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

The fixed Farey bound

Theorem 9.30. If $S$ is rational, its reduced denominator exceeds $7.963 964 664 670 137 532 335 577 487 583 105 3 \times 10^{34}$ . This is the finite denominator exclusion in the corpus, logically independent of the certificate-supply reduction.

Remark 9.31 (What a larger Farey bound requires). Unbounded valid denominator exclusions would imply $S \notin Q$ . For the single window $K = 240$ , however, the first-failure computation (gap_check_window_1_240_first_failure) proves that the residue inequality fails at the mediant denominator $b + d = Q_{0} + 1$ . This exact first failure prevents a larger cutoff from that same inequality at that same window. It is not an obstruction to other windows or to other tests using the finite data.

The existing proofs for individual windows use explicit totient residues and rational endpoint calculations. Extending that implementation calls for new finite data; a uniform proof of a growing bound need not use hard-coded convergents. If $S$ had reduced denominator $q$ , every valid exclusion bound would be smaller than $q$ . Thus unbounded exclusions suffice for irrationality; rationality would bound them, but would not force eventual constancy. No converse or comparison with the continued-fraction denominators of $S$ is proved here.

Observation 9.32 (The limited gain from a unit-gap refinement). The finite unit-gap condition of Observation 6.114 (ReducedDenominatorUnitGapCert) and its quantified version (PrimitiveReducedDenominatorUnitGapSupply) are definitions; the latter is not asserted to hold. Two separate results (reducedDenominatorUnitGapCert_prime_pow_iff, reducedDenominatorCandidateCount_prime_pow_le_one) show that, for $p$ prime and $e \geq 1$ , a window at denominator $p^{e}$ that passes the refined test contains at most one candidate integer. A window with two or more candidates fails the test. This is a condition for successful certification, not a bound on all windows or on the improvement in the denominator cutoff. It does not apply to all odd denominators: Observation 6.114 gives an actual totient window with two admissible nonunit candidates modulo $15$ .

Proposition 9.33 (A sufficient extension). Suppose $g (K) \to \infty$ and, for every $K$ , the $(N = 1, K)$ gap check excludes every rational of reduced denominator at most $g (K)$ . Then $S$ is irrational: any rational value of $S$ would have a fixed finite denominator, contradicted at a sufficiently large $K$ . Only unbounded exclusion bounds along a sequence of windows are needed for this argument. No equivalence with a bound for the continued-fraction denominators of $S$ is asserted; such a comparison would require its own proof connecting those convergents to the certified intervals.

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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