The Binary Totient Series

LCM tails and rational comparisons

We first specialise the tail congruence to LCM shifts, then compare it with rational series preserving selected coefficient data. The source declarations accompany the formulas; references point to the complete constructions and their hypotheses in the preceding sections.

1. Tail differences at least common multiples. Let $H_a=\mathrm{lcm}(1,\dots ,2^a)$. Then $S$ is irrational if and only if, beyond every threshold $a_0$, some $a\ge a_0$ satisfies ${\mathrm{\Delta }}_{H_a}(H_a)\notin \mathbb{Z}$. This is the exact cofinal nonintegrality criterion for the actual LCM diagonal, not a quantitative separation estimate and not a proof that such indices occur; the forward implication is recorded separately.

2. Positive tail differences need not be nonintegral. For $a\ge 8$ and $J+(a+6)<2\cdot 2^a$, one has ${\mathrm{\Delta }}_{H_a}(H_a+J)>0$. Suppose this difference is an integer. If a truncation depth $K$ satisfies

   $$J+K+(a+6)<2\cdot 2^a,2H_a+J+K+2<2^K,$$

   then

   $$D_{H_a,H_a+J,K}mod{2}^K=2^K-{\mathrm{\Delta }}_{H_a}(H_a+J+K)$$

   lies strictly between $2^K-(2H_a+J+K+2)$ and $2^K$. The later tail difference is a positive integer; its negative is the representative near zero. Thus positivity determines which endpoint contains the residue, but does not exclude an integral tail difference. Both displayed bounds on $K$ are required for this conclusion. Source declarations: positive tail differences, negative endpoint representative, residue near the upper endpoint.

3. Divisibility and finite linear combinations do not suffice. A dyadic coboundary is a sequence of the form $2u_j-u_{j+1}$; its weighted sum telescopes. For every $t\ge 3$, with $H=\mathrm{lcm}(1,\dots ,t)$, the linked source constructs a nonzero comparison sequence of this form. Its coefficients reproduce the actual totient differences at $t-2$ prescribed indices and are divisible by $\phi (H)$. Every finite integer linear combination of forward shifts preserves the telescoping identity and divisibility by $\phi (H)$ and the corresponding lower-level factors. The bounds remain uniform in the index, but are multiplied by the sum of the absolute values of the combination’s coefficients. A variant preserves sparse prescribed values along entire arithmetic progressions. These properties alone do not force a contradiction: the comparison need not consist of actual totient differences. The theorem concerns linear combinations of shifts, not nonlinear operations, and does not supply certificates at arbitrarily large scales. Source declarations: obstruction for finite integer combinations of shifts.

If $S=a/v$ with integers $a$ and $v\ge 1$, the scaled tails $u_N=v{R}_N$ are integers and satisfy

In particular $u_N/2^N\to 0$. Its dyadic sections $n\mapsto u_{2^{j}n+r}$ have rational rank at least $2^e-1$ through level $e$ and a common eventual period modulo $v$. Both properties are the corresponding theorem for the same carry; they are compatible, and neither supplies a contradiction.

Section 12.6 constructs a rational comparison sequence with $c(n)=\phi (n)$ for odd $n$, $|c(n)-\phi (n)|\le 2$ for even $n$, $0\le c(n)\le n$, and

This comparison sequence has an integral carry satisfying $u_N/2^N\to 0$, with dyadic rank at least $2^e-1$ at every level. Thus the same lower bound for a hypothetical totient carry is compatible with rationality. For one fixed prime $p$ and $g:\mathbb{N}\to \mathbb{Z}$, the exact laws $g(pn)=pg(n)$ when $p\mid n$ and $g(pn)=(p-1)g(n)$ otherwise, together with $g-\phi =o(n)$, force $g=\phi$: a nonzero defect keeps a nonzero ratio to the argument along $p^{j}n$. The comparison sequence fails these dilation laws. The unchanged odd-residue sections also give coefficient rank at least $2^e-1$ at every level. This is a lower bound, not an exact formula for the full coefficient rank.