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

Tail integrality on an LCM grid

The comparison. The diagonal condition (Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_lcm_diagonal_certificate_supply) uses the pair $(H_{t}, 2 H_{t})$ . Under rationality, the same fractional part occurs at every positive multiple of $H_{t}$ once $t$ is sufficiently large. A finite set of these multiples therefore gives another possible place to test for a nonintegral difference. The criterion below does not assert a smaller successful depth than the diagonal test.

Eventual equality of fractional parts.

Proposition 10.5 (Tail integrality on an LCM grid). $S \in Q ⟹ \exists t_{1}, \forall t \geq t_{1}, \forall q, m : N, 0 < q ⟹ R (q \cdot H (t) + m \cdot H (t)) - R (q \cdot H (t)) \in Z$ .

Rationality makes the fractional parts of $R_{k H_{t}}$ equal for all $k \geq 1$ at each scale $t \geq t_{1}$ . The corresponding implication, Theorem 6.53, shows that a certificate $C (m H_{t}, q H_{t}, L)$ with $q \geq 1$ at arbitrarily large $t$ implies $S \notin Q$ .

(b $^{'}$ ) A finite test on the grid.

Proposition 10.6 (A finite-grid certificate gives a nonintegral pair). Let $Q \subseteq N_{> 0}$ be finite and nonempty. For each $q \in Q$ , put $A_{q} = \sum_{j = 1}^{L} φ (q H + j) 2^{L - j}$ and $B_{q} = q H + L + 2$ . Suppose $B_{q} < 2^{L}$ for every $q \in Q$ and

\forall q_{i} \in Q \exists q_{j} \in Q, B_{q_{j}} < (A_{q_{i}} - A_{q_{j}}) mod 2^{L} .

Then $R_{q_{j} H} - R_{q_{i} H} \notin Z$ for some $q_{i}, q_{j} \in Q$ . The proof and a four-point example are given in Theorem 6.56.

Scope. Eventual integrality on the LCM grid is a necessary consequence of rationality, not a claim about the actual series made without that hypothesis. The finite-grid test uses the individual one-sided bounds $B_{q} < 2^{L}$ , rather than charging the larger radius to both ends of a pairwise interval. This compares modulus bounds, not binary depths. A finite nonempty set and every stated size inequality are needed; three or more points and pairwise compatibility are not.

Neither statement supplies the unbounded family required to decide #249. Exhibiting a nonintegral tail difference on the LCM grid at arbitrarily large $t$ would refute rationality. The finite-grid test is a way to certify such a difference, not a proof that these tests succeed at arbitrarily large scales.

The implication from rationality to grid flatness (rational_totient_series_forces_lcm_cone_flatness) should not be confused with an assertion of flatness for the actual series. The exact diagonal integrality reformulation and its quantified converse (diagonal_int_iff_foreignDiagonalDefect_hits_fullTarget, irrational_totientSeries_of_full_target_avoidance_supply) give other equivalent forms of the problem. They are not the only such forms; Proposition 1.8 lists further equivalent certificate conditions.

Consequence. Choosing the least deep tail is the step that turns a common-fractional-part assumption into an intersection of arcs. A related finite comparison concerns first and second differences: at $(h, N) = (1, 8)$ the first-difference test holds at depth $8$ , while no second-difference certificate exists at depth at most $8$ (totient_tail_rank_two_kill_sound_but_not_shallower_cell). This disproves a universal claim of smaller depth for second differences; it does not show that they can never improve a certificate.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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