Determination of the fifth Busy Beaver value (2509.12337v1)

Abstract: We prove that $S(5) = 47,176,870$ using the Coq proof assistant. The Busy Beaver value $S(n)$ is the maximum number of steps that an $n$-state 2-symbol Turing machine can perform from the all-zero tape before halting, and $S$ was historically introduced by Tibor Rad\'o in 1962 as one of the simplest examples of an uncomputable function. The proof enumerates $181,385,789$ Turing machines with 5 states and, for each machine, decides whether it halts or not. Our result marks the first determination of a new Busy Beaver value in over 40 years and the first Busy Beaver value ever to be formally verified, attesting to the effectiveness of massively collaborative online research (bbchallenge$.$org).

• The paper presents the first formal proof that S(5) equals 47,176,870 via exhaustive enumeration in Coq.
• It details a robust decider pipeline—from loops detection to NGramCPS and WFAR—that processes all 181,385,789 unique Turing machines.
• The collaborative approach, with over 27,000 lines of Coq code and formal proofs, sets a new benchmark in verifying uncomputable function boundaries.

Determination of the Fifth Busy Beaver Value: A Formal, Collaborative Milestone

The paper "Determination of the fifth Busy Beaver value" (2509.12337) presents the first formal proof that $S(5) = 47,\!176,\!870$, where $S(n)$ denotes the maximum number of steps a halting $n$-state, 2-symbol Turing machine can execute from the all-zero tape. This result, obtained via a massive collaborative effort and formalized in the Coq proof assistant, marks a significant advance in the paper of uncomputable functions and the boundaries of algorithmic decidability. The work also establishes $S(2,4)$ and reconfirms $S(4)$, providing a comprehensive, formally verified landscape of small Busy Beaver values.

Figure 1: 5-state 2-symbol Busy Beaver winner. This machine was discovered by Marxen and Buntrock in 1989 and is now formally proven to be the maximal halting 5-state 2-symbol Turing machine.

Theoretical Context and Motivation

The Busy Beaver function, introduced by Rado in 1962, is a canonical example of a simple yet uncomputable function. $S(n)$ is uncomputable because, if it were computable, it would yield a solution to the halting problem for $n$-state Turing machines. Historically, only $S(1)$ through $S(4)$ had been established, with $S(5)$ remaining open for over four decades. The challenge is twofold: the combinatorial explosion in the number of machines and the inherent undecidability of the halting problem, which, for larger $n$, can encode arbitrarily complex mathematical statements.

Formalization and Enumeration: Tree Normal Form

A central technical contribution is the exhaustive enumeration of all 5-state, 2-symbol Turing machines in Tree Normal Form (TNF), which eliminates unreachable transitions and state/symbol permutations. This reduces the search space from $1.67 \times 10^{13}$ to $181,\!385,\!789$ unique machines. The TNF enumeration is implemented directly in Coq, ensuring that no machine is omitted and that the enumeration itself is formally verified.

Figure 2: Transition table of the 2-state 4-symbol Busy Beaver winner found by Ligocki and Ligocki in 2005, also formally verified in this work.

Decider Pipeline: Automated and Formal Halting Analysis

The proof employs a multi-stage pipeline of deciders—algorithms that attempt to decide the halting status of a Turing machine. The pipeline is as follows:

1. Loops: Detects machines that enter periodic or spatially translated cycles (Cyclers and Translated Cyclers).
2. NGramCPS: Uses $n$-gram-based local configuration analysis, with alphabet augmentations (fixed-length history, LRU) to capture more complex behaviors.
3. RepWL: Employs regular expressions over repeated tape blocks to generalize and close over infinite families of configurations.
4. FAR/WFAR: Utilizes (weighted) finite automata to construct regular or nonregular over-approximations of reachable configurations, providing certificates of nonhalting.
5. Individual Proofs: For 13 "Sporadic Machines" not captured by automated deciders, bespoke Coq proofs are constructed.

This pipeline, implemented and verified in Coq, decides the halting status of all $181,\!385,\!789$ machines, with only 13 requiring manual intervention.

Figure 3: Space-time diagrams of the first 45 steps of a Cycler, illustrating a machine that eventually repeats the same configuration forever.

Figure 4: 10,000-step space-time diagram of a Translated Cycler not decided by the decider for loops in Coq-BB5 (it is decided by NGramCPS).

Figure 5: 10,000-step space-time diagram of a "fractal-looking" 5-state Turing machine that is solved by the LRU augmentation but has no known solution with standard NGramCPS or the fixed-length history augmentation.

Formal Verification in Coq: Proof by Reflection

The entire enumeration, decider pipeline, and verification are implemented in Coq, leveraging proof by reflection. This approach ensures that both the algorithms and their correctness proofs are machine-checked. The Coq development comprises over 27,000 lines of code and 638 lemmas, with an additional 10,000 lines for imported proofs. The proof compiles in under an hour on commodity hardware, demonstrating the feasibility of large-scale formal verification for combinatorial problems.

Handling Irregular and Pathological Machines

While the majority of machines are handled by regular CTL-based deciders, a small set of "irregular" machines require nonregular techniques (WFAR) or individual analysis. Notably, the 13 Sporadic Machines exhibit behaviors such as extremely long pre-periods before entering cycles (e.g., $5.4 \times 10^{51}$ steps), double Fibonacci counters, and obfuscated Gray code transformations. These cases highlight the diversity and complexity of behaviors even in small Turing machines.

Figure 6: Family picture of the 5-state Sporadic Machines (20,000-step space-time diagrams) which required individual Coq nonhalting proofs.

Implications for Computability and Mathematical Logic

The formal determination of $S(5)$ provides a concrete boundary for the knowable in the context of Busy Beaver values. For $n > 5$, the halting problem for $n$-state machines can encode open mathematical conjectures (e.g., Goldbach, Riemann), and for sufficiently large $n$, even the consistency of ZF set theory. The work identifies "Cryptids"—machines whose halting status is believed to be mathematically hard, such as the 6-state "Antihydra" machine, which encodes a Collatz-like problem.

Collaborative and Open Research Model

The project was conducted as a massively collaborative, open research effort via bbchallenge.org, involving hundreds of contributors, most without academic affiliation. The infrastructure included a Discord server, a wiki, and a public codebase, with contributions in a wide range of programming languages. The collaborative model facilitated rapid development, cross-verification, and the integration of diverse expertise, culminating in a formally verified result.

Scaling, Performance, and Future Directions

The Coq-based approach demonstrates that large-scale, combinatorial proofs can be both feasible and trustworthy. The enumeration and verification pipeline is parallelizable and can be further optimized. However, the exponential growth in the number of machines and the undecidability barrier imply that $S(6)$ and beyond are likely intractable with current methods, especially given the presence of Cryptids.

Figure 7: Main zoological families that were identified among 5-state Turing machines, together with Cyclers and Translated Cyclers.

Conclusion

This work establishes $S(5) = 47,\!176,\!870$ and $S(2,4)$ via a fully formal, collaborative, and reproducible process, setting a new standard for rigor in the paper of uncomputable functions. The integration of advanced decider pipelines, formal verification, and open collaboration provides a template for future research at the interface of computability, logic, and large-scale formal mathematics. The dataset and methods developed here will serve as benchmarks for future AI reasoning systems and as a foundation for further exploration of the boundaries of algorithmic knowledge.