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

Separation of a rational approximation

Theorem 9.19. Unconditionally, for every $a$ and $q$ ,

| Ω_{a} - ρ_{a, q} | < \frac{4 H + 2 (2 q + 1) + 4}{2^{2 q + 2}},

where $ρ_{a, q}$ is an explicit finite computable rational block. For fixed $a$ , the error tends to zero as $q$ grows. In the quantified condition below, however, the depth is prescribed, so increasing $q$ is not a free way to satisfy the separation hypothesis.

Remark 9.20 (Scope of the implication). A cofinal $1 / 32$ -separation of the raw approximants at the prescribed indices $q = q_{a}$ suffices for #249. Allowing arbitrary depths would be a different hypothesis. The cited short-window theorem gives nonintegrality of $Ω_{a}$ at $a = 4, 6$ (Proposition 6.94). It neither asserts the $1 / 32$ margin at those exponents nor says that these are the only successful exponents. Indeed, the diagonal table also gives short-window certificates $(a, L) = (2, 7)$ and $(3, 15)$ , since $H_{4} = 12$ , $H_{8} = 840$ , and $7 < 8$ , $15 < 16$ . All these are finite nonintegrality examples, not the cofinal approximation-separation condition.

Definition 9.21 (The additional hypothesis). The cofinal approximation-separation condition, which is unproved, is

\begin{matrix} \forall a_{0} \exists a \geq max (2, a_{0}) \exists q, q = q_{a} \land \\ \forall z \in Z, \frac{1}{32} + ε_{a, q} \leq | Ω_{a} - z | . \end{matrix}

This hypothesis is about the real quantity $Ω_{a}$ , not just a finite rational calculation. It implies $| ρ_{a, q_{a}} - z | > 1 / 32$ for every integer $z$ , by the displayed error bound and the triangle inequality. A cofinal $1 / 32$ -separation condition on the finite approximants is another sufficient input, through the residue-band implication. These sufficient conditions are not asserted to be equivalent.

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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