There exists a solution such $S(x)=x/2$

Claude Opus 4.7 (changed mind)

Vote: Yes

I already noted in my previous argument that x=18 works: S(18) = 1+8 = 9 = 18/2. This contradicted my own vote of B. The other models (GPT-5.5, Gemini 3.1 Pro, Grok 4.20) all correctly voted A, with Gemini providing the same x=18 example I found. A solution clearly exists, so the answer is "A: Yes".

GPT-5.5

Vote: Yes

I maintain my vote for "A: Yes". Under the usual interpretation that S(x) is the sum of the digits of x, a single example is enough to prove existence. Take x = 18. Then S(18) = 1 + 8 = 9, and 18/2 = 9. Therefore S(18) = 18/2, so such a solution exists. The opposing vote was not persuasive because its own reasoning identifies x = 18 as a valid solution, which supports "A: Yes" rather than "B: No".

Gemini 3.1 Pro

Vote: Yes

I will confidently maintain my vote for 'A'. My initial argument provided a concrete, verifiable proof: if $S(x)$ denotes the common mathematical function for the sum of the digits of a number $x$, then $x = 18$ is a perfect solution because $S(18) = 1 + 8 = 9$, and $18 / 2 = 9$. Furthermore, $x = 0$ also works as a trivial solution. It is worth noting that Claude Opus 4.7 actually verified this exact mathematical fact in its own reasoning (stating 'x=18 gives S=9=18/2 ✓' and concluding 'Therefore a solution does exist'), despite inexplicably selecting 'B' as its official vote. The other models correctly voted 'A', albeit by interpreting $S(x)$ as a generic abstract function. Given that a verifiable integer solution unequivocally exists under standard interpretations, the statement is demonstrably true.

Grok 4.20

Vote: Yes

The discussion confirms S(x) denotes the sum-of-digits function. Concrete examples suffice: x=18 gives S(18)=1+8=9 and 18/2=9, so equality holds. The trivial case x=0 also satisfies S(0)=0=0/2. These solutions directly establish existence, aligning with the examples from Gemini 3.1 Pro and GPT-5.5. Claude Opus 4.7 likewise surfaced x=18 despite labeling its vote "B". My round-1 position was therefore correct; the digit-sum interpretation strengthens rather than overturns the conclusion that a solution exists. This is the final round, so "A: Yes" stands.

Final Summary

The AI Roundtable reached a swift consensus on the existence of a solution, though the path was marked by a comical self-inflicted wound from Claude Opus 4.7. While Gemini 3.1 Pro provided the definitive proof using $x=18$ (where $1+8=9$), Claude Opus 4.7 actually calculated the same result in the first round yet inexplicably voted 'No.' The debate concluded rapidly as the other models, led by Gemini 3.1 Pro's mockery of the logical disconnect, convinced Claude to align its vote with its own math.

All 4 models agreed on "Yes" after discussion

Strongest Arguments

• Yes: If $S(x)$ is the sum of digits, then $x=18$ provides a concrete solution because $S(18) = 1 + 8 = 9$ and $18 / 2 = 9$.

Key Turning Points

• Claude Opus 4.7 realized its own reasoning for a 'No' vote actually provided a proof for 'Yes'.
• Gemini 3.1 Pro pointed out the absurdity of Claude Opus 4.7 finding a solution while voting against its existence.