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 diagonal certificate table

Theorem 9.24.

\forall t \in {1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17}, \exists L, C (H t) (H t) L

at depths ${6, 5, 7, 7, 9, 14, 15, 14, 21, 22, 23, 26}$ respectively, extended by separate computations to 28 historical values through $t = 64$ . The later aggregate theorem closes every scale $t \leq 82$ with no holes. Each witness is checked by exact integer arithmetic on two finite totient windows. For example, $H_{17} = 12252240$ and $L = 26$ use $φ (H_{17} + i)$ and $φ (2 H_{17} + i)$ for $1 \leq i \leq 26$ , not every totient between the endpoints $12252241$ and $24504506$ . These computations verify the listed cases; they do not give a formula producing a certificate for every $t$ .

Remark 9.25 (The observed depths are finite data). The necessary depth inequality of Lemma 6.17 forces $2 (2 H_{t} + L + 2) < 2^{L}$ , that is, $L ≳ \log_{2} (4 H_{t})$ at any certified depth. Comparing that bound with the recorded depths shows that every recorded certificate depth lies within a small additive constant of that minimum, at every tested scale. This is an observation from finite computation, and no general theorem asserts it.

Proposition 9.26 (A sufficient extension).

\exists C \forall t_{0} \exists t \geq t_{0} \exists L \leq \log_{2} (4 \cdot H t) + C, C (H t) (H t) L .

This asks for a certificate within a fixed additive constant of the necessary logarithmic depth bound. The uniform constant is an additional requirement, not a consequence of the finite data.

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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