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

Guide to the detailed statements

The remainder records the formulas and proofs behind the preceding arguments. To use a statement, distinguish its conclusion from its input. A finite certificate excludes a family of denominators; a cofinal supply excludes every rational value. Positivity of a tail difference is not nonintegrality, and a conditional implication still needs its arithmetic premise.

The following entry points answer different questions; they are not successive steps of one proof of irrationality.

Question	Where to read
Which section identities hold?	Section 10.8: independent sections and integral relations.
What does a finite computation exclude?	Section 5: tail residues; Section 5.3: the denominator cutoff.
Can one nonintegral difference yield more certificates?	Appendix A.1: propagation with growing shift and depth.
Which data still allow a rational value?	Section 2: comparison sequences, with their hypotheses kept separate.
Which estimate would suffice?	Section 12.3: the four-term decomposition; Theorem 3.5: the weaker good-index condition.
What was formally checked?	Appendix B: the declaration and source-version guide.

The detailed statements retain their source references so that a proof can be recovered without treating a source label as another hypothesis.

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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