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

Certificates with depth equal to the shift

The shift and truncation depth can be required to agree, provided the shift is allowed to run through positive multiples of a fixed $d$ . The propagation argument in Appendix A.1 explains why this restriction preserves the equivalence below.

Definition 3.7 (Depth equal to a shift multiple). For every $d \geq 1$ and every $N$ there is $t \geq 1$ with $C (t d, N, t d)$ ; unpacked,

N + 2 t d + 2 < D (t d, N, t d) mod 2^{t d} < 2^{t d} - (N + 2 t d + 2) .

It is equivalent to $S \notin Q$ . ApFullDepthEscape apFullDepthEscape_iff_irrational

For a fixed pair $(d, N)$ , a nonintegral tail difference supplies a certificate in every sufficiently late adjacent pair of multipliers. Appendix A.1 proves this propagation and hence the equivalence in Definition 3.7 (apFullDepthEscape_iff_irrational). The finite examples verify the starting hypothesis for individual pairs, not for every $d, N$ . That universal supply of nonintegral differences is the missing arithmetic input. With these quantifiers, the condition has the full strength of the irrationality question; it is not a mild regularity assumption. The open problem remains unproved.

Formal sources: windowDiscrepancy_self_eq_totientBlock_sub

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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