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.

Start here, by the editorial route

In this paper

Depth equal to a period multiple

The quantified residue condition.

full-depth condition :\equiv \forall d \geq 1 \forall N \exists t \geq 1 : C (t d, N, t d),

unpacked, $(N + 2 t d + 2 : Z) < D (t d, N, t d) mod 2^{t d} < 2^{t d} - (N + 2 t d + 2)$ . This condition is unproved. It is equivalent to irrationality (ErdosProblems.Erdos249.PeriodMultipleEscape.ApFullDepthEscape, ErdosProblems.Erdos249.FullDepthRayAmplifier.apFullDepthEscape_iff_irrational); both directions are proved, and the remaining gap is a nonintegral tail difference $R_{N + d} - R_{N}$ for every $d \geq 1$ and basepoint $N$ .

The condition in terms of binary digits.

Writing $h = t d$ and using $D (h, N, L) / 2^{L} = (R_{N + h} - R_{N}) - (R_{N + L + h} - R_{N + L}) / 2^{L}$ at $L = h$ , together with the proved strip $| R_{M + h} - R_{M} | < M + h + 2$ (Erdos249257.TotientTailPeriodKiller.abs_tail_diff_lt), applied at $M = N + h$ , one gets:

Proposition 12.9. $C (h, N, h)$ holds whenever $‖ 2^{N + h} S - 2^{N} S ‖_{R / Z} > 2 (N + 2 h + 2) / 2^{h}$ .

Proof. $‖ D (h, N, h) / 2^{h} ‖ \geq ‖ R_{N + h} - R_{N} ‖ - | R_{N + 2 h} - R_{N + h} | / 2^{h} > 2 (N + 2 h + 2) / 2^{h} - (N + 2 h + 2) / 2^{h} = (N + 2 h + 2) / 2^{h}$ , and $R_{N + h} - R_{N} \equiv 2^{N} (2^{h} - 1) S = 2^{N + h} S - 2^{N} S (\mod 1)$ by Lemma 12.1. A residue at distance more than $(N + 2 h + 2) / 2^{h}$ from $Z$ is exactly a residue certificate at depth $h$ . ◻

Proposition 12.9 gives a sufficient analytic criterion for the condition with depth equal to the shift: for each $d \geq 1$ and $N \geq 0$ , it is enough to find $h = t d$ , $t \geq 1$ , for which the displayed distance exceeds the stated threshold. The criterion is not asserted to be equivalent to each individual finite certificate. It involves the phase of $S$ itself, and the threshold tends to zero exponentially as $t$ grows. The propagation theorem in Appendix A.1 explains why one nonintegral tail difference produces late certificates. It does not supply such tail differences for all $d, N$ .

Comparison with digit complexity.

The phase inequality above is a quantitative statement at specified positions and period multiples. A general block-complexity assertion is a different quantified statement, of the kind treated by the Adamczewski–Bugeaud method for algebraic numbers [27]. No implication from such an assertion to the required inequality is established here. In particular, this record does not justify describing one existing method as the only possible route, or algebraicity as the sole missing input. The arithmetic nonintegrality obligation must be stated and proved in its own right.

Finite examples.

At $N = 300$ , the least $t$ with $C (t d, 300, t d)$ exists and is small for every $d \leq 24$ : $t = 10$ for $d = 1$ , $t = 5$ for $d = 2$ , $t = 4$ for $d = 3$ , $t = 3$ for $d = 4$ , $t = 2$ for $5 \leq d \leq 9$ , and $t = 1$ for $10 \leq d \leq 24$ ; in every case at or near the first depth admitted by the lower bound on the depth. The residues are not marginal: $r / 2^{h} = 0.3594$ at $d = 1$ , $0.5977$ at $d = 3$ , $0.6853$ at $d = 7$ , $0.2620$ at $d = 17$ , $0.5887$ at $d = 23$ , against edge radii $3.1 \cdot 10^{- 1}$ , $8.0 \cdot 10^{- 2}$ , $2.0 \cdot 10^{- 2}$ , $2.6 \cdot 10^{- 3}$ , $4.1 \cdot 10^{- 5}$ . This is a finite verification at the displayed basepoint and shifts. It neither bounds the required multiplier for arbitrary $d, N$ nor establishes the universally quantified condition.

A possible heuristic, not an estimate.

If a phase $U$ were uniform on $R / Z$ , then $Pr (‖ U ‖ \leq ϵ) = min (1, 2 ϵ)$ for $ϵ \geq 0$ . At $ϵ = 2 (N + 2 h + 2) 2^{- h}$ this gives the marginal model $min (1, 4 (N + 2 h + 2) 2^{- h})$ . The phases $2^{N} (2^{t d} - 1) S$ for fixed $d$ are arithmetically dependent; no independence model or uniform distribution theorem for this orbit is proved here. Consequently the marginal estimate cannot be multiplied across depths, and no summability conclusion over all basepoints follows from it. The finite experiments are finite evidence; the condition for every $d$ and $N$ remains unproved.

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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