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

Finite observations and quantified conditions

The finite observations can be compared with the quantified conditions, but they do not establish an asymptotic digit distribution. They also supply no basis for ranking possible proof methods.

Comparison of conditions.

The following table records the conditions themselves, rather than sufficient digit proxies. Lemma 12.1 explains how their phases are related.

Condition	Required assertion	Indices and quantifiers
Block norm	Norm at most $21 X / 25$ for the truncated phase sum	All $N \in [X, 2 X)$ at one depth; arbitrarily large $X$ for each $h$
Four terms	The four separate budgets of Definition 3.4	One common $X, L, s, η$ ; $s, η$ fixed before $X_{0}$
Prime gap	Distance from $Z$ greater than $\arccos (9 / 10) / (2 π)$	The phase $2^{p - h - 1} α_{h}$ at arbitrarily large primes for each $h$
Full depth	$C (t d, N, t d)$ for some $t \geq 1$	Every $d \geq 1$ and every $N \geq 0$
Generic rank bound (false)	A uniform or subexponential upper bound for carries of rational coefficient series	All finite dyadic levels; refuted by the $5 / 4$ comparison

The first four conditions concern the same constant but impose different averaging, sampling and separation requirements. The $11 / 100$ digit-change test in Corollary 12.3 is sufficient for a real-part bound; it is not a replacement for the norm or four-term rows. Only the full-depth row is proved here to be equivalent to irrationality. The last row records a false generic upper bound, not an additional open hypothesis.

Quantitative estimates.

The counterexamples identify missing assumptions in the specific arguments tested here. They do not imply that the binary digits of $S$ are random, normal, or unusually structured. A constructed counterexample can disprove an implication without resembling $S$ in other respects. Likewise, failure of one congruence construction does not exclude every use of the Chinese remainder theorem.

Additional arithmetic information.

Table 1 records finite observations: the sampled digit-change proportions are near $0.50$ , above the sufficient threshold $0.11$ , and the displayed normalized cosine sums are below $0.89$ . These computations are compatible with the proposed conditions. They do not establish those conditions for every shift and arbitrarily large blocks, or prove a fair-coin model for the digits. The relevant quantified estimates remain unproved here.

The recorded small margins near a forbidden endpoint also require careful interpretation. One small margin disproves a proposed lower bound larger than that margin at that index. It does not rule out every positive margin on a suitable subsequence, or every estimate with scale-dependent bounds. No numerical census in this record determines which of these possibilities holds asymptotically.

Scope of the comparison.

Corollary 12.3 specifies one sufficient digit statistic: for every fixed $h$ , the digit-change proportion must reach at least $11 / 100$ on arbitrarily large blocks. A statement for one shift, a smaller positive proportion, or finitely many blocks is not this hypothesis. An upper bound $o (N)$ on the longest run alone would not imply irrationality; a periodic alternating expansion has bounded runs. The prime-index condition in (5) is another proposed sufficient estimate. Its exact algebraic decomposition does not establish the required correlation 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.

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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