is our first testing ground, and we are expanding Épi into other domains, including research on recursive self-improvement. As we develop its agents, verifiable environments, and sandbox infrastructure together, our goal is for Épi to become a more effective research system through the work it carries out.

Our next evaluations will examine how efficiently Épi explores different approaches and whether it can combine discoveries to address more complex research questions. We also want to move beyond tasks whose objectives are already defined, toward identifying worthwhile questions and deciding where further investigation is most likely to matter.

In French, an épi is an ear of grain, carrying the seeds of the next season. It captures our hope for Épi: that each discovery leaves behind methods, ideas, and new questions from which further research can grow.

Models provide capability. Environments make progress verifiable. Épi turns compute into improvement.

We work with teams building verifiable research environments, deploying research agents, and running controlled experiments at scale on sandbox infrastructure that provides unified execution and traceable results. Get in touch to work with us.

Related work

Épi builds on a growing body of work on models as research systems. FunSearch uses execution feedback to guide program search.¹ AlphaEvolve extends evolutionary coding to scientific and algorithmic problems.² The AI Scientist-v2 coordinates experiments through agentic tree search.³ TTT-Discover trains the model during its search.⁴ Hyra has released constructions across mathematics, scientific computing and AI research.⁵ Our focus is the environment, agent and infrastructure that let this work continue across investigations, with a verifiable record of what has improved. Most of the results reported here were obtained with OpenAI’s GPT-5.6 Sol and GPT-6-Astra.