Claims

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2012_09_19_pollack_pomerance_trevino: Theorem 1.5 of the 2013 paper gives the longest monotone run of consecutive totients below x the length (1+o(1)) log_3 x/log_6 x, so the threshold F(n) is o(log_3 n) and no positive constant c works in the first question.

2026_04_19_chojecki: Chojecki's manuscripts of 2026, written with OpenAI models, claim the strict pattern threshold (1+o(1)) log_3 x/log_6 x, a counterexample at 826 to the decreasing pattern failing first, and a no to the natural-ordering question.

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