Plectis

Long paper

The Binary Totient Series

Erdős #249 By July 2026; revised 1 October 2026 Authorship, AI use and citation
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Summary. The dyadic totient sections through level e≥1 have an explicit rational basis of size 2e+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

Conditions that would imply irrationality

Below are quantitative sufficient conditions, an equivalent residue formulation, and a proposed generic rank bound. The rank bound is false, as its rational counterexample shows. The other arithmetic conditions remain open for the totient; the proofs establish their stated implications. They are examples of possible approaches, not an exhaustive list.

Cancellation on a full block

A four-term bound for the real part

One bound on the good indices

A fixed separation at prime positions

Certificates with depth equal to the shift

A false generic rank bound

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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