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

The short-window examples through exponent 6

Theorem 9.22.

\forall a_{0} \leq 6, \exists a, L, a_{0} \leq a \land L < 2 \cdot 2^{a} \land C (H_{2^{a}}, H_{2^{a}}, L)

The single witness $(a, L) = (6, 93)$ works for every threshold $a_{0} \leq 6$ . Its certificate is verified by exact integer arithmetic on the finite totient windows. This proves the displayed bounded statement. It does not construct witnesses for unbounded thresholds; further finite checks would extend the verified range, not establish the universally quantified condition below.

Definition 9.23 (The additional hypothesis). The cofinal short-window condition, which is unproved, is

\forall a_{0} \exists a, L, a_{0} \leq a \land L < 2 \cdot 2^{a} \land C (H_{2^{a}}, H_{2^{a}}, L) .

The depth restriction gives a useful description of the offsets: if $1 \leq j < 2 \cdot 2^{a}$ and $j ∤ H_{2^{a}}$ , then $j$ is a prime power greater than $2^{a}$ . Indeed, some prime-power divisor $p^{r}$ of $j$ exceeds $2^{a}$ ; if $j / p^{r} \geq 2$ , then $j > 2 \cdot 2^{a}$ , a contradiction. This describes the possible offsets, not the size of their weighted sum.

The condition restricts the depth in the ordinary diagonal certificate criterion. It is not established by pointwise completeness, which permits arbitrarily large depths. Nor does a successful short-window certificate by itself yield the fixed $1 / 32$ separation required by the preceding approximation condition. These are distinct sufficient hypotheses; no implication from this condition to that fixed-margin condition is proved here.

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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