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Claim. Przemek Chojecki, Consecutive totient patterns, manuscript dated 13 July 2026 (the second preprint link), posted on the problem's discussion thread as the full solution. It reads the ordering patterns of Problem 415 as the $k!$ strict orderings of $k$ distinct values, writes $F_{\mathrm{str}}(x)$ for the largest $k$ such that every strict pattern of length $k$ occurs in some block $\phi(n+1),\ldots,\phi(n+k)$ with $n+k\le x$, and answers the three questions as follows.

• Theorem 1.1: as $x\to\infty$,

$$F_{\mathrm{str}}(x)=\frac{\log_3x}{\log_6x} +(\alpha-\gamma+o(1))\frac{\log_3x}{(\log_6x)^2},$$

with $\log_j$ the $j$-fold iterated logarithm, $\gamma$ Euler's constant and $e^\alpha=\prod_p(1-1/p)^{-1/p}$; the longest strictly decreasing block has the same asymptotic, and every fixed strict permutation occurs infinitely often. So no positive constant $c$ has $F_{\mathrm{str}}(x)\sim c\log_3x$; the manuscript adds that if $c=0$ were allowed the literal answer would be vacuously yes, since $F_{\mathrm{str}}(x)=o(\log_3x)$.

• Proposition 1.2: $F_{\mathrm{str}}(826)=3$, while $(\phi(823),\phi(824),\phi(825),\phi(826))=(822,408,400,348)$. The decreasing pattern of length four occurs below $826$ while only $15$ of the $24$ strict patterns of that length do, the increasing one among the nine missing; so the decreasing pattern is not always the first to fail. The manuscript checks this by an exact sieve printed in its Appendix A.
• Proposition 1.3: for $k=2$ the natural ordering of $\phi(1),\phi(2)$ is the tie $\phi(1)=\phi(2)$, which has density $0$ by Erdős's 1936 theorem (Proc. Cambridge Philos. Soc. 32, 530--540) that $\phi(m)<\phi(m+1)$ and $\phi(m)>\phi(m+1)$ each hold on a set of density $1/2$, while each strict ordering has density $1/2$; so under the density reading the natural ordering is not the most likely to appear.

The upper bound in Theorem 1.1 is the decreasing-block bound of Pollack, Pomerance and Treviño (their claim page), which the manuscript reproves; the lower bound is a permutation-equivariant refinement of the construction in their Section 8. The manuscript says that its proof of Theorem 1.1 is self-contained apart from the prime number theorem and Mertens's theorem.

Earlier draft. A strict-order resolution of Erdős Problem #415 for consecutive totients, a circulation draft dated 18 April 2026 (the first preprint link), names no author on its title page and was posted by Chojecki on 19 April 2026. Its Theorem 1.2 states the same asymptotic for $F_{\mathrm{str}}(x)$, taking the upper bound from Theorem 1.5 and Remark 8.1 of Pollack, Pomerance and Treviño; its Corollary 1.3 and Remark 4.3 say that it answers the first question, shows the monotone patterns asymptotically extremal without identifying the first missing pattern at a given $x$, and leaves the third question to the weak-order setting, where its Proposition 5.3 notes that realizing the equality pattern infinitely often would settle Problem 1003. A reply on the thread the same day reported that an automated check had found a mismatch between the draft's Proposition 3.3 and its Theorem 1.2. The July post says only that it closes some loose ends from the previous draft; neither the post nor the July manuscript names that mismatch.

AI systems. The April post says the note was written with GPT-5.4 Pro. The July post says the problem and the GPT-5.4 Pro output were run through GPT-5.6 Sol, which produced the completion. The July manuscript's disclosure says that OpenAI's ChatGPT assisted with drafting, proof checking and typesetting. The site's commentary credits Chojecki and GPT-5.4 with sketching how the proof of Pollack, Pomerance and Treviño adapts to an arbitrary strict pattern.

Standing. Claimed. Neither manuscript is refereed or on arXiv, no outside reader is recorded as having reviewed the July manuscript, the site's proof-claims tab for the problem carries no entry, and the site's commentary credits the April sketch on a problem it labels OPEN, which is not acceptance. Proposition 1.2 rests on the finite computation printed in the manuscript's Appendix A; no proof in either manuscript has been independently reviewed. The reading of the three questions follows the manuscript's and is recorded in the Formulation on the problem page.

Depends on. Nothing in this wiki.