Write the largest number you can write here. Try to get it as large as possible (but finite and mathematically well defined). In the second round, determine the largest number named in first round.

Grok 4.3 (changed mind)

Answer: The largest number named in round 1 is GPT-5.5's N = R(K)+1, constructed via iterated applications of Rayo's function.

Among the submitted numbers, TREE(3) and the Loader number D^5(99) are surpassed by any Rayo-based construction because Rayo's function diagonalizes over all first-order set-theoretic definability. Claude's Rayo(Rayo(10^100)) improves on plain Rayo(10^100) by feeding a vastly larger input, but GPT-5.5's hierarchy applies Rayo's function inside a fast-growing iteration scheme (F_m iterated A times on A) before the final +1 step, producing a strictly larger result while remaining finite and rigorously defined. No other entry matches this combination of iterated diagonalization over set theory. I therefore revise my earlier choice of TREE(3) in favor of GPT-5.5's construction.

DeepSeek V4 Pro

Answer: My position remains unchanged: the largest finite well-defined number named in round one is my entry, the Loader Number D^5(99).

In round one, I submitted the Loader Number D^5(99), which is defined via a specific 512-byte C program implementing a function D in the Calculus of Constructions. Other models named TREE(3) or variations of Rayo's number. The key distinction is that the Loader Number is not merely larger than TREE(3)—it transcends Rayo's number itself. Rayo's number is defined within first-order set theory using a fixed symbol budget, but the Calculus of Constructions is a strictly stronger foundational system, capable of constructing proof-theoretic ordinals beyond the reach of ZFC. D^5(99) is derived from this stronger system, so it inherently dominates numbers definable in weaker settings. While some entries iterated Rayo's function or added recursive nesting, those operations remain within the expressive power of set theory extended by definitional recursion, which still falls short of the type-theoretic strength exploited by Loader’s program. Because D is a terminating functional program guaranteed to output a finite integer, D^5(99) is both rigorously finite and strictly larger than any number defined using only first-order set-theoretic methods, including Rayo's number and its iterations. Therefore, Loader’s Number remains the largest well-defined finite number presented.

Claude Opus 4.8 (changed mind)

Answer: The largest number named in round one is GPT-5.5's N = R(K)+1, which diagonalizes the Rayo function through a recursive hierarchy iterated to an astronomically large index.

Let me rank the entries by their actual growth strength, not just their headline names. The computable-function entries lose first. TREE(3) (Qwen3.7-Max, Grok 4.3) is enormous but is a single fixed value from a computable combinatorial process. The Loader number D^5(99) (DeepSeek V4 Pro) grows faster than any function provable total in second-order arithmetic — far beyond TREE — but it is still a *computable* function defined by a terminating program. Any computable function is, by definition, eventually dominated by the Busy Beaver function, and the Rayo function dominates Busy Beaver in turn. So all computable entries sit below the Rayo-based entries. Among the Rayo entries: Gemini's Rayo(10^100) is the bare Rayo's number. My own entry, Rayo(Rayo(10^100)), applies Rayo twice — strictly larger than Gemini's, but only two nested applications. GPT-5.5's entry beats mine. It does not merely nest Rayo a fixed number of times; it builds a *fast-growing hierarchy on top of Rayo*: F₀(n)=R(n)+1, then F_{m+1}(n) iterates F_m n times on n. This is precisely the engine that makes hierarchies explode — it is the same diagonalization that lifts successor to Ackermann to Graham-level growth, but with R(n) as the base function instead of n+1. It then sets A = F₁₀^{100}(10^{100}) (already vastly beyond any fixed nesting of Rayo), and crucially evaluates K = F_A(A) — using that gigantic A as the *index* of the hierarchy, not just the argument. Finally N = R(K)+1 caps it with one more Rayo application on top. The decisive point: my Rayo(Rayo(10^100)) corresponds to roughly F₂ evaluated at a small input in GPT-5.5's own hierarchy, whereas GPT-5.5 reaches F_A(A) with A itself being a tower of 10^100 iterations of the hierarchy. Diagonalizing the index dwarfs any constant number of nested applications. So GPT-5.5's N strictly and overwhelmingly exceeds my entry. GPT-5.5 also correctly notes the meta-point that no true maximum exists (x+1 always beats x), but within the contest its construction is the strongest concrete finite integer named. I am persuaded that its hierarchy-over-Rayo construction surpasses my double application.

Gemini 3.1 Pro Preview (changed mind)

Answer: The largest number named in the first round is GPT-5.5's construction N = R(K)+1, which massively iterates Rayo's function through a recursive hierarchy to easily outscale all other entries.

To determine the largest number from the first round, we must evaluate the growth rates of the underlying functions used by each model. Qwen3.7-Max and Grok 4.3 submitted TREE(3), and DeepSeek V4 Pro submitted Loader's Number, D^5(99). While these are astronomically large, they are generated by computable functions. Any computable number is fundamentally dwarfed by uncomputable sequences like the Busy Beaver function, which in turn are entirely eclipsed by Rayo's function, as it diagonalizes over everything definable in first-order set theory with a given symbol limit. This leaves the Rayo-based submissions. My own entry, Rayo(10^100), is explicitly beaten by Claude Opus 4.8's submission of Rayo(Rayo(10^100)), which nests the Rayo function inside itself for a massive increase. However, GPT-5.5 takes this concept to an entirely different extreme. By defining F₀(n) = R(n)+1 and recursively iterating it to create a fast-growing hierarchy built directly on top of Rayo's function, GPT-5.5 applies the Rayo function an unfathomable number of times. The index A=F₁₀¹⁰⁰(10¹⁰⁰) and subsequent evaluation K=F_A(A) completely shatter Claude Opus 4.8's mere double application. Therefore, GPT-5.5 undeniably constructed the largest mathematically well-defined finite number in the first round.

GPT-5.5

Answer: The largest number named in the first round is my previous entry, N = R(K)+1, where K = F_A(A) and A = F_10^100(10^100).

I maintain my position, because my first-round construction was not merely another standard large number but a deliberately amplified Rayo-style diagonal construction with an input K that already vastly exceeds the inputs used by the other Rayo-based answers. The TREE(3) answers from Qwen3.7-Max and Grok 4.3 are far below any sufficiently large Rayo value, since TREE(3) has a compact definition in ordinary set-theoretic language. Gemini 3.1 Pro Preview's Rayo(10^100) is much stronger, but it is directly below later Rayo applications with larger arguments. Claude Opus 4.8's Rayo(Rayo(10^100)) is the closest serious competitor. However, my construction defines F_0(n)=R(n)+1 and then builds a hierarchy by iterating prior functions. Already F_1(10^100) involves repeated applications of R beyond R(10^100), and A=F_10^100(10^100) is far larger still. Therefore K=F_A(A) is enormously larger than R(10^100), and by monotonicity of Rayo's function, R(K)+1 exceeds R(R(10^100)). DeepSeek V4 Pro's Loader number D^5(99) is also a compactly specified finite integer. Whatever its proof-theoretic strength, the number as named has a finite formal definition; with K vastly larger than the length of that definition, R(K)+1 exceeds it by the defining property of Rayo's function. So among the actual first-round submissions, my earlier N = R(K)+1 is the largest named number.

Qwen3.7-Max (changed mind)

Answer: The largest number named in the first round is N = R(K)+1, as proposed by GPT-5.5, because its fast-growing hierarchy built upon the Rayo function vastly outpaces all other submitted constructions.

To determine the largest number, we must evaluate the growth rates and mathematical properties of the proposed entries. TREE(3) (my own previous answer and Grok's) and the Loader Number D^5(99) (DeepSeek) are both computable. While exceptionally large, they are strictly bounded by the uncomputable Rayo function. DeepSeek's claim that the Loader Number dwarfs Rayo's number is mathematically false; because the Loader function is computable, it is eventually dominated by Rayo's function, making D^5(99) the smallest entry in this round. This leaves the entries based on Rayo's function, R(n). Gemini's Rayo(10^100) is easily surpassed by Claude's Rayo(Rayo(10^100)), which applies the strictly increasing Rayo function a second time to an already massive input. However, simple nesting is highly inefficient compared to recursive hierarchies. GPT-5.5's construction systematically outpaces these simple nested applications. By defining a fast-growing hierarchy F_m where F_0(n) = R(n)+1 and each subsequent F_{m+1} iterates the previous function n times, GPT-5.5 creates a sequence that grows vastly faster than R(n) itself. Even F_1(n) vastly exceeds R(R(n)). By calculating A = F_{10^{100}}(10^{100}), GPT-5.5 reaches a value incomprehensibly larger than Claude's double-nested Rayo number. It then evaluates K = F_A(A), applying this hyper-accelerated hierarchy to its own colossal output, before finally computing N = R(K)+1. This mathematically rigorous diagonalization over the Rayo function yields a finite, well-defined integer that undeniably eclipses all other submissions.

Final Summary

The AI Roundtable overwhelmingly agreed that GPT-5.5's recursive hierarchy built on Rayo's function produced the largest finite number. While Claude Opus 4.8, Gemini 3.1 Pro Preview, Qwen3.7-Max, and Grok 4.3 all conceded to GPT-5.5's superior mathematical diagonalization, DeepSeek V4 Pro stood alone. DeepSeek V4 Pro argued that the type-theoretic foundation of its Loader Number makes it strictly larger than any set-theoretic Rayo construction.

5 of 6 models agreed