The Binary Totient Series

The hypothetical rank bound.

Unconditionally, for every level $e\ge 1$, the dyadic totient basis of $2^e+1$ sequences is linearly independent over $\mathbb{Q}$ (via CRT and Dirichlet’s theorem on primes in arithmetic progression) (Erdos249257.linearIndependent_canonicalTotientKernelFamily). Consequently, if $S\in \mathbb{Q}$, there is an integer $v>0$ with $vS\in \mathbb{Z}$. Its integral carry $u_N=v{R}_N$ satisfies, for every $e\ge 0$,

It obeys $u_{N+1}=2u_N-v\phi (N+1)$ and $0\le u_N\le v(N+2)$. Erdos249257.not_irrational_totientSeries_implies_unbounded_carryRank_unconditional

A bound independent of $e$ for these same carry-section ranks would contradict the displayed lower bound; so would an upper bound that is $o(2^e)$. Rationality and linear coefficient growth alone do not give either conclusion: the fixed sequence constructed in Section 12.6 has sum $5/4$ and the same rank lower bound. An incompatible upper bound for the actual totient carry would have to use information not shared by that comparison sequence.

The most direct generic construction is also impossible: no integer certificate satisfying the following conditions ($Q\cdot v\cdot A=\mathrm{boundary}$ with $|\mathrm{boundary}|<Q\cdot v$ and $A\ne 0$) can exist (Erdos249257.false_of_compressedAdjointCertificate); and the full family is proved infinite-dimensional in span, the case $k=2$ of Coons’s non-regularity theorem (§10.8; Erdos249257.not_finiteDimensional_span_fullTotientKernel).

Proposition 1.7 gives a common eventual period modulo $v$ for the sections of this same $u$, together with their unbounded rational ranks. These conclusions concern different properties of the same carry; they supply no rank upper bound.

Growth of the Farey denominator bound.

The finite denominator exclusion for $S$ comes from a Farey-gap denominator exclusion at a single fixed window $K=240$:

Erdos249257.tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053 This bound is sharp at $K=240$; the mediant denominator $q=79639646646701375323355774875831054$ is the exact first failing denominator (GapFareyBound.gap_check_window_1_240_first_failure); so the same interval cannot exclude every rational at the next denominator. A different window requires a new certificate or a uniform argument; the fixed-window calculation supplies neither.

A sufficient extension is a function $g(K)\to \mathrm{\infty }$ for which every $(N=1,K)$ gap check excludes all rationals of reduced denominator at most $g(K)$. More generally, unbounded valid exclusion bounds along a sequence of windows suffice. This is a statement about the certified intervals. The record does not prove it equivalent to a lower bound on the continued-fraction denominators of $S$. This extension is unproved.

For $p$ prime and $e\ge 1$, a successful unit-gap certificate at denominator $p^e$ has zero candidates or a single candidate divisible by $p$ (Erdos249257.reducedDenominatorCandidateCount_prime_pow_le_one). The success hypothesis is essential: the bound is not a statement about every window at that denominator. Nor does it bound the eventual denominator cutoff. The prime-power restriction cannot be omitted, as the modulo-$15$ example in Observation 6.114 shows. No obstruction to a uniform estimate as $K$ varies is proved here.