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

The block exponential-sum condition

A bound for an exponential sum is sufficient for the quantified residue condition. With $e (x) = \exp (2 π i x)$ , the assertion is

block norm condition :\equiv \forall h > 0, \forall X_{0}, \exists X, L, max (X_{0}, 1) \leq X \land 16 (2 X + h + L + 2) \leq 2^{L} \land ‖ \sum_{N \in [X, 2 X)} e (D (h, N, L) / 2^{L}) ‖ \leq \frac{21}{25} X .

Erdos249257.TotientTailPeriodKiller.DTWFirstHarmonicNormGap The universally quantified assertion is not proved.

block norm condition ⟹ S \notin Q .

Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_first_harmonic_norm_gap The threshold is $X_{0}$ , while $X$ and $L$ are chosen witnesses and may depend on both $h$ and $X_{0}$ . Taking $X_{0} = N_{0}$ gives a certificate at some $N \in [X, 2 X)$ , hence at $N \geq N_{0}$ . The elementary reason is that, if every residue failed the test, each would be within $(2 X + h + L + 2) / 2^{L} \leq 1 / 16$ of an integer. Every summand would then have real part at least $\cos (π / 8) > 21 / 25$ , contradicting the norm bound. The implication is proved; the bound on suitable blocks beyond every threshold remains to be established.

The underlying finite lemma is more general: it applies to any nonempty finite $T \subseteq [0, 2 X)$ with the stated size bound and $‖ \sum_{N \in T} e (D (h, N, L) / 2^{L}) ‖ \leq (21 / 25) | T |$ . Taking $T = [X, 2 X)$ gives the full-block implication. Allowing a chosen subset weakens the hypothesis; it does not strengthen it. The Lean proof (Erdos249257.TotientTailPeriodKiller.exists_certifiedKill_of_first_harmonic_norm_gap) uses the elementary block lemma of Theorem 9.4, which is unconditional as an implication. The real-part version uses $Re \sum_{N \in T} e (D (h, N, L) / 2^{L}) \leq (9 / 10) | T |$ . Its proof uses $\cos (π / 8) > 9 / 10$ and averaging. An unspecified saving below $| T |$ is not the stated hypothesis.

What supplying this form would take: genuine cancellation for the additive character $e (D (h, N, L) / 2^{L})$ at the growing dyadic moduli permitted by the size bound. The quantifiers require suitable blocks beyond every threshold for each fixed $h > 0$ ; they do not require one threshold that works simultaneously for all $h$ . A determinant identity (Erdos249257.ResidualGaugeObstruction.det_residualMonomialMatrix) gives a comparison in which every weighted phase equals $1$ while the specified determinant remains nonzero. Thus nonvanishing of that determinant alone cannot prove cancellation. A successful use of a determinant would need further information connecting its entries to the phase sum; the comparison does not exclude every such argument.

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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