Sparse small changes of the totient give the binary series any value

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_proof: Greedy rounding at each changed place keeps the remaining error below 2^(s-e-1), so the next change is at most 2^(s+g), which is at most e^(1/2).

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Statement

Let $\varphi$ be Euler's totient, with $\varphi(1)=1$. Let $g,s$ be nondecreasing functions from the positive integers to the nonnegative integers and $n_0\ge1$ an integer such that

$$s(n)+g(n)\le\tfrac12\log_2n\qquad(n\ge n_0).$$

Let $e_1<e_2<\cdots$ be integers with $e_1\ge n_0$ and $e_{k+1}-e_k\le g(e_k)$ for every $k\ge1$. Then for every real number $r$ there are integers $\delta_1,\delta_2,\ldots$ such that, with $\psi(n)=\varphi(n)$ for $n\notin\{e_k\}$ and $\psi(e_k)=\varphi(e_k)+\delta_k$,

1. $2^{s(e_k)}\mid\delta_k$ for every $k\ge1$;
2. $|\delta_k|\le e_k^{1/2}$ for every $k\ge2$;
3. $\sum_{n\ge1}\psi(n)2^{-n}$ converges absolutely and equals $r$.

Only the first change $\delta_1$ may be large. The proof is a greedy rounding.

Significance

Problem 249 asks whether $\sum_{n\ge1}\varphi(n)2^{-n}$ is irrational. With $s(n)=g(n)=\lfloor\frac14\log_2n\rfloor$ for large $n$ and gaps exactly $g(e_k)$, the sequence $\psi$ agrees with $\varphi$ modulo $2^{\lfloor\log_2n/4\rfloor}$ at every $n$, differs from it by at most $n^{1/2}$ except at the single place $e_1$, and differs from it only on a set that windows of $o(\log N)$ consecutive integers near $N$ meet for a vanishing proportion of $N$. Since $r$ may be any rational number, no property shared by $\varphi$ and all such $\psi$ can decide the problem. The [[research/totient_binary_new_attack/perturbation_barrier|perturbation barrier page]] derives these window statements and explains what they exclude and what they leave open, in particular arguments that use exact multiplicative identities of $\varphi$; the [[research/totient_binary_new_attack/proof_sketch|proof sketch of the fresh attack]] records the limitation on arguments using only these shared data. The claim does not prove or disprove the irrationality of the totient series.

Current standing and verification

The complete proof is author-recorded at tier 0, with no independent acceptance, formal verification, or novelty claim. No native L-claim is consumed. The proof uses only $0<\varphi(n)\le n$, rounding to the nearest integer, and absolute convergence. Check 6 of the research evidence runs the construction on a finite prefix for four rational targets and verifies the residue, size and error bounds of the proof at every step there, with a control that must fail; that finite exercise does not certify the claim, which rests on the proof.