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

One common certificate for sixteen shifts

Theorem 9.27.

C (h, 14, 9) (h \in {1, \dots, 16}) .

The basepoint $14$ and depth $9$ are common to all sixteen shifts. Consequently, $S \neq r$ whenever $r \in Q$ and $den (r) ∣ 2^{14} (2^{h} - 1)$ for some integer $h$ with $1 \leq h \leq 16$ . The exact computation uses two nine-term windows for each shift and reduces their discrepancy modulo $2^{9}$ . The earlier eight-shift computation uses basepoint $12$ and depth $16$ .

Remark 9.28 (Finite simultaneity and the unbounded condition). The displayed computation concerns only $N = 14$ , $L = 9$ and $1 \leq h \leq 16$ . It does not provide certificates beyond every basepoint threshold. Simultaneous certification of finitely many shifts at a sufficiently large, unrestricted depth is not an additional obstacle: under irrationality it follows from the same truncation estimate, as shown next. The argument below does not prescribe the common depth in terms of the basepoint and number of shifts.

Proposition 9.29 (Simultaneous certificates with unrestricted depth). The existence of a function $f : N \to N$ with $f (N) \to \infty$ such that

\forall N_{0} \exists N \geq N_{0} \exists L \forall h \in {1, \dots, f (N)}, C (h, N, L)

is equivalent to irrationality of $S$ . Under irrationality, one can take $f (N) = N + 1$ , and every $N$ has a suitable depth.

Proof. The displayed condition supplies Proposition 6.71’s quantified certificate condition: for fixed $h \geq 1$ , choose the basepoint threshold so large that $f (N) \geq h$ beyond it. That proposition gives irrationality.

Conversely, suppose $S$ is irrational. For every $h \geq 1$ and $N \geq 0$ ,

R_{N + h} - R_{N} = 2^{N} (2^{h} - 1) S - (Φ_{N + h} - Φ_{N})

is nonintegral. Fix $N$ and a positive integer $M$ . Choose $L$ so large that

\frac{2 (N + M + L + 2)}{2^{L}} < min_{1 \leq h \leq M} dist (R_{N + h} - R_{N}, Z) .

The finite minimum is positive, whereas the left side tends to zero. For each $h \leq M$ , the truncation error between $D (h, N, L) / 2^{L}$ and $R_{N + h} - R_{N}$ is at most $(N + h + L + 2) / 2^{L}$ . The triangle inequality therefore puts $D (h, N, L) / 2^{L}$ at distance greater than $(N + h + L + 2) / 2^{L}$ from every integer, which is precisely $C (h, N, L)$ . Taking $M = N + 1$ proves the converse. ◻

This proof gives no quantitative lower bound for the finite minimum as the basepoint or number of shifts grows. It selects a common depth after those parameters are fixed; it does not impose depth equal to the period.

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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