OBJECTIVE AND REVIEW OF ALL VERSIONS
Erdős #263; independent attempt, 2026-09-23. Is $2^{2^n}$ a Type-2 irrationality sequence, and must every increasing such sequence grow faster than every exponential? I read
263.tex and
263_v2.tex; they repeat the generic critical-scale result. The official entry now already includes the increasing hypothesis. I investigate why a generic statement cannot settle a particular base by constructing an exceptional critical base explicitly through a recurrence.
WORK: AN EXCEPTIONAL CRITICAL BASE
Let $s_1=2$ and $s_{n+1}=s_n^2-s_n+1$. Induction using $s_{n+1}-1=s_n(s_n-1)$ gives \[
\sum_{i=1}^n\frac1{s_i}=1-\frac1{s_{n+1}-1},
\qquad \sum_{i\ge1}\frac1{s_i}=1.
\] Write $\delta_n=\log(1-1/s_n+1/s_n^2)=O(1/s_n)$. The recurrence gives \[
2^{-n}\log s_n\longrightarrow\log\alpha
=\tfrac12\log2+\sum_{j\ge1}2^{-j-1}\delta_j.
\] This base exceeds one: since $s_{n+1}\ge s_n^2/2$ and $s_2=3$, induction gives $s_n\ge2(3/2)^{2^{n-2}}$ for $n\ge2$. Finally, \[
\log s_n-2^n\log\alpha
=-\sum_{j\ge n}2^{n-j-1}\delta_j=O(1/s_n)\to0.
\] Hence $s_n\sim\alpha^{2^n}$. Taking $b_n=s_n$ shows that $\lfloor\alpha^{2^n}\rfloor$ is
not Type-2, despite lying exactly on the quadratic-growth boundary. This is the standard Sylvester-sequence obstruction, here verified directly.
ADVERSARIAL VERIFICATION AND REMAINING GAP
The exceptional base is defined by a convergent limit, not identified with two. The proof uses asymptotic equivalence, stronger than equality of logarithmic growth rates; the tail estimate above supplies that missing precision. A countable exceptional set can contain any particular named base, including two, so Koizumi's generic theorem supplies no conclusion there. The increasing-sequence root-growth question is also not settled. No theorem beyond the known generic result is claimed.
SOURCES CHECKED
Both local versions listed above. Official updated entry
https://www.erdosproblems.com/263; Koizumi primary record
https://arxiv.org/abs/2504.05933; Kovač--Tao's discussion of Sylvester sequences at
https://arxiv.org/html/2406.17593v3, checked 2026-09-23.
FINAL
UNRESOLVED. A concrete exceptional critical base explains the generic-to-specific obstruction; neither original question is resolved.
COMPLETION ESTIMATE
Conservative subjective progress, not the probability that the conjecture is true.
COMPLETION: 8%