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

Additional mathematics required by the sufficient conditions

We turn from the proved reductions to the estimates needed to apply them. The doubling identity relates window phases to exact tails; the subsequent decompositions separate the weighted sums for which cancellation is required. The suggested analytic applications remain proposals where those estimates are missing. Failure of one application does not exclude the analytic method.

The section contains a doubling-orbit formulation with explicit thresholds, a lacunary comparison sequence for the strength of a block estimate, and sufficient estimates for the prime-factor decomposition. The discussion of a rank upper bound must also be read with the rational $5 / 4$ comparison sequence: rationality alone does not give bounded rank. These are the scope distinctions needed to interpret the individual statements, not priority claims.

The proved reductions identify explicit sufficient hypotheses. Those hypotheses remain to be established; the reductions do not prove them.

The doubling identity

The full-block exponential-sum estimate

The four-term decomposition

Separation along prime positions

Depth equal to a period multiple

A rational series with unbounded carry rank

Finite observations and quantified conditions

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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