The Binary Totient Series

Will Cook

Long paper

The Binary Totient Series

Erdős #249 By Will Cook July 2026; revised 1 October 2026 Authorship, AI use and citation

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Summary. The dyadic totient sections through level $e \geq 1$ have an explicit rational basis of size $2^{e} + 1$ . Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ $φ (n) 2^{- n}$ .

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open.

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In this paper

A false generic rank bound

The coefficient rank might suggest a bound on carries under rationality. The next proposed assertion makes that suggestion precise; the rational comparison following it disproves both versions.

Definition 3.8 (A counterfactual rank criterion). The proposed assertion is a constant $C$ bounding the rank for every $c : N \to N$ with $c (n) \leq n$ and $T_{c} \in Q$ , every integer $v > 0$ , and every integer sequence $u$ satisfying

u (n + 1) = 2 u (n) - v c (n + 1), u (n) / 2^{n} ⟶ 0.

Specifically, the proposed bound is

\dim_{Q} {span}_{Q} {n \mapsto u (2^{j} n + r) : 1 \leq j \leq e, 0 \leq r < 2^{j}} \leq C (e \geq 0) .

The alternative replaces $C$ by a function $g (e)$ growing strictly slower than $2^{e} - 1$ , with the same quantifiers. Either assertion would contradict the lower bound of Proposition 1.6 under rationality of $S$ . Both assertions are false, as the following comparison shows.

Both assertions in Definition 3.8 are false. The comparison sequence with sum $5 / 4$ in Section 10.8 satisfies $0 \leq c (n) \leq n$ and has an integral carry with $u (n) / 2^{n} \to 0$ and rank at least $2^{e} - 1$ at every depth (Section 12.6). It therefore contradicts both proposed upper bounds. This leaves open bounds under further hypotheses specific to the totient.

Periodicity modulo $v$ alone does not provide the additional rank bound.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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