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 rational sequence with any prescribed totient prefix

A finite prefix leaves the remaining coefficients free. We construct a rational continuation satisfying the same linear size bound, then show why it cannot satisfy the quantified residue condition. First we record the two tail facts needed for a general coefficient sequence. For $c : N \to N$ with $c (n) \leq n$ , write $T_{c} := \sum_{n \geq 1} c (n) / 2^{n}$ , $R_{N}^{c} := \sum_{j \geq 0} c (N + 1 + j) / 2^{j + 1}$ , $D_{c} (h, N, L) := \sum_{j < L} (c (N + h + 1 + j) - c (N + 1 + j)) 2^{L - 1 - j}$ , and let $C_{c} (h, N, L)$ and ${Sep}_{c}$ be $C$ and $Sep$ with $D_{c}$ in place of $D$ . For $c = φ$ these are the objects of Definition 1.2.

Lemma 2.1 (A residue certificate excludes an integral tail difference). Let $c : N \to N$ satisfy $c (n) \leq n$ for all $n$ . For $h, N, L \in N$ , $C_{c} (h, N, L) \Rightarrow R_{N + h}^{c} - R_{N}^{c} \notin Z$ .

Proof. Splitting both tails after $L$ terms gives

2^{L} (R_{N + h}^{c} - R_{N}^{c}) - D_{c} (h, N, L) = R_{N + h + L}^{c} - R_{N + L}^{c} .

Both tails on the right are nonnegative, and $R_{M}^{c} \leq \sum_{j \geq 1} (M + j) 2^{- j} = M + 2$ . Their difference therefore has absolute value at most $N + h + L + 2$ . The corresponding totient bound is formalised at Erdos249257.TotientTailPeriodKiller.tail_after_le and used there to prove Erdos249257.TotientTailPeriodKiller.tail_diff_notMem_int_of_certifiedKill. If $R_{N + h}^{c} - R_{N}^{c} = k \in Z$ then $2^{L} k \equiv 0 (\mod 2^{L})$ and $D_{c}$ lies within $N + h + L + 2$ of $2^{L} k$ , so $D_{c} mod 2^{L}$ lies within $N + h + L + 2$ of $0$ or of $2^{L}$ , contradicting $C_{c}$ . ◻

Lemma 2.2 (Generic tail-period law). Let $c : N \to N$ satisfy $c (n) \leq n$ for all $n$ . If $T_{c} = p / (2^{e} m)$ with $p \in Z$ , $e \geq 0$ and $m$ a positive odd integer, and if $h \geq 1$ satisfies $m ∣ 2^{h} - 1$ , then $R_{N + h}^{c} - R_{N}^{c} \in Z$ for every $N \geq e$ .

Proof. Since $2^{N} T_{c} - R_{N}^{c} = \sum_{n = 1}^{N} c (n) 2^{N - n}$ is an integer,

R_{N + h}^{c} - R_{N}^{c} \equiv 2^{N} (2^{h} - 1) T_{c} (\mod) Z .

The right side is integral because $2^{e} ∣ 2^{N}$ and $m ∣ 2^{h} - 1$ . ◻

Remark 2.3. Lemmas 2.1 and 2.2 are also checked for every coefficient sequence satisfying $c (n) \leq n$ . The formal proofs use the true-tail recurrence and the bound $0 \leq R_{N}^{c} \leq N + 2$ ; the residue certificate keeps the full finite discrepancy. The tail-period proof uses $m ∣ 2^{h} - 1$ directly, so oddness of $m$ is only needed when choosing a period from the denominator. The following rational continuation and its finite-prefix consequence are also checked.

Theorem 2.4 (A rational sequence agreeing with any finite totient prefix). Let $B \geq 1$ and $P > B$ be integers. Define $γ : N \to N$ by

γ (n) := φ (n) (n \leq B), γ (n) := {\begin{cases} n - 1, & P ∣ n \\ n, & P ∤ n \end{cases} (n > B) .

Then:

$γ (n) \leq n$ for all $n$ , so $γ$ lies in the same coefficient class as $φ$ ;
$γ (n) = φ (n)$ for every $n \leq B$ , and consequently $D_{γ} (h, N, L) = D_{φ} (h, N, L)$ for every $(h, N, L)$ with $N + h + L \leq B$ ;
$T_{γ} = 2 - \sum_{n \leq B} (n - φ (n)) / 2^{n} - 1 / (2^{P} - 1) \in Q$ , and the odd part of its reduced denominator is exactly $2^{P} - 1$ ;
${Sep}_{γ}$ is false; indeed it fails already at $h = P$ .

Hence for every $B$ there is a coefficient sequence in the same class, agreeing with $φ$ on all of $[1, B]$ , whose quantified condition is false.

Proof. (i) is immediate; (ii) holds because $D_{φ} (h, N, L)$ reads $φ$ only at indices $\leq N + h + L$ . For (iii), since $P > B$ every multiple of $P$ exceeds $B$ , so

T_{γ} = \sum_{n \geq 1} \frac{n}{2^{n}} - \sum_{n \leq B} \frac{n - φ (n)}{2^{n}} - \sum_{k \geq 1} 2^{- k P} = 2 - \sum_{n \leq B} \frac{n - φ (n)}{2^{n}} - \frac{1}{2^{P} - 1} .

The first two terms have denominator dividing $2^{B}$ . On putting the expression over $2^{B} (2^{P} - 1)$ , its numerator is congruent to $- 2^{B}$ modulo $2^{P} - 1$ . Since $gcd (2^{B}, 2^{P} - 1) = 1$ , no odd factor cancels. Thus the reduced denominator has odd part exactly $2^{P} - 1$ and its power-of-two part divides $2^{B}$ . For (iv), apply Lemma 2.2 with $h = P$ : $R_{N + P}^{γ} - R_{N}^{γ} \in Z$ for every $N \geq B$ , so by Lemma 2.1 no $C_{γ} (P, N, L)$ holds for any $N \geq B$ and any $L$ . Taking $N_{0} = B$ refutes the inner existential of ${Sep}_{γ}$ at $h = P$ . ◻

Corollary 2.5 (The limit of a finite-prefix argument). No proof rule uniform over all $c : N \to N$ with $c (n) \leq n$ can establish $Sep$ from a single fixed prefix ${c (n) : n \leq B}$ : Theorem 2.4 supplies a rational countermodel with that same prefix. This does not invalidate an argument that uses the fixed arithmetic sequence $φ$ together with compatible information at arbitrarily large horizons; the theorem gives a different $γ_{B}$ for each $B$ , not one sequence agreeing with $φ$ at every $B$ .

Remark 2.6 (Four distinct finite restrictions). Theorem 2.4 concerns the information in a finite prefix. It does not identify $φ$ with one of the comparison sequences. It also does not exclude a proof that combines finite computation with an arithmetic theorem applying at all larger indices. Four different restrictions must be kept separate: a bound on the indices inspected, a bound on the depth $L$ , a bound on the carry states retained, and a bound on matrix rank. The depth inequality $2 (N + h + L + 2) < 2^{L}$ , for example, concerns the second restriction, not all four.

Proposition 1.9 gives another rational comparison sequence, this time preserving parity and specified aperiodicity properties. The two examples test different assumptions and must not be combined into a single stronger example.

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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