The Binary Totient Series

Proposition 1.4 (Finite certificate computations). $\mathcal{C}$ has been verified at: the $28$ diagonal instances of the least-common-multiple diagonal through $t=64$; all shifts $h\in [1,16]$ simultaneously at $(N,L)=(14,9)$, by decide; and eight further period examples at $N=300$. These are historical subsets of the examples. The supplied source also proves a complete diagonal band for $1\le t\le 82$, with no omitted scales. This is bounded coverage, not a cofinal diagonal family. Note the quantifier order: the simultaneous small-shift example uses one pair $N,L$ for every $1\le h\le 16$. In symbols, it proves $\mathrm{\exists }N\mathrm{\exists }L\mathrm{\forall }h(1\le h\le 16\Rightarrow \mathcal{C}(h,N,L))$. In contrast, $\mathrm{Sep}$ requires arbitrarily large $N$ for each positive $h$; neither the shift range nor the basepoint threshold is bounded.

Proposition 1.5 (Positivity does not exclude an integer). Put $H_a=\mathrm{lcm}(1,\dots ,2^a)$. For every $a\ge 8$ and $J\ge 0$ with $J+(a+6)<2\cdot 2^a$,

This needs no irrationality hypothesis. If this difference is an integer and $K\ge 0$ satisfies

then

lies strictly between $2^K-(2H_a+J+K+2)$ and $2^K$. The later tail difference is a positive integer. Its negative, not the carry itself, is the representative near zero. Thus positivity locates the residue near the upper endpoint; it does not establish the central-residue inequalities. Both restrictions on $K$ are part of this conclusion.

Proposition 1.6 (Rationality forces unbounded carry rank). If $S$ is rational then, for every $e$, the carry sections $n\mapsto u_{2^{j}n+r}$ with $1\le j\le e$ and $0\le r<2^j$ span a rational vector space of dimension at least $2^e-1$. The lower bound holds at every depth. It comes from the linear independence of the $2^e+1$ retained dyadic totient sections for $e\ge 1$, proved using the Chinese remainder theorem and Dirichlet’s theorem, so the full family spans an infinite-dimensional space, the case $k=2$ of Coons’s non-regularity theorem (§10.8).

Proposition 1.7 (Periodicity modulo an integer does not bound rational rank). If $S$ is rational, the same carry $u$ has unbounded rational section rank and one eventual period modulo $v$ valid for all its dyadic sections. More precisely, there are $h\ge 1$ and $N_0$ such that, for every $j,r\ge 0$ and $n\ge N_0$,

The rank and periodicity assertions hold together. This conditional theorem alone is not a counterexample to a general periodicity-to-rank implication: its antecedent is not established. The rational comparison sequence with sum $5/4$ gives a counterexample to a rank upper bound based on rationality alone; see Section 10.8.

Proposition 1.8 (Equivalent certificate conditions). Put $H_t=\mathrm{lcm}(1,\dots ,t)$. Several variations of Definition 1.2 are equivalent to $S\notin \mathbb{Q}$. One may allow a positive multiple of each prescribed shift, still requiring certificates beyond every basepoint threshold (Theorem 6.48). Alternatively, one may use the original quantified condition or restrict to $h=N=H_t$ at arbitrarily large $t$ (Theorem 6.49). Replacing the symmetric residue test by

also gives equivalent quantified conditions, including its restriction to $h=N=H_t$ at arbitrarily large $t$. Finally, the counted phase-separation condition in Proposition 2.7(b) is equivalent to irrationality. These equivalences change the form of the arithmetic question, not its logical strength.

Proposition 1.9 (A rational sequence preserving size bounds, parity and aperiodicity). There is $c:\mathbb{N}\to \mathbb{N}$ with $c(n)\le 6$ and $c(n)\le n$ for all $n$, $c(n)\equiv \phi (n)(\mathrm{mod}2)$ for every $n$, and $c$ not eventually periodic; indeed for every $N,G,K$ there are $K$ pairs of coefficients $(6,0)$ starting beyond $N$, with starting positions pairwise separated by more than $G$; and $\sum _{n}c(n)/2^n=3/2\in \mathbb{Q}$.