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

Rationality and carry rank

The totient sections have a dimension lower bound, as do the sections of an integral carry under the rationality hypothesis. An incompatible upper bound would finish a rank argument. The rational comparison sequences show why rationality by itself gives no such bound.

Theorem 9.47. For every $e \geq 1$ , the retained dyadic totient sections form a basis for all sections through level $e$ , of dimension $2^{e} + 1$ . The proof uses the Chinese remainder theorem and Dirichlet’s theorem. The complete level-zero truncation, consisting only of $φ$ , has dimension one.

Theorem 9.48. If $S \in Q$ , choose an integer $v > 0$ with $v S \in Z$ and put $u_{N} = v R_{N}$ . Then $u_{N} \in Z$ , $u_{N + 1} = 2 u_{N} - v φ (N + 1)$ , and $0 \leq u_{N} \leq v (N + 2)$ . For every $e \geq 0$ ,

\dim_{Q} span {n \mapsto u_{2^{j} n + r} : 1 \leq j \leq e, 0 \leq r < 2^{j}} \geq 2^{e} - 1.

The proof is given in Section 10.8.

Remark 9.49 (The generic upper-bound inference is false). The bound $2^{e} - 1$ holds at every level. An upper bound contradicting it would need additional information about the totient, since the rational $5 / 4$ comparison has the same lower bound. One proposed system of integer equalities and strict inequalities is inconsistent: there are no integers $Q, v > 0$ , $A \neq 0$ and $b$ with $Q v A = b$ and $| b | < Q v$ , since a nonzero integer multiple of $Q v$ has absolute value at least $Q v$ (false_of_compressedAdjointCertificate). This excludes those simultaneous requirements, not every argument using a matrix or its adjugate. Separately, the span of all dyadic totient sections is infinite-dimensional (Proposition 1.6).

Proposition 9.50 (An additional hypothesis for a totient-specific rank argument). A rank bound that contradicts the lower bound $2^{e} - 1$ would have to use additional arithmetic of the actual totient coefficients. The generic proposal that rationality bounds the rank of every integral carry satisfying $u (n) / 2^{n} \to 0$ is disproved by the comparison sequence recorded in the short paper: its carry rank is at least $2^{e} - 1$ at every level. It is therefore not a remaining general lemma from which totient irrationality follows. The conditional Proposition 1.7 records eventual periodicity modulo $v$ together with unbounded rational rank. Periodicity of the residue sequences concerns values in a finite quotient; it does not bound the rational span of the integer-valued sections. The $5 / 4$ comparison shows that this distinction persists for a rational coefficient series.

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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