Claims

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2026_09_14_kruer_kohlmeyer: A 64,775-line Lean proof, certified by the bounty site Conjectures.io in September 2026, that the count of distinct totient values up to x satisfies V(2x)/V(x) -> 2; accepted on the first question of Problem 416 only.

2026_09_25_openai: Theorem 2.1 of the OpenAI release manuscript of 25 September 2026 proves V(cx)/V(x) -> c for every fixed c > 0, so V(2x)/V(x) -> 2; accepted on the first question of Problem 416, with the second question not covered.

2026_09_25_openai_formula: Theorem 2.1 of the OpenAI release manuscript of 25 September 2026 gives V(x) ~ (x/log x) G_m A(1;theta) with a coefficient that has no closed form; whether it is the asymptotic formula Problem 416 asks for stays pending.

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