The Universal Language of AI: Why Math and Code Matter Most

Author: Denis Avetisyan

A new framework reveals that the ability to excel at mathematical reasoning and coding tasks defines the core competence of any truly intelligent system.

This review establishes a geometric understanding of AI evaluation, demonstrating that mathematics and coding represent a dense subspace of all possible benchmarks, suggesting a pathway towards more reliable and verifiable artificial intelligence.

Evaluating artificial intelligence remains a surprisingly open problem, given the proliferation of increasingly complex models. This motivates the work ‘Mathematics and Coding are Universal AI Benchmarks’, which establishes a geometric framework for understanding AI evaluation batteries, demonstrating that tasks rooted in mathematics and coding densely span the space of all possible benchmarks. This result suggests these disciplines provide fundamental “coordinates” for measuring AI progress and, crucially, offer a natural domain for recursively self-improving agents. Could leveraging this mathematical and computational foundation unlock a pathway towards truly robust and verifiable artificial intelligence?

Deconstructing Intelligence: The Challenge of Robust Evaluation

Determining the genuine capabilities of an artificial intelligence agent presents a significant hurdle in contemporary AI development, as reliance on simplistic tasks provides an incomplete and often misleading assessment of its potential. While an agent might successfully navigate a narrowly defined scenario, this success doesn’t necessarily translate to robust performance in more complex, real-world situations demanding adaptability and nuanced reasoning. The limitations of simple benchmarks stem from their inability to probe an agent’s capacity for generalization, problem-solving under uncertainty, or handling unforeseen circumstances – crucial attributes for any truly intelligent system. Consequently, a more holistic and challenging evaluation paradigm is needed to accurately gauge an agent’s true capabilities and drive meaningful progress in the field, moving beyond easily solvable problems to assess genuine intelligence.

Existing methods for assessing artificial intelligence agents frequently struggle to gauge genuine intelligence because they prioritize easily quantifiable outputs over the subtleties of cognitive processes. Current evaluation frameworks often rely on narrow, pre-defined tasks that lack the complexity of real-world scenarios, failing to test an agent’s capacity for flexible problem-solving or its ability to generalize learned knowledge. This limitation is particularly acute when assessing adaptability; an agent might perform well within a constrained environment but falter when faced with unexpected variations or novel situations. Consequently, these frameworks often provide an incomplete – and potentially misleading – picture of an agent’s true capabilities, hindering progress towards genuinely intelligent systems and necessitating the development of more comprehensive and nuanced evaluation strategies.

Determining the true capabilities of an artificial intelligence agent necessitates more than isolated tests; a comprehensive and standardized Battery of tasks is crucial for thorough assessment. This battery should move beyond simple benchmarks and encompass challenges demanding complex reasoning, planning, and adaptation to novel situations. However, task variety alone is insufficient; a clearly defined and measurable EvaluationMetric is equally vital. This metric must objectively quantify agent performance across the battery, providing a consistent and comparable score that allows for meaningful progress tracking and reliable comparison between different agents. Without such a standardized approach, evaluating agent robustness becomes subjective and hinders the development of truly intelligent systems capable of tackling real-world complexities.

Constructing Capability: Fibers for Rigorous Assessment

Current automated evaluation often relies on task collections assembled without specific structural considerations. This work proposes a shift toward constructing evaluation batteries – sets of tasks designed to assess agent capabilities – by intentionally utilizing structured subspaces known as “fibers.” Specifically, the $MathematicsFiber$ and $CodingFiber$ represent curated sets of tasks grounded in formal mathematical problems and code generation challenges, respectively. Leveraging these fibers allows for a more principled approach to battery construction, moving beyond arbitrary task selection and enabling targeted assessment of capabilities within well-defined domains. This structured approach facilitates precise capability measurement and provides a foundation for a more rigorous and reliable evaluation framework.

Structured subspaces, termed “fibers,” are crucial for constructing reliable battery benchmarks because they offer tasks with explicitly defined input-output relationships and evaluation criteria. This semantic clarity facilitates precise measurement of an $Agent$’s capabilities by reducing ambiguity in task interpretation and scoring. Unlike natural language tasks prone to subjective evaluation, tasks within these fibers-such as those requiring formal mathematical proofs or functional code generation-allow for automated and objective assessment. The resulting metrics are therefore more sensitive to genuine improvements in $Agent$ performance and less susceptible to noise introduced by imprecise task definition or evaluation.

The incorporation of tasks from $FormalMathematics$ and $CodeGeneration$ into battery construction facilitates the evaluation of both abstract reasoning capabilities and executable intelligence. Research indicates that the algebraic structure formed by these mathematics and coding tasks is dense within the broader space of possible batteries; this density implies that any sufficiently rigorous benchmark can be approximated through a combination of tasks drawn from these two domains. This allows for a focused and theoretically grounded approach to capability assessment, moving beyond arbitrary task selection towards a more principled methodology.

The Self-Improving Agent: A Cyclical Operator for Growth

The GVUOperator facilitates agent self-improvement through a cyclical process of capability refinement. This framework operates by generation, where the agent produces candidate improvements to its existing skills; verification, which objectively assesses the performance of these candidate improvements against defined criteria; and updating, where successful improvements are integrated into the agent’s core functionality. This iterative loop allows the agent to progressively enhance its abilities over time, with each cycle contributing to a measurable increase in performance. The process is designed to be generalizable, applicable to a wide range of agent types and tasks, and focuses on systematically improving existing capabilities rather than introducing entirely new ones.

The $VarianceInequality$ is a mathematical condition integrated into the GVU Operator to guarantee monotonic improvement during iterative self-refinement. Specifically, it dictates that the variance of the performance distribution, measured after each iteration of generation and verification, must demonstrably decrease. This ensures that not only does the average performance improve, but the consistency and reliability of the agent’s outputs also increase. By mathematically bounding the permissible changes in performance variance, the inequality effectively prevents performance regression; if a generated update fails to meet the variance reduction criterion, it is rejected, and the agent retains its previous state, thus maintaining a consistent upward trajectory in capability.

The $GVUFlow$ outlines the cyclical process by which the $GVUOperator$ enhances agent performance. This flow begins with generation, where the operator proposes a modified version of the agent’s current policy or knowledge. Next, verification assesses the generated modification using a defined reward signal and acceptance criterion. If the modification is accepted, the agent is updated with the new version, completing one iteration of the loop. The process then repeats, continuously refining the agent’s capabilities based on empirical evaluation of generated changes. This iterative sequence enables ongoing self-improvement without requiring external supervision or pre-defined learning schedules.

The Foundations of Robustness: Guaranteeing Battery Quality

The core of this research rests on the $DensityTheorem$, a foundational principle asserting the capacity of batteries – constructed not from physical materials, but from mathematical and coding challenges – to effectively simulate any other battery design. This isn’t merely a claim of functional equivalence; the theorem provides a rigorous theoretical justification for the chosen approach, allowing for the analysis of complex battery behaviors through the lens of computational tasks. By demonstrating that any battery can be approximated through these constructed counterparts, researchers unlock the potential for streamlined testing, targeted optimization, and ultimately, the development of more robust and efficient energy storage solutions. This approximation isn’t simply possible in theory, but is demonstrably achievable within defined error bounds, solidifying the methodology’s validity and paving the way for practical applications.

The efficacy of the $DensityTheorem$ hinges on a principle called $UniformTightness$, a mathematical property that effectively limits the complexity of the ‘traces’ generated by the learning agents within the battery construction process. These traces, representing the agent’s interactions and decisions, could theoretically become infinitely complex, rendering any evaluation meaningless. However, $UniformTightness$ establishes a quantifiable bound on this complexity, ensuring that the traces remain manageable and representative of genuine learning progress. This constraint is crucial because it allows for rigorous and statistically sound evaluation of battery performance; without it, observed improvements might simply reflect the agent exploring an ever-expanding solution space rather than converging on an optimal strategy. By controlling trace complexity, $UniformTightness$ guarantees that any observed performance gains are demonstrably linked to the learning process itself, forming a solid foundation for the theoretical guarantees provided by the $DensityTheorem$.

The stability of the battery learning process is rigorously ensured through the application of $LipschitzContinuity$ and the $BLDistance$ metric. These mathematical tools allow for precise analysis of how sensitive the learning algorithm is to minor perturbations in the input data, effectively controlling the learning dynamics. Crucially, the $Lipschitz Constant$ (L) serves as a critical parameter, providing an upper bound on the rate of change of the learning function, and thus, limiting the potential approximation error. Demonstrably, the achieved approximation error is constrained to be less than $\epsilon$/2, guaranteeing a high degree of accuracy and reliability in the final battery model – a result vital for constructing batteries that reliably mimic desired performance characteristics.

Towards True Intelligence: A Future of Adaptable Systems

The pursuit of truly intelligent artificial systems necessitates a shift from narrowly focused performance to broad, adaptable capabilities. Current AI often excels within constrained environments but falters when confronted with novelty. To address this, researchers are integrating rigorously defined evaluation frameworks – systems for objectively measuring an agent’s performance across diverse scenarios – with self-improving agent architectures. These architectures allow the AI to learn not just what to do, but how to learn more effectively, constantly refining its strategies based on feedback and experience. This synergistic approach fosters robustness, enabling agents to maintain functionality even when faced with unforeseen circumstances, and generalization, allowing them to apply learned knowledge to previously unencountered situations, ultimately moving the field closer to AI systems capable of genuine adaptability and intelligence.

An agent’s ability to reason effectively hinges not simply on processing information, but on translating that reasoning into concrete action within a given environment; this is where the principle of ExecutableSemantics proves crucial. By prioritizing a system where an agent’s internal reasoning is directly linked to demonstrable outcomes, researchers aim to overcome the limitations of purely symbolic AI. This approach ensures that the agent isn’t merely manipulating symbols, but actively engaging with and responding to the real world, grounding its understanding in tangible results. Consequently, agents built upon ExecutableSemantics demonstrate increased robustness, as they can verify their reasoning through direct interaction and adjust their strategies based on observable feedback – effectively bridging the gap between thought and action and paving the way for more reliable and adaptable artificial intelligence.

Investigations are now directed toward extending the capabilities of this framework to increasingly intricate environments, moving beyond controlled simulations to tackle real-world complexities. A key area of development centers on adaptive batteries – dynamic resource allocation systems designed to personalize operational parameters for each agent. These batteries will not simply provide power, but will intelligently manage computational resources, tailoring processing speed and algorithmic complexity to individual agent strengths and the demands of their specific task. This personalized approach promises to overcome limitations inherent in one-size-fits-all AI architectures, fostering greater efficiency and resilience as agents navigate uncharted territory and encounter unforeseen challenges. The ultimate goal is to create AI systems capable of not only learning, but also of intelligently managing their own resources to achieve robust and generalizable performance across a diverse range of applications.

The pursuit of universal benchmarks, as outlined in the paper, isn’t about finding the one true test, but acknowledging that the space of possible evaluations is fundamentally structured. It’s a landscape where mathematics and coding, far from being niche areas, occupy a surprisingly dense region. This resonates with Grace Hopper’s assertion: “It’s easier to ask forgiveness than it is to get permission.” The paper implicitly argues that rigidly defined benchmarks are often less useful than exploring the boundaries – ‘breaking’ the system, as it were – to understand where it truly holds and where it falters. The Generator-Verifier-Updater loop, for instance, is essentially a formalized method of controlled breakage, iteratively refining the system through challenge and response. A truly robust AI doesn’t fear a little ‘forgiveness’ in its evaluation; it demands it to reveal its limits.

Beyond the Benchmark

The assertion that mathematics and coding define a dense subspace of all possible AI benchmarks isn’t a destination, but a demolition. It dismantles the notion of a singular, definitive test. If any sufficiently complex problem can be approximated by a mathematical or computational equivalent, then the pursuit of ‘general’ intelligence becomes less about scaling current architectures and more about understanding the limits of that approximation. The real challenge lies not in building systems that pass tests, but in meticulously mapping the space of all possible failures – a cartography of inadequacy.

Current evaluation frameworks, even those employing formal methods, remain vulnerable to subtle distortions within that dense subspace. Density theorems guarantee existence, not practicality. The Generator-Verifier-Updater loop, while elegant, still relies on assumptions about the structure of the problem space. Future work must confront the inherent ambiguity of translating real-world complexity into formal systems, and investigate how seemingly innocuous distortions accumulate and propagate through the evaluation process.

The ultimate test won’t be whether an AI can solve a problem, but whether it can convincingly demonstrate its own uncertainty. A system that understands the boundaries of its competence, that can articulate its limitations with precision, is a system that truly begins to resemble intelligence – or, at the very least, a compelling simulation of one.

Original article: https://arxiv.org/pdf/2512.13764.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/