Ildar Shar, the former StackAdapt chief executive who founded FirstPrinciples in 2024, is building the research company around a difficult proposition: AI systems can participate in original science without asking researchers to trust a black box. A new mathematics result offers his clearest test of that thesis so far.
In a July 29th account, FirstPrinciples said its Theo Conjecture system and OpenAI Codex helped graph theorist Randy Davila develop a proof of a conjecture associated with mathematicians Paul Erdos and William Staton. The conjecture dates to 1989, making it roughly 37 years old, rather than the 35 years stated in FirstPrinciples' headline.
Davila uploaded the research manuscript on July 28th. It presents a proof of Staton's prediction and derives a previously unspecified second-order term. The manuscript is a preprint and has not completed peer review, so the result currently rests on Davila's proof and FirstPrinciples' account of how it was produced.
That qualification matters because the work is being presented as evidence for a new kind of AI-assisted research. The manuscript describes a supervised loop in which Davila used Theo Conjecture and OpenAI Codex to propose, test and revise mathematical ideas. The theorem itself is supported by ordinary mathematical arguments, according to the paper, rather than certified by a model output or a finite set of numerical experiments.
A graph built from prime factors
The problem begins with a graph whose vertices are the integers from 2 through a chosen number n. Two vertices are connected when the corresponding numbers share a factor greater than one. In the graph built through 30, for example, 6 connects to 10 because both are divisible by 2.
The prime numbers form a largest set of vertices with no connections among themselves. In graph theory, that is a maximum independent set. Its size equals pi(n), the number of primes up to n.
The conjecture concerns a simpler quantity called the Havel-Hakimi residue. It is calculated from the number of edges touching each vertex and provides a lower bound on the maximum independent set. The procedure discards much of the graph's structure, including the identity of each vertex's neighbors, while retaining its degree sequence.
Erdos and Staton studied whether that compressed representation still captured the asymptotic scale of prime counting. Staton predicted that the residue would approach:
(zeta(2) - 1)n / log n
Davila's preprint claims a matching proof and a more precise expansion:
`R(G_n) = c0*n/log(n) + (c0 - A)*n/log(n)^2 + O(n/log(n)^3)`
Here, `c0 = zeta(2) - 1`
, while the newly identified constant A
is approximately 0.3201986326
. The extra term describes how the finite graphs approach the predicted limit. FirstPrinciples says the result does not provide a faster way to count primes or address the Riemann hypothesis.
Davila built the system for this kind of work
Davila is the technical protagonist behind the result. He is a FirstPrinciples member of technical staff with an affiliation at Rice University, and he has worked on automated mathematical conjecturing since 2016.
FirstPrinciples describes his earlier TxGraffiti toolkit as a modern successor to Graffiti, the program Siemion Fajtlowicz began developing in the 1980s. Graffiti generated relationships between mathematical properties, tested them against examples and filtered out statements judged too weak or uninteresting. Its conjectures fed a body of subsequent graph-theory research.
Davila independently reconstructed the common-divisor graph problem in 2025 before tracing it to that earlier work, according to FirstPrinciples. He then placed the problem inside Theo Conjecture, which stores mathematical objects, properties, candidate relationships and counterexamples in an inspectable registry. A language-model agent can revise the registry, launch exact tests and use failed conjectures to direct another search.
The system then focused on the Caro-Wei sum, a degree-based lower bound that produced both the expected leading constant and the additional term.
That still left an upper bound to prove. The paper's central construction rewires the graph while preserving every vertex's degree. Prime vertices with equal degrees are grouped into cliques, and displaced connections are absorbed among composite vertices. The resulting graph has a simpler independent-set structure, allowing Davila to establish the required upper estimate.
The workflow assigns the tools a substantial research role while keeping the argument open to human scrutiny. Exact computation rejected false paths. The manuscript lists Davila as its author and says each promoted theorem is proved independently of the computational search. That division of labor is more informative than the claim that an AI system "solved" the conjecture on its own.
Shar's move from advertising software to fundamental science
Shar previously co-founded advertising technology company StackAdapt, serving as its CEO from 2013 through 2023 before remaining on its board.
At FirstPrinciples, Shar has shifted from commercial advertising infrastructure to research systems for physics and mathematics. FirstPrinciples says its broader Theo program includes five specialized models, more than 120 billion parameters and training on a corpus of more than 3 million scientific papers. Those are company-reported figures, and the mathematics manuscript does not establish that the broader Theo model stack produced Davila's result. It specifically names Theo Conjecture and OpenAI Codex as tools used during the work.
That distinction gives FirstPrinciples a credible path for evaluating its research claims. A model benchmark can be optimized, selected or rendered obsolete by the next release. A published proof creates a durable artifact that other mathematicians can inspect line by line. Peer review and independent checking are the next tests for Davila's argument.
The work also leaves behind another conjecture. Davila's computations indicate that the Havel-Hakimi residue may remain within two integers of the rounded-up Caro-Wei sum across the tested range, including selected values up to one million. The preprint labels that relationship an open Theo Conjecture problem.
For Shar, the business bet is that systems organized around counterexamples, exact tools and human judgment can produce discoveries that general-purpose chat interfaces miss. Davila's preprint supplies a serious first case. Its lasting value will depend on whether mathematicians outside FirstPrinciples confirm the proof and whether the same workflow can repeatedly turn machine-generated leads into results that survive scrutiny.