Program Announcement (PA) Mathematics of Boosting Agentic Communication

(MATHBAC)

DEFENSE SCIENCES OFFICE

DARPA-PA-26-05

AMENDMENT 01

11 JUNE 2026

DARPA-PA-26-05

PROGRAM ANNOUNCEMENT OVERVIEW INFORMATION:

Federal Agency Name: Defense Advanced Research Projects Agency (DARPA), Defense

Sciences Office

Funding Opportunity Title: Mathematics of Boosting Agentic Communication

(MATHBAC)

Solicitation Type: Amendment

Funding Opportunity Number: DARPA-PA-26-05

Assistance Listing Number: Not applicable

Dates/Time - All Times are Eastern Time Zone (ET):

o Posting Dates:

o Initial Announcement: April 7, 2026 o Amendment 1: June 11, 2026 o Proposers Day: April 21, 2026 o Proposal Abstract Due Date: April 30, at 4:00 p.m.

o Question Submittal Closed: June 4, 2026, at 4:00 p.m.

o Proposal Due Date: June 18, 2026, at 4:00 p.m.

o Estimated Period of Performance Start: September 15, 2026

Anticipated individual awards: Multiple awards are anticipated

Period of Performance: The total period of performance is anticipated to be 34 months. The program will be broken into two phases. Phase I will be 16 months. Months 10-16 of Phase I will either involve wrapping up Phase I work, and/or beginning the transition from Phase I to the Phase II of 18 months. Months 17-34 will be the Phase II.

Types of instruments that may be awarded: Other Transaction Agreements for Research

Anticipated Funding Available for Award: Phase I awards will be limited to $2,000,000 per award

NAICS Code: 541715

Agency contact:

The PA Coordinator for this effort may be reached at:

MATHBAC@darpa.mil

DARPA/DSO

ATTN: DARPA-PA-26-05

675 North Randolph Street Arlington, VA 22203-2114 mailto:MATHBAC@darpa.mil

SECTION I: FUNDING OPPORTUNITY DESCRIPTION

The Defense Sciences Office (DSO) at the Defense Advanced Research Projects Agency (DARPA) is soliciting innovative research proposals that advance the foundational mathematics, systems theory, and information theory required to enable/accelerate and understand science-discovery by autonomous agents and agent collectives. Proposed research should investigate innovative approaches that promote revolutionary advances in science, devices, or systems.

Specifically excluded is research that primarily results in incremental improvements to the existing state of practice.

Mathematics of Boosting Agentic Communication (MATHBAC) will enable the next generation of Agentic Artificial Intelligence Platforms that systematically evolve to transform the nature of scientific discovery for defense. MATHBAC will develop the science of AI communication, leveraging/deriving mathematical theory, systems theory, and information theory to govern how AI agents collaborate and share information, as well as how they disentangle complex data streams to extract generalizable principles. MATHBAC enables agentic platforms to not only solve complex problems but also to understand the fundamentals behind their own functioning and solutions.

MATHBAC will understand/design/improve the communication protocols of interacting agents, discover new efficient protocols, and develop agentic coordination mechanisms (such as the distilling of shareable, generalizable principles from data). This will enable individual agents and agent collectives to effectively and efficiently address an array of tasks, from science-discovery to problem solving and instruction. It aspires to lower/remove barriers to understanding and help tackle challenges across domains ranging from the physical and mathematical sciences to military strategy, drastically accelerating and potentially reinventing their “ways of doing.” If successful, MATHBAC will accelerate science discovery and create tools to enable the systematic evolution/invention of new science itself.

In Phase I, MATHBAC will accelerate science by developing/deriving the mathematics for (1) understanding/designing agentic communication protocols and for (2) improving communication content; the latter will involve improving understanding by disentangling common from case-specific domain data; extracting generalizable domain principles from volumes of data that exceed human processing limits (i.e., from novel data streams pertaining to a science subdomain); and discovering novel generalizable principles, laws, or correlations. Successes in these steps will be used in Phase II, whose focus is to create tools that enable systematic evolution/invention of new science; this will be achieved by orchestrating agentic self-evolution towards improved science-discovery task-solving, distilling a curriculum for agent finetuning, and/or updating agent-shared core knowledge modules towards solving new tasks. This program will formulate and solve fundamental scientific and mathematical problems underpinning collective agentic intelligence; it is not aimed at building applications or specific products.

1. BACKGROUND

The cutting edge of machine learning (ML)-assisted scientific discovery continues to explode, driven by the ability of artificial intelligence (AI) agents to write quality code1 and execute

1 https://www.anthropic.com/news/claude-opus-4-5 https://www.anthropic.com/news/claude-opus-4-5 workflows using scientific tools.2 The explosion is fueled by agentic, interacting ML platforms that communicate, adapt, and are capable of rewriting their own code based on performance: Large Learning Models (LLM), Large Reasoning Models (LRM), Small Science Models (SSM), and smaller Foundation/Knowledge models (i.e., transformers).

Recent advances in AI for domain sciences like chemistry, for example, focus on autonomous systems capable of hypothesis generation, experimentation/computation, and refinement without human intervention.3 Examples include SparksMatter4 for materials discovery and Virtual Lab5 for biomedical research. Relevant recent research in LLMs has evolved along four key directions:

standalone LLMs fine-tuned for domain tasks,6 tool-augmented LLMs integrating external resources for autonomous workflows,7 multi-agent systems with specialized roles collaborating on complex tasks,8 and multimodal LLMs combining text and visual data for enhanced experimental analysis.9 The potential advantages of these advances led to a recent partnership between Google DeepMind and the UK government to support prosperity and security in the AI era10 and the Genesis Mission,11 a partnership between the Department of Energy and a swath of AI and AI-adjacent companies including Amazon, AMD, Anthropic, FutureHouse, NVIDIA, Google, OpenAI, and Scale AI, to name a few. Simultaneously, several Agent Protocols have been developed to establish communication standards for multi-agent systems, enabling them to collaborate on complex tasks. Protocols such as Agent2Agent (A2A),12 Agent Communication Protocol (ACP),13 and Model Context Protocol (MCP)14 define the syntax, structure, and sequences of messages underpinning agentic cooperation.

Despite the impressive accomplishments of the agentic models and systems mentioned above, these approaches remain fundamentally heuristic, lacking a rigorous mathematical foundation.

Current methodologies rely heavily on ad hoc, trial-and-error processes, which resemble an "Edisonian" approach to optimization, focusing on end-to-end performance evaluation. While this has led to notable successes, it highlights a critical limitation: the absence of principled frameworks for understanding, predicting, and systematically improving the various stages15 of behavior of these systems. Without sound mathematical underpinning, the optimization of agentic communication and coordination will remain inefficient, inconsistent, and difficult to generalize across domains even when occasionally successful.

A prescient 2021 sentence in a review article by DARPA performers excelling in AI for chemical synthesis highlights this gap (paraphrased as): “We can find new molecules, and we can find new

2 https://www.elagente.ca/ 3 https://sakana.ai/ai-scientist-first-publication/ 4 https://arxiv.org/abs/2508.02956 5 https://www.nature.com/articles/d41586-024-01684-3 6 https://www.nature.com/articles/s42256-025-01066-y 7 https://www.nature.com/articles/s42256-024-00832-8 8 https://www.nature.com/articles/s41467-024-54457-x 9 https://arxiv.org/abs/2408.07246 10 https://deepmind.google/blog/strengthening-our-partnership-with-the-uk-government-to-support-prosperity-and-security-in-the-ai-era/ 11 https://genesis.energy.gov/ 12 https://github.com/a2aproject/A2A 13 https://agentcommunicationprotocol.dev/introduction/welcome 14 https://modelcontextprotocol.io/docs/getting-started/intro 15 https://arxiv.org/pdf/2508.14111 https://www.elagente.ca/ https://sakana.ai/ai-scientist-first-publication/ https://arxiv.org/abs/2508.02956 https://www.nature.com/articles/d41586-024-01684-3 https://www.nature.com/articles/s42256-025-01066-y https://www.nature.com/articles/s42256-024-00832-8 https://www.nature.com/articles/s41467-024-54457-x https://arxiv.org/abs/2408.07246 https://deepmind.google/blog/strengthening-our-partnership-with-the-uk-government-to-support-prosperity-and-security-in-the-ai-era/ https://deepmind.google/blog/strengthening-our-partnership-with-the-uk-government-to-support-prosperity-and-security-in-the-ai-era/ https://genesis.energy.gov/ https://github.com/a2aproject/A2A https://agentcommunicationprotocol.dev/introduction/welcome https://modelcontextprotocol.io/docs/getting-started/intro https://arxiv.org/pdf/2508.14111 reactions, but we cannot find the periodic table, and we cannot find Bronsted-Lowry acid-base theory – because the space of solutions is different than the space of hypotheses.”16 This underscores a fundamental challenge: while AI excels at navigating solution spaces, it struggles to systematically explore hypothesis spaces, which are essential for generating transformative and generalizable scientific insights. A recent survey on autonomous scientific discovery notes:

“Effective collaboration enables multi-agent systems to address complex scientific problems that surpass the capabilities of any single agent.”17 Protocols enabling such a collaborative approach are key to generating novel scientific hypotheses. It is precisely the rate of discovery of these important new hypotheses that MATHBAC aims to systematically accelerate by facilitating AI communication so as to enable breakthroughs in the efficiency of agentic scientific reasoning.

If successful, MATHBAC will fundamentally change the “ways of doing,” whether for scientific discovery or for instruction, by exploiting/synthesizing both established and novel tools/results from mathematics and systems/information theory. This synthesis will provide a new vantage point for observing data and information propagated within and communicated across AI agents – not based on the beginning-to-end performance success but rather based on the nature and content of the interactions. Under current state of the art (SOA), the agents interact with each other, their environment, and tools under ad hoc protocols.

One way of leveraging mathematics and systems/information theory is by considering, for example, an ML agent as a mathematical operator, operating on inputs (e.g., token strings, number strings, also possibly multimodal input including images); processing these inputs in its internal latent space (e.g., through the attention heads, routing heads, and various architectural components of a transformer); and producing outputs – new token strings, number strings, or images. This echoes the “input-state-output” view of a system in control theory and opens the door to tools and techniques developed over decades to identify the nature of the operator.

Such an approach empowers the evaluation and scheduling of interactions between heterogeneous operators and codes, enabling the efficient integration of different fidelity models, scales, partial system components, and operational stages. Planning, monitoring, manipulating, and understanding these interactions will allow for more intrinsic comparisons of communication protocols beyond merely evaluating final results. This will require developing and using advanced algorithms for system (e.g., agent) identification and interaction planning. It will also require exploiting analogies with applied mathematics algorithms harnessing the communication between heterogeneous operators/codes, such as (1) preconditioning operators with other operators; (2) operator splitting; (3) domain decomposition; and/or (4) multifidelity, multiscale, or mixed precision computations. These types of numerical algorithms enable powerful tools to quantify communication dynamics and optimize the associated protocols.

Combining two operators—such as a low-fidelity operator that preconditions a high-fidelity one— creates a new composite operator whose spectrum reflects the synergy of the constituent operators.

Analyzing this new operator reveals how the strengths of the individual components are combined to achieve faster convergence and improved solutions, demonstrating the transformative potential of applied mathematics in advancing computational efficiency and precision. Understanding and exploiting such identification technology/numerical technology holds the promise of better-

16 https://onlinelibrary.wiley.com/doi/abs/10.1002/anie.201909987 17 https://arxiv.org/pdf/2508.14111 https://onlinelibrary.wiley.com/doi/abs/10.1002/anie.201909987 https://arxiv.org/pdf/2508.14111 informed design and control of agentic communication to accelerate discovery without relying exclusively on trial-and-error experimentation or “black box” reinforcement learning to discover optimal communication protocols (and eventually, optimal agents).

The overall boosting of agentic communication task is facilitated by both the broad use of natural language as a common communication tool and multimodal inputs to a transformer embedded as numbers in its internal latent space, so we can use data analysis techniques to understand the evolution of the problem-solving campaign. We will take advantage of this by studying the role of different communication architectures and protocols on what is exchanged between agents, especially as a function of the length of exchanged prompts. In Phase I MATHBAC will start with fixed pretrained agents (with no finetuning) because we believe that current knowledge and its intelligent communication, even with fixed agents, can already lead to nontrivial successes; the significantly larger number of abstractions an agent can simultaneously buffer compared to the corresponding physiological limits of an expert human18 supports this conviction. In Phase II, MATHBAC will explore how adaptively evolving agents may specialize or even decide to disaggregate into multiple agents with distinct knowledge domains/skills.

In exploring agentic communications, we clearly see an expansion beyond traditional Shannon information (and so-called “syntactic” communication protocols), where we are interested in the accuracy of the transmission of bit strings to the two higher levels of information – semantic information/communication and communication effectiveness.19,20 Currently, agents interact with each other, their environment, and tools under ad hoc protocols, prescribed by the platform designer. Agents need to coordinate their understanding of the communication signals, of the requests/goals of the communication they undertake with each other (and the overarching problem target), and of their shared knowledge bases—the shared long-term “memory” in memory networks.21,22 These all relate to semantic communication and to communication effectiveness.

Currently, this coordination is all hand-curated.

Beyond evaluating and optimizing interaction protocols, MATHBAC will also focus on features of the content of agentic communication: in particular, it will focus on the discovery of “principles” (laws, correlations) from data–the extraction of compact, generalizable “nuggets” that should become part of the common knowledge module (“memory”) of cooperating agents. This is analogous to common core knowledge between cooperating humans and the ways we adjust it (e.g., through targeted supplementary training) towards joint success.

An example: when reaction progression data from different acid-base chemical experiments become embedded in a transformer latent space, we can study using new data science methods what is common across all studied reactions and what pertains to each particular experiment separately. The low-dimensional manifold pertaining to commonality across the cases pertains then to the acid-base nature of the reactions; parametrizing this manifold should “rediscover” the fundamental similarity of acid-base reactions and embody its acid-baseness. Data-driven studies

18 Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63(2), 81–97. https://doi.org/10.1037/h0043158 19 https://arxiv.org/pdf/2212.01485 20 Shannon, C. E., & Weaver, W. (1949). The mathematical theory of communication. University of Illinois Press.

21 https://arxiv.org/abs/1410.3916 22 https://arxiv.org/pdf/2405.06394 https://psycnet.apa.org/doi/10.1037/h0043158 https://arxiv.org/pdf/2212.01485 https://arxiv.org/abs/1410.3916 https://arxiv.org/pdf/2405.06394 can help us reversely articulate such “principles” in natural language,23 possibly inventing new terms (like click chemistry, high entropy alloys, or even reticular chemistry—to include this year’s Nobel prize) when these terms have not yet been coined.

In the case of generalizable principle discovery from data, we propose to focus on (1) tracking the exchange streams of information between agents, especially as a function of the length of the prompts (the exchanged token streams) and (2) exploring what is “the part of the communication that matters” in a domain context, i.e., what is semantically shared in the communications that arise in processing similar phenomena/queries versus what is case-dependent. The fact that multimodal information (from sentences to experimental data) after tokenization becomes jointly embedded in a transformer’s latent space provides an avenue towards the systematic data-driven study in the digital latent space of the transformer.

Improving coordination between different cooperating agents includes possibly developing a completely new domain language. Chemists use words and equations but also diagrams; architects and construction engineers use words and blueprints; the military uses doctrine in manuals as well as laminated instruction cards for operating drones. There exist beautiful literature cases illustrating the evolution of a new language (both from Facebook24 in the case of chatbots working to equitably share a treasure and in the Google Neural Machine Translation Program25 that came up with its own Interlingua) pointing towards what may well develop into the “Neuralese” of AI2027.26 The development of useful domain languages is part of semantic communication, which will be a component of the second phase of MATHBAC. Phase I will focus on natural language and floating-point numbers as the communication substrates.

Fifty years ago, DARPA27 guided the development of the TCP/IP protocol for transmitting Shannon information between computers and adopted it forty years ago in ARPANET. The future of MATHBAC is to move beyond the already ambitious semantic extension of TCP/IP to the case where agents can adapt themselves and their knowledge through experience, new data, communication, and shared knowledge/memory. Phase II will focus on agent evolution as part of improving agentic communication. Up to now the evolutionary pressure on AI and agentic platforms comes from evolutionary pressures on the humans involved in their development.

MATHBAC wants to move this (not Darwinian, but Lamarckian) pressure onto the agents themselves and their cooperation skills. MATHBAC will systematically explore, understand, and design the best protocols, content, and possibly even language of collaborative AI agent communication.

2. PROGRAM DESCRIPTION AND SCOPE

MATHBAC is founded on the observation that, in current practice, autonomous scientific discovery is mostly treated as an end-to-end black-box performance optimization problem. By

23 “the interaction of RL and LLMs is not merely additive, it is multiplicative: RL equips LLMs with the ability to learn from interaction and feedback, while LLMs equip RL with contextual awareness, explainability, accessibility, and code generation capability” - Bertsekas, D., A Course in Reinforcement Learning, 2nd-edition, Athena Scientific, 2025, page 136.

24 https://engineering.fb.com/2017/06/14/ml-applications/deal-or-no-deal-training-ai-bots-to-negotiate/ 25 https://research.google/pubs/googles-multilingual-neural-machine-translation-system-enabling-zero-shot-translation/ 26 https://ai-2027.com/ 27 https://www.darpa.mil/about/innovation-timeline https://engineering.fb.com/2017/06/14/ml-applications/deal-or-no-deal-training-ai-bots-to-negotiate/ https://research.google/pubs/googles-multilingual-neural-machine-translation-system-enabling-zero-shot-translation/ https://research.google/pubs/googles-multilingual-neural-machine-translation-system-enabling-zero-shot-translation/ https://ai-2027.com/ https://www.darpa.mil/about/innovation-timeline focusing on distinct components of agentic communication, MATHBAC will make the overall task tractable and efficient by first individually and then collectively optimizing the components to clarify when and how the current process fails. In Phase I, the program focuses on optimizing distinct components of the communication process (communication protocols, communication content) for a fixed set of agents. Knowing how to routinely optimize these components in the short time scale should allow the agents in Phase II to gradually, slowly (adiabatically!) self-evolve their capabilities more efficiently than a simultaneous multi-objective Reinforcement Learning optimization.

The MATHBAC program in Phase I requires advances in two technical areas (TAs):

TA1: The Mathematics of Agentic Communication Protocols TA2: Geometry and Evolution of Shared Knowledge - Generalizable Principles from Data

In Phase I, TA1 and TA2 development will be separate. In Phase II the two TAs will be combined into “Evolution Teams” (defined below) via a Phased Acquisition Approach (details below). As such , proposals must address TA1 or TA2 in Phase I; they should also briefly describe how they anticipate capabilities developed by both TA1 and TA2 researchers in Phase I could be orchestrated to address Phase II in a draft Phase II Statement of Work. Proposals that do not describe how they could address Phase II by leveraging anticipated Phase I capabilities may be considered non-conforming and removed from consideration. Proposers may propose to both TAs, but they must be submitted as separate proposals.

MATHBAC will also fund a Federally Funded Research and Development Center (FFRDC), University-Affiliated Research Center (UARC), or Government entity as the Independent Verification & Validation (IV&V) team (not solicited under this Program Announcement).

As noted above, MATHBAC is dedicated to advancing the foundational mathematics, systems/control theory, and information theory required to enable/accelerate (and understand) science-discovery by autonomous agents and agent collectives. In order to make this effort tractable, the MATHBAC program will restrict its focus to science domains since (1) science domains have a vast, structured curated knowledge corpora; (2) groundbreaking discoveries often require interdisciplinary synthesis and communication; (3) discoveries require hypothesis generation via analogical reasoning; and (4) science domains provide a clear method for validation and measuring success.

TA1 and TA2 proposers are expected to identify their selected science subdomain(s) and two families of scientific tasks in their subdomain(s) they plan to address–for example, for a chemistry subdomain, two such task families might be (1) Structure Elucidation (e.g., predict spectra from structures, identify structures consistent with spectra, deconvolute, and analyze complex mixtures); and (2) Reaction Development (e.g., predict new products of known reactions); TA2 performers must identify at least one well-established scientific principle, law, or correlation to be rediscovered in their domain(s) (e.g., rediscovering the Karplus equation or coupling rules in Nuclear Magnetic Resonance spectra); and their curated science corpora. TA2 proposers must also convincingly demonstrate that these corpora do not explicitly contain nor purposefully seed the targeted principle(s), law(s), or correlation(s). TA1 and TA2 proposers must (i) specify the best performance they have achieved with their current methods, as a baseline and (ii) identify potential additional metrics to be adopted by the program. For each potential additional metric, the proposer must provide the following:

A description of what it measures The method for its calculation A description of how it measures progress against program goals An explanation of how it supplements current program metrics An argument for why the metric should be adopted (e.g., what insights it will provide) Literature sources providing metric provenance and application (if available)

Lastly, proposers must identify and provide rationale about other scientific domain(s)/subdomain(s) to which they believe their current approach can generalize.

In TA1, Phase I approaches will be compared to a SOA model, e.g., a Mixture of Experts model.

In TA2, Phase I will include two evaluations. In the first evaluation, proposers will rediscover at least one known generalizable principle, law, or correlation with agents trained on domain literature chronologically prior to the original discovery. In the second evaluation, agents will be required to discover at least one (but ideally, more than one ) previously unknown principle, law, or correlation from novel data streams (a DARPA-hard goal, for instance, would start with the data-driven Mendeleev-level rediscovery of the periodic table for atoms and proceed to a “multi-dimensional analog” of a periodic table for molecules).

Toward the end of Phase I, DARPA will host a PI meeting for performers to demonstrate their current capabilities and share their test results to facilitate potential Phase II collaborations between TA1 and TA2 performers. Specifically, performers are highly encouraged to collaborate in what will be termed an “Evolution Team” for their Phase II proposals. An Evolution Team is defined as:

A single performer or a team of performers that addresses both TA1 and TA2 in Phase II; no performer can be part of more than a single Evolution Team.

In Phase II, Evolution teams’ approaches will be compared to (and expected to significantly outperform) a collective of domain experts with hand-crafted protocols and access to SOA models.

While performers will have the option not to collaborate with other performers in Phase II, collaboration is strongly encouraged. It should be noted that the decision to work independently or collaborate with others will not impact Phase II determinations.

For Phase 1 only, organizations/PIs may participate as a non-lead, contributing team member on multiple proposals submitted by different lead (prime) entities. Please review Section 2, Page 9 of PA-26-05 for details related to Phase 2.

DARPA will use a Phased-Acquisition Approach for the MATHBAC program. Proposers should provide a fully-definitized proposal for Phase I and a draft Statement of Work (SOW) and Rough Order of Magnitude (ROM) cost for Phase II. The SoW should be included in Attachment_D _Volume_I and the ROM should be included in Attachment_E_Volume_II. For the Phase II ROM, proposers may use the expected capabilities and programmatic cost assumptions, such as potential teammate size, estimated labor rates, and historical costs, to generate the estimate. Proposed price for Phase I must not exceed $2,000,000. DARPA anticipates releasing proposal instructions for Phase II prior to month 10 of Phase I; Phase II awards are anticipated by month 16. Phase II proposals will require the Evolution team to have one performer acting as the prime performer and the rest as subperformers. It is incumbent upon the members of the Evolution Teams to choose and negotiate their own prime-sub relationships. Phase II awards are subject to availability of funds, assessed Phase I performance, and the evaluation of Phase II proposals. Evaluation criteria for Phase II proposals will be in the Phase II Proposal Instructions and will be consistent with the criteria in this solicitation. DARPA may invite one or more Evolution Teams to continue to Phase II of the program. Phase II may be negotiated as a modification to an existing Phase I award or may result in a new award.

Phase II Agentic Evolution towards New Science This section describing MATHBAC Phase II is provided for planning purposes only. Proposers should use this information to justify how their Phase I effort will support Phase II and how they plan to integrate into an “Evolution Team” for Phase II. Initial proposals should describe potential Phase II approaches, with supporting technical details, assuming all of the expected outcomes from Phase I TA1/TA2 are successful. During Phase I, DARPA will issue proposal requests to Phase I performers detailing the content and evaluation criteria DARPA will use to determine Phase II awardees.

The expertise developed in both TA1 and TA2 of Phase I will be on fixed agents, i.e., agents that do not change/adapt during (or after) a campaign. The TA2 Orchestrator tool is the precursor of what will be the main goal of Phase II: allowing the agents to determine what, when, how, and where to evolve the agents. Currently, evolutionary pressure for agent platform design is mainly applied on the platform creators (individuals or companies). Agent self-evolution (rewriting the agent code based on performance, e.g., fine-tuning, updating the agent common knowledge modules, updating the agent communication protocols, disaggregating existing agents to subspecialized ones, and/or creating new agents and placing them in the protocol) may occur either in a punctuated (inter-campaign) or continuous (intra-campaign) manner. Performers will implement agentic self-evolution via optimization (reward based, possibly using reinforcement learning) that may be general or focus on a particular science domain, driven by agent performance (time to task completion, communication token bandwidth, energy requirements, etc.). The proposers must identify the rewards they anticipate to leverage in Phase II and why.

Phase II focuses on techniques that allow the agentic interactions—and even the agents themselves—to adapt and evolve, possibly differentiating or even disaggregating themselves as the collaboration campaigns progress.

Proposers are expected to identify the mathematics/systems and information theory algorithms and tools that will be leveraged and advanced to allow the protocols and the agents/agent collectives themselves to adapt and evolve. Proposals must describe the (1) methods that will be used to (a) identify and understand evolutionary pressures from monitoring evolving systems, (b) create evolutionary pressure mechanisms for directed evolution, (c) measure and monitor agents knowledge (and latent space if accessible), and (d) evolve and measure efficiency of the agents’ selected language; (2) the experiments to be used to provide baseline improvement comparison;

(3) how the team will implement an oracle (what information is needed, how much, what type, etc.) to confirm successful reception/understanding of a semantic communication by the receiving agent; (4) how TA1 and TA2 will be extended to address Phase II; and (5) how the proposed approach will achieve the Phase II notional outcomes, see Table 1: Program Metrics. It is expected that proposals will include a grounded example in their selected subdomain, demonstrating the expected capabilities of their approach.

It is assumed that in Phase II performers will need to develop new methods that will be used to implement shared latent space representation across agents (or identify/infer it, when not accessible) for a given task, understand and engineer the form (language) of the content for particular pairs in a collective, and design agentic pressures that force agent collectives to converge to optimal representations.

3. PROGRAM STRUCTURE

The MATHBAC program is a 34-month, two phase effort with two TAs that will run concurrently in Phase I and become combined in Phase II. Phase I is 16 months (Base), and Phase II is planned to be 18 months. Proposals must address a single TA in Phase I and describe how the results of both TA1 and TA2 combined will address the agentic evolution towards new science in Phase II.

Any changes to the Phase II focus will be delineated in the Phase II Proposal Instructions.

TA1 –The Mathematics of Agentic Communication Protocols

TA2 – Geometry and Evolution of Shared Knowledge—Generalizable Principles from Data

Phase I will hold agents static and not allow them to evolve, and the two TAs may be completed separately. In Phase II the agents are planned to evolve, and the TAs are expected to be unified within an Evolution team.

In Phase II performers are expected to work on both TAs as part of a unified Evolution Team that need not be defined until performers submit proposals for Phase II. These Evolution Teams should possess expertise in a range of topics including mathematics, information theory, proposer selected science subdomain, and science-discovery agents. The performers should also have access to agentic platforms allowing access to latent spaces and communication logs.

Phase I Accelerating Agentic Communication towards Scientific Discovery Technical Area 1: The Mathematics of Agentic Communication Protocols

Goal: To establish a quantitative mathematical framework for designing, optimizing, and establishing guarantees for the performance of communication protocols within a collective of fixed pretrained AI agents, moving beyond current ad-hoc methods.

This TA focuses on methods to understand/design/improve the communication protocols of interacting agents in agent platforms. The protocols will include how many and what type of agents are needed on the team and which agents “talk to” which agents at each stage of the science-discovery campaign.

Proposers are expected to identify the mathematics, systems theory, and information theory methods that will be leveraged and advanced to understand/design and improve the communication protocols of interacting agents. These may include input-output attempts to create for each agent approximate “internal models” of the agents with which it interacts (a sort of “theory of mind” of agents). Proposers must state:

(1) the methods that will be used to

(a) evaluate and schedule interactions between heterogeneous agents

(b) understand and enable more efficient (e.g., lower communication prompt length/faster completion) agent interactions

(b1) without access to any latent spaces (b2) with access to transformer latent spaces (b3) with mixed access agents (with/without accessible latent spaces)

(c) characterize and quantify the progress of communication dynamics throughout the campaign

(d) optimize communication protocols.

(2) the experiments and baseline protocol(s) (which can be handcrafted) to be used for comparison and to quantify improvement.

(3) which “small science” foundation models will be used; these must be science models whose latent spaces can be accessed. Proposers must describe how they will access these latent spaces and how they will use the information they access.

(4) the method in which data from agentic communication stream and agent latent spaces during a campaign will be captured, processed to improve communication protocols, and how they will be shared with IV&V.

(5) how the proposed approach will achieve and/or exceed the TA1 metrics (see Table 1:

Program Metrics for a description).

Proposals must include a clearly documented example in their selected science subdomain, demonstrating the expected capabilities of their approach as well as two families of scientific tasks in their subdomain(s) they plan to address. Additionally, as noted above, proposers must identify and provide rationale about other scientific domain(s) / subdomain(s) to which they believe their current approach can generalize.

TA1 posits that by systematically designing agent communication protocols we can markedly improve speedup and percentage success in agentic task completion.

Proposals must state how they will address the following anticipated (and possibly additional expected) mathematical challenges:

Agents as Operators/Systems: Develop mathematical techniques, such as system identification and input-output based reduced-order modeling, to locally approximate/represent individual AI agents as (1) mathematical operators operating on token strings (or more generally, multimodal input streams) and/or (2) systems with inputs, internal states (in the agent latent space), and outputs.

Communication Protocols in Operator Networks/System of Systems: Adapt and extend the fidelity (e.g., via hierarchical spectral augmentation of the rank) of the identified models of the operators/systems as well as graph theory tools to analyze, evaluate the performance of, and design efficient complex networked interactions (including internal cascades and recurrences) toward the desired goals.

Performance Prediction: Formulate discrete mathematical models to estimate/analyze how different successive combinations/permutations of agent

(operator, system) actions systematically affect the collective's efficiency, stability, and convergence over broad classes of related domain tasks.

Protocol Optimization: Formulate and solve discrete/continuous optimization problems (with defined cost functions (e.g., token usage, solution time) and performance metrics) to minimize end-to-end overall campaign cost while maximizing task success frequency.

Construction of an Oracle: What information is needed, how much, what type, etc.

to confirm semantic communication between agents was understood.

Deliverables:

A Quantitative Mathematical Framework: A theory to describe, analyze, and evaluate the design and implementation of multi-agent communication.

A Computational Design Tool: A software suite implementing the above framework in specific science-discovery subdomains to allow researchers (and even, in Phase II, the agents themselves) to implement and validate optimal protocols without ad hoc large-scale experimentation.

A Catalog of Protocol Design Principles and of Domain-Specific Optimal Protocols: A set of demonstrably superior communication protocols for specific tasks (the program focuses on scientific discovery), verifiably proven through the mathematical framework. Additionally, a distillation of the interpretable guiding principles leading to optimal protocol design is highly desirable.

Technical Area 2: Geometry and Evolution of Shared Knowledge—Generalizable Principles from Data.

Goal: To create a mathematical foundation based on disentanglement for discovering and incrementally sharing new, abstract, generalizable principles, laws, and correlations at meaningful spatiotemporal scales (in contrast to principles discovered through wisdom/serendipity).

This TA focuses on methods to monitor and usefully analyze multiple data streams (e.g., experimental data streams or streams recording the content of agentic communication), quantify its efficiency, discover new generalizable principles and abstractions, and use inductive and/or abductive reasoning in science-discovery.

Proposers are expected to identify the mathematics, information theory, and data science methods that they intend to leverage and advance towards discovering new generalizable principles by monitoring and analyzing multiple data streams. Proposals must describe:

(1) Methods the proposer will use to

(a) enable combination/comparisons of data streams across modalities and time scales.

(b) identify common and uncommon/particular features of the data streams, e.g., experimental data streams or streams arising from agentic interactions.

(c) extract generalizable principles, laws, or correlations; explicitly describe cost function/rewards used to drive principle/law/correlation extraction.

(d) enable interpretability of the extracted principles, laws, or correlations.

(e) generate and validate a minimum curriculum (content) for succeeding in a given task/science domain, if possible.

(2) How the proposed approach will achieve the TA2 metrics (see Table 1: Program

Metrics for a description).

(3) The computations/experiments to validate the generalizability of the discovered principle(s), law(s), or correlation(s).

Proposals must include a clearly documented example in their selected domain, demonstrating the expected capabilities of their approach.

TA2 posits that agentic interactions improve reasoning and thinking capabilities28 that should make common concepts “mineable” across a collective of data streams/agentic communication streams/agentic latent spaces. To test this, the TA2 metric will involve “understanding” that measures the ability to extract, distill, and share generalizable principles (laws or correlations).

Proposals must state how they will address the following anticipated (and possibly additional expected) mathematical challenges:

Develop algorithms for learning to identify consistent, low-dimensional structures shared across multiple, distinct latent space point clouds pertaining to the same knowledge subdomain, separating (disentangling) them from case-dependent or random features. One way of approaching this is through joint manifold learning techniques. The data come from (1) prior domain knowledge, (2) data collected by the collective’s tools during a campaign, and/or (3) data from observation of the agentic communication streams and agent latent spaces during the campaign.

Geometric and Topological Characterization of these common data manifolds, using tools from computational differential geometry/harmonic analysis as well as Topological Data Analysis (TDA) to compute meaningful distances in the “common” submanifolds and in their complements.

Interpretability as an Inverse Problem: Develop mathematical methods to solve the inverse problem of projecting the discovered “common” geometric object (in transformer latent space) back into an interpretable (e.g., natural language) space.

Deliverables:

A “Generalizable Principle Discovery Tool from Observations”: An algorithm suite capable of extracting the shared principles, laws, and correlations that connect a family of problems

A “Principle Speaker”: A tool that meaningfully projects the geometrically discovered principles into human-interpretable expressions

28 https://arxiv.org/pdf/2601.10825 https://arxiv.org/pdf/2601.10825

An “Orchestrator”: A tool that selects either (1) the fine-tuning curriculum of agents so that they embody the new discoveries or (2) the update of the agent shared core memory modules in view of these discoveries.

As noted above, MATHBAC will have an IV&V team who will prescribe and administer baseline comparisons for team performance. The IV&V team will by month 7 of Phase I (1) design bespoke challenge problems for each performer team in their scientific subdomain and (2) create or identify a current SOA scientific discovery agentic platform as the basis of comparison. This SOA platform will provide the basis of a “living metric”; IV&V will assess performer progress against the best and most recent version of the platform available at the time of comparison. The bespoke challenge problems will be released at month 7 of Phase I and can be completed on a rolling basis. The IV&V team will adopt a rubric, in consultation with DARPA, to assess the performance for TA1 and TA2. The final rubrics will be shared with the performers by Month 4 of Phase 1. TA1 and TA2 performers are expected to report on their progress using the IV&V rubric.

The TA1 rubric shall include, at a minimum,

(1) % success rate improvement for agentic campaigns:

Performers will need to demonstrate that their multi-agent communication design and optimization tool(s) will improve the end-to-end success rate for a campaign, which starts with a human positing an unsolved scientific task/problem and ends when the agentic platform returns one or more confirmable (computationally or experimentally) solutions. This will be compared to the TA1 baseline protocols until the IV&V living metric is created/identified.

(2) speedup of task completion(s):

Performers will need to demonstrate that their multi-agent communication design and optimization tool(s) will speed up the completion of their scientific subdomain tasks (e.g., characterization of the natural transition orbitals of 4-(dimethylamino) benzonitrile). This will be compared to the TA1 baseline protocols until the IV&V living metric is created/identified.

(3) adaptability/generalizability to related scientific-discovery tasks:

Performers will need to demonstrate that their multi-agent communication design and optimization tool(s) will adapt/generalize to nearby tasks. Until the IV&V living metric and bespoke challenge problems are available, performers will demonstrate this capability using tasks from their two proposed families.

The TA2 rubric shall include, at a minimum,

(1) # of data streams

Performers will need to demonstrate that their data fusion and analysis platform can successfully integrate and process multiple concurrent data streams. These streams could include diverse data types such as sensor outputs, experimental results, and computational models.

(2) amount of data, # of samples, and richness Performers will need to demonstrate that their data processing and analysis tool(s) can effectively handle and extract meaningful information from large and complex datasets.

This includes processing a high volume of data (amount), a large number of individual data points (samples), and data with high dimensionality or multiple modalities (richness).

(3) time to disentangle Performers will need to demonstrate that their causal inference and data disentanglement tool(s) will reduce the time required to separate and identify the underlying common factors or signals within a complex, multimodal dataset.

(4) Interpretability of results Performers will need to demonstrate that their data analysis and modeling platform produces outputs that are interpretable and explainable to a human scientific expert.

The goal is to ensure that the disentangled factors, causal relationships, or generated models are not merely predictive but also provide clear insights into the underlying scientific phenomena.

IV&V will further develop the rubrics to appropriately challenge each team according to the chosen subdomain.

Phase I baselines: All results will be evaluated by IV&V and DARPA. (On the left) In months 0-4, TA1 performers will use their own data, agentic platform, and families of tasks as their baseline; in months 5-7 TA1 performers will have to achieve the IV&V rubric targets using their own data, agentic platform, and families of tasks; in months 8 – the end of Phase I, performers will compare to the living metric, that is IV&V data, IV&V agentic platform, IV&V families of tasks, and the IV&V rubric targets for the bespoke challenge problem. (On the right) In months 0-4,TA2 teams will use their own selected correlation, principle, or law, and their own curated science corpora, that excludes their target known correlation, principle, or law, to be rediscovered; in months 5-7, TA2 performers will discover a new correlation, principle, or law and achieve the IV&V rubric targets; in months 8 – the end of Phase I, performers will use the IV&V data for their bespoke challenge problem and achieve the IV&V rubric targets.

In addition to meeting and or exceeding the rubric, performers must explain why their approach worked (or, equally importantly, why it did not work) and the IV&V team must validate, through implementation in collaboration with the team, the proposed methodology. In the Table below, an example of a family of tasks could be the characterization of the natural transition orbitals of various organic compounds.

In Phase II, we anticipate that the rubric will have to be updated based on program results and progress published/shared by the broader community. Instead of metrics, we provide here a notional idea of expected outcomes for Phase II. An updated rubric will be included in the Phase II Proposal Instructions.

Phase I Metrics Phase II Notional Outcomes

Accelerating Agentic Communication towards Scientific Discovery

Agentic Evolution towards New Science

For TA1:

Meet or exceed IV&V rubric and mathematically explain/rationalize why approach worked (or did not)

For TA 2:

Demonstrate ability to identify common and uncommon features across novel data streams pertaining to a science subdomain

Discover at least one new generalizable principle, law or unknown surprising correlation

Meet or exceed IV&V rubric and mathematically explain/rationalize why approach worked (or did not)

For Combined TA1 and TA2: Distill curriculum for a family of new tasks

Demonstrate effectiveness of new agent collective, capable of solving tasks from unknown (to the agent) problems

Table 1: Program Metrics

Program Schedule

Milestones and Deliverables

Proposers must address the following fixed payable milestones in their proposals. Proposers must complete the “Schedule of Milestones and Payments” tab of the DARPA Streamlined Cost Buildup Workbook Excel Attachment F provided with this solicitation.