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

Finite data and rational continuations

We begin with a rational continuation of an arbitrary finite prefix (Theorem 2.4). Proposition 1.9 instead preserves parity at every index, a linear size bound and aperiodicity. The carry and linear descriptions in Propositions 2.11 and 2.12 impose different requirements again. Their counterexamples concern the displayed descriptions only; they do not exclude all finite-state or finite-dimensional arguments.

Two other limitations do not require a rational comparison sequence. A congruence construction can increase the index so much that its tail error exceeds the residue it produces (Proposition 2.9). An exact reformulation or a finite list of examples still needs the quantified arithmetic assertion (Proposition 2.7).

The finite certificates in Proposition 1.4 and the $K = 240$ Farey calculation in Theorem 1.3 have bounded ranges. Theorem 2.4 excludes an inference from one such fixed range uniformly over the whole coefficient class. It does not exclude a proof using the arithmetic of $φ$ at arbitrarily large horizons. Likewise, the cofinal positivity theorem does not by itself separate the residue from the top edge: that is the additional information identified by Proposition 1.5.

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.

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