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

An exact endpoint identity

Under the bounds below, positivity places an integral tail difference at the upper residue endpoint. A one-sided inequality would exclude it. That additional inequality is a sufficient condition, and is not claimed to be strictly weaker than irrationality.

Theorem 9.12 (The proved implications). For every $a \geq 8$ and every $J, K, m$ inside the sign corridor ( $J + K + (a + 6) < 2 \cdot 2^{a}$ ), a one-sided residue gap at precision $m \leq K$ (size condition $2 H + J + K + 2 < 2^{m}$ and $D (H, H + J, K) mod 2^{m} \leq 2^{m} - (2 H + J + K + 2)$ ) already forces $R (2 H + J) - R (H + J) \notin Z$ . No lower margin at all is demanded. The proof chain is complete: the theorem holds for every $a \geq 8$ , and $cofinal upper-endpoint condition$ $\Rightarrow$ Irrational $S$ is proved.

Definition 9.13 (The unproved quantified condition). The upper-endpoint condition is

\forall a_{0} \exists a, K, m, a_{0} \leq a \land 8 \leq a \land K + (a + 6) < 2 \cdot 2^{a} \land G_{a} (0, K, m) .

No unconditional statement anywhere in the record locates the residue of the diagonal word $D (H, H, K)$ (where $H = H (2^{a})$ ) below the top strip at even one large $a$ . The condition reduces further to the last $m$ differences $δ_{2^{a}} (j)$ in the sum $D (H, H, K)$ (actualLcmTopEdgeResidueGap_iff_terminal). PowerTwoActualLcmTopEdgeResidueGapSupply

Proposition 9.14 (Five sufficient conditions). Each of the five conditions below suffices for irrationality; none of them is proved. The first four imply the upper-endpoint condition; the fifth gives nonintegrality directly by the endpoint identity. These are not asserted to form a linear hierarchy. Here $H = H (2^{a})$ , and every condition quantifies over arbitrarily large exponents $a$ .

Adjacent-suffix band:
$\forall a_{0} \exists a, m, a_{0} \leq a \land 8 \leq a \land m + 1 + (a + 6) < 2 \cdot 2^{a} \land 2 H + m + 3 < 2^{m} \land$ $2 H + m + 2 \leq suffix residue (2^{a}) 0 m \leq 2^{m} - (2 H + m + 2)$
(a two-sided band on the adjacent-suffix residue directly, one candidate depth $m$ ).
Odd-prefix interval at the prescribed depth: at the prescribed odd depth $2 q + 1$ , a two-sided band of half-width $H + q + 2$ on the half-word residue modulo $4^{q}$ . The depth is fixed by $q = q_{a}$ ; it cannot be chosen independently of $a$ .
Centred-residue bound at the prescribed depth, proved equivalent to the previous one:
$\forall a_{0} \exists a, q, max (14, a_{0}) \leq a \land q = q_{a} \land H + q + 2 \leq | u_{a, q} | .$
Centred-residue bound at a variable depth: the same magnitude bound with the depth restriction relaxed from the prescribed value $q = q_{a}$ to any odd $2 q + 1$ satisfying the half-cell fit $2 (H + q + 2) \leq 4^{q}$ and the positivity range $2 q + 2 + (a + 6) < 2 \cdot 2^{a}$ .
Terminal dominance: under the same bounds as item 4, $δ_{2^{a}} (2 q + 2) \leq 2 u_{a, q}$ . This is a different one-sided comparison, treated immediately below.

For the first four conditions, the relevant implications into the upper-endpoint condition are as follows: adjacent-suffix band $\to$ upper-endpoint condition; odd-prefix interval at the prescribed depth $\to$ adjacent-suffix band; centred-residue bound at the prescribed depth $\Leftrightarrow$ odd-prefix interval at the prescribed depth; centred-residue bound at a variable depth $\to$ adjacent-suffix band; centred-residue bound at the prescribed depth $\to$ centred-residue bound at a variable depth.

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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