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The direct substitution $\theta(n)=\varphi(n)$ into Duverney–Tachiya's Theorem 1.1 fails its coefficient-growth hypothesis. This tests a source's applicability to Problem 249; it gives no irrationality or rationality conclusion for $S=\sum_{n\ge1}\varphi(n)2^{-n}$. The elementary calculation below is author-recorded by Plasma AI and has no independent review.

The failed hypothesis

Condition $(H_2)$ asks for a fixed $\nu>0$ such that, uniformly over coprime positive integers $a,b$,

$$\sum_{i=0}^{n}|\theta(ai+b)| \le n(2+\log n)^\nu \qquad(n\ge\max\{a,b\}).$$

The pair $a=b=1$ already prevents the proposed substitution. If $m\ge1$ has distinct prime factors $r_1<\cdots<r_s$, then $r_j\ge j+1$ and $2^s\le m$. The empty product covers $m=1$. Hence

$$\frac{\varphi(m)}m =\prod_{j=1}^s\left(1-\frac1{r_j}\right) \ge\prod_{j=1}^s\frac{j}{j+1} =\frac1{s+1} \ge\frac1{1+\log_2m}.$$

For $n\ge2$, at least $n/2$ integers in $[1,n+1]$ are at least $n/2$. Keeping just those summands gives

$$\sum_{i=0}^{n}\varphi(i+1) \ge\frac{n^2}{4(1+\log_2(n+1))}.$$

For every fixed $\nu>0$, the ratio of this lower bound to $n(2+\log n)^\nu$ tends to infinity. Thus $(H_2)$ fails even if its right side is multiplied by any fixed constant. Checking the separate divisibility condition $(H_1)$ cannot repair this direct application.

Scope of another application

A transformed integer coefficient sequence could still be relevant, but it must meet two separate obligations: an implication from its irrationality to that of $S$, and the actual hypotheses of the theorem being used. The [[research/leads/totient_binary_series/analytic_sources|analytic source tests]] record other restrictions on fixed dyadic filters and sparse transfers; those restrictions are not assertions about every possible transformation. No such new transformation is supplied here.

The source statement and both hypotheses were read in the held transcription and the linked statement page. This applicability check does not require, and does not supply, an independent audit of the source proof. It concerns only the original coefficients $\varphi(n)$.