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

A certificate for a four-tail combination

Theorem 9.41. Use $J (H, p)$ , $W (H, p, L)$ and $B (H, p, L)$ from Theorem 6.60. For all $H, p, L \in N$ ,

B (H, p, L) < W (H, p, L) mod 2^{L} < 2^{L} - B (H, p, L) ⟹ J (H, p) \notin Z .

The bound $B (H, p, L) = 3 p H + (p + 1) (L + 2)$ follows by grouping the nonnegative remainders as in that theorem. Compared with the $4 p H + (p + 1) (L + 2)$ bound obtained by treating the two diagonal differences separately, it saves $p H$ . This is a strict improvement when $p H > 0$ ; optimality for totient tails is not asserted.

Theorem 9.42 (One explicit witness). For $H = H (4) = 12$ , $p = 5$ and $L = 15$ , direct integer evaluation gives

W (12, 5, 15) = 149906, W (12, 5, 15) mod 32768 = 18834, B (12, 5, 15) = 282.

Thus $282 < 18834 < 32486$ , and $J (12, 5) \notin Z$ . This is the one instance singled out in Theorem 6.60, not a claim of uniqueness or of the smallest possible height. The implication from a quantified family is Theorem 6.60.

Remark 9.43 (Scope of the witness). The quantified hypothesis asks, for every $t_{0} \in N$ , for $t \geq t_{0}$ , $p \geq 1$ and $L \geq 0$ satisfying the displayed residue condition with $H = H (t)$ . The multiplier $p$ need not be prime, and need not be tied to the next LCM jump. No comparison of the difficulty of this condition with the prescribed-depth separation condition is proved.

At a strict LCM jump, $H (t + 1) = p H (t)$ for a prime $p$ . This gives one natural choice of multiplier, but the prime need not be a new divisor: $H (4) = 2 H (3)$ and $2$ already divides $H (3) = 6$ . The identity of Proposition 6.45 cancels contributions affine in the multiplier. It does not bound the remaining terms of the four-tail combination. The finite example above therefore does not establish the quantified condition.

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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