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

Proved results and the additional assumptions they need

Besides the exponential-sum condition, the sufficient hypotheses involve tail signs, endpoint residues, rational approximation, short windows, Farey bounds, divisibility, denominator growth and carry rank. For each, we state the missing estimate used by the proposed application. These are specific possibilities rather than a classification of methods for Problem #249.

Positivity and the remaining residue inequality

An exact endpoint identity

A bound on the last totient difference

Separation of a rational approximation

The short-window examples through exponent 6

The diagonal certificate table

One common certificate for sixteen shifts

The fixed Farey bound

A two-adic congruence that does not give a certificate

The residue margin at an LCM jump

A certificate for a four-tail combination

Denominators of Möbius sums

Rationality and carry rank

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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