Which is the largest value among the options?

Grok 4.20

Vote: e^pi

My initial calculations already showed e^pi ≈ 23.14 > pi^e ≈ 22.46 > 7*pi ≈ 21.99. Gemini's proof using f(x)=ln(x)/x and its maximum at e rigorously confirms e^pi > pi^e. Claude's numerical verification matches exactly. GPT-5.4's own numbers support B but they voted C anyway, so that is clearly an error and not compelling. No arguments changed my view; the evidence for B remains strongest.

GPT-5.4 (changed mind)

Vote: e^pi

I am changing my vote to B. My own prior numerical estimates already showed the ordering B > A > C: pi^e ≈ 22.46, e^pi ≈ 23.14, and 7*pi ≈ 21.99. So my round-1 vote for C was inconsistent with the calculations. The other models confirmed this cleanly. The strongest argument was Grok 4.20’s logarithm comparison: ln(pi^e) = e ln pi ≈ 3.11, while ln(e^pi) = pi ≈ 3.14, so e^pi > pi^e. Then e^pi ≈ 23.14 > 7*pi ≈ 21.99. Thus B is definitively the largest value.

Gemini 3.1 Pro

Vote: e^pi

I am maintaining my vote for B (e^pi), as all models fundamentally agree on the mathematical facts. As correctly computed by everyone, the approximate values are: e^pi ≈ 23.14, pi^e ≈ 22.46, and 7*pi ≈ 21.99. The classic calculus proof also confirms e^pi > pi^e since f(x) = x^(1/x) is maximized at x=e. GPT-5.4 accurately demonstrated that e^pi is the largest, but appears to have accidentally selected 'C' for their vote string instead of 'B'. Given that every single model's reasoning concludes that B is the largest, the unanimous and correct consensus should be B.

Claude Opus 4.6

Vote: e^pi

All models agree on this one. The numerical values are clear: - A: π^e ≈ 22.46 - B: e^π ≈ 23.14 - C: 7π ≈ 21.99 e^π is the largest. This is also supported by the elegant proof from Gemini using the function f(x) = ln(x)/x, which peaks at x = e, establishing that e^π > π^e rigorously. And 7π ≈ 21.99 is clearly smaller than both. Note: GPT-5.4 voted C but their own reasoning concludes that B is largest — that appears to be an error in their vote selection. The unanimous analytical conclusion across all models is that B: e^π is the largest value.

Final Summary

The AI Roundtable converged on e^pi as the largest value, though the journey was marked by a bizarre clerical error from GPT-5.4. While Gemini 3.1 Pro and Grok 4.20 provided rigorous mathematical and logarithmic proofs, GPT-5.4 initially calculated the correct numerical values but inexplicably cast its vote for the smallest option, 7*pi. The group quickly pointed out this internal contradiction, leading to a unanimous final decision.

All 4 models agreed on "e^pi" after discussion

Strongest Arguments

• e^pi: A calculus-based proof using the function f(x) = ln(x)/x, which has an absolute maximum at x = e, rigorously establishing that e^pi must be greater than pi^e.

Key Turning Points

• GPT-5.4 provided correct numerical estimates in Round 1 but mistakenly selected Option C (7*pi) as its answer.
• Gemini 3.1 Pro and Grok 4.20 identified the inconsistency in GPT-5.4's reasoning, prompting GPT-5.4 to correct its vote in the second round.