% SPDX-FileCopyrightText: 2026 Will Cook
% SPDX-License-Identifier: CC-BY-4.0
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\documentclass[11pt]{article}
\input{problem-note-preamble}
\renewcommand{\commit}{9e231ce4371fcda607f81c4520d29b16dcb4482e}
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\hypersetup{
  pdftitle={A basis for the 2-kernel of Euler's totient},
  pdfsubject={Erdős Problem Notes: Problem 249},
  pdfkeywords={irrationality, Euler totient, k-regular sequences, 2-kernel, Lambert series, Möbius inversion, linear independence, Lean 4},
}

\newcommand{\Stot}{S}
\newcommand{\EB}{E}
\newcommand{\Aw}{A}
\newcommand{\Ker}{\mathcal K}
\newtheorem{equivform}[theorem]{Equivalent formulation}

% Status tag from the claim taxonomy, printed on every numbered result.
\newcommand{\stag}[1]{{\normalfont\footnotesize\textsc{[#1]}}}

\runninghead{Erd\H{o}s \#249: a basis for the dyadic totient sections}
\title{A Basis for the 2-Kernel of Euler's Totient}
\subtitle{Exact finite-level ranks and consequences for Erd\H{o}s
Problem~\#249}
\author{Will Cook}
\date{July 2026}

\begin{document}
\maketitle
\noteprovenance

\begin{abstract}
\noindent
For $j\ge0$ and $0\le r<2^j$, write
$\ph_{j,r}(n)=\ph(2^jn+r)$.  We prove that
\[
  \{\ph_{0,0},\ph_{1,0}\}
  \cup
  \{\ph_{j,r}:j\ge1,\ 0<r<2^j,\ r\ {\rm odd}\}
\]
is a $\Q$-basis for the span of the $2$-kernel of Euler's totient.
Consequently, the span of the dyadic sections through level $e$ has dimension
exactly $2^e+1$ for every $e\ge1$, and the two elementary reduction identities
generate every $\Q$-linear relation among the sections.  This determines the
entire dyadic-section span; it does not decide the irrationality question below.

Let $\Stot=\sum_{n\ge1}\ph(n)/2^n$.  A rational value would induce a tempered
integral tail orbit whose dyadic sections through level $e$ have rank at least
$2^e-1$; no conflicting upper bound is proved.  Independently, every rational
representation $\Stot=a/q$ must satisfy
\[
  q>79\,639\,646\,646\,701\,375\,323\,355\,774\,875\,831\,053,
\]
the sharp exclusion for the chosen $240$-binary-digit Farey window.  The
current repository also verifies a diagonal tail certificate at every natural
scale $t\le82$.  This aggregate is finite: it supplies neither an instance at
$t=83$ nor an unbounded family.

We also record exact equivalences between irrationality of $\Stot$ and
cofinal tail-nonintegrality or certificate supplies.  These are lossless
reformulations of the open problem, not evidence for the missing cofinal
producer.  Erd\H{o}s Problem~\#249 remains open; formalisation alone is not
presented as a novelty claim.
\end{abstract}

\externalreviewobject{The paper gives an explicit basis and relation normal
form for the dyadic totient kernel, proves level rank $2^e+1$, derives the
all-base rank $k^e+1$ from Martin plus explicit reductions, and checks a finite
denominator exclusion together with its first failing
certificate scale.}{The rank theorems do not supply a
rationality-to-finite-rank bridge, while the finite certificate does not
supply an unbounded producer.  Problem~\#249 remains open.}{Comparator selects
the explicit odd-core basis and the exact dyadic finite-level rank theorem.
Full-kernel infinite-dimensionality is prior-art context from Coons, while the
all-base linear-independence input uses Martin externally; Lean checks the
rank conclusion conditional on that explicit input.  The finite arithmetic
certificate retains its separate evidence route.}

\section{Introduction and main results}\label{sec:results}

Erd\H{o}s Problem~\#249 asks whether
\[
  \Stot=\sum_{n\ge1}\frac{\ph(n)}{2^n}
\]
is irrational; see Erd\H{o}s and Graham~\cite[p.~61]{erdosgraham1980} and
Erd\H{o}s~\cite[p.~102]{erdos1988}.  Bloom's current catalogue record
reproduces this question and labels it open, while explicitly warning that
the status is the website owner's present assessment and may omit relevant
literature~\cite{erdosproblems}.  We therefore use the catalogue for
numbering and current reported status only; the two original sources carry
the problem statement.

The principal result determines the dyadic sections of Euler's totient at
every finite level: the two zero-residue sections $\ph_{0,0}$ and
$\ph_{1,0}$, together with the odd-residue sections, form a rational basis,
and the two reduction identities below generate every rational relation among
the sections.  Its connection with Erd\H{o}s Problem~\#249 is through the
integral scaled-tail recurrence displayed below.  The basis theorem gives a
lower bound for the dyadic-section rank of every such recurrence supplied by a
hypothetical rational value of $\Stot$, but no corresponding upper bound is
known.

The other unconditional statements are a finite Farey denominator exclusion,
finite diagonal certificates at every $t\le82$, identities for the Lambert
coefficient sequence $\Aw=\ph*\mu$, and a theorem that no fixed common
denominator clears every $\Aw(n)/n$.  The certificate equivalences have a
different logical status: \ref{res:equivalences} gives exact characterisations
of irrationality, while \ref{res:supply} gives a sufficient criterion.  Their
unbounded hypotheses are not proved, so the finite certificate band is not
progress on the cofinal obligation.

\statusboundary

Each \emph{Checked} line below links the relevant declarations at
commit~\texttt{\commitshort}.  The bracketed tags reproduce the public status
recorded by the claim registry; where no registry row owns the cited
declarations, the text says so rather than assigning a neighbouring status.

Throughout $\N=\{0,1,2,\ldots\}$ and $\ph(0)=0$.  Thus
$\ph_{j,r}(n)=\ph(2^jn+r)$, for $j\ge0$ and $0\le r<2^j$, is the
$(j,r)$ \emph{dyadic section} of Euler's totient, a vector in $\Q^{\N}$, and
$j$ is its \emph{level}; the family of all of them is the $2$-kernel
$\Ker_2(\ph)$ in the sense of Allouche and Shallit~\cite{allouche-shallit}.
Two elementary identities relate them: the \emph{reductions}
$\ph_{j+1,0}=2^{j}\ph_{1,0}$
(\mword{Erdos249257/TotientMahlerDefect.lean}{160}{totientKernel_zero_residue}{checked})
and $\ph_{j,\,2^{t+1}s}=2^{t}\ph_{j-t-1,\,s}$ for $s$ odd with $2^{t+1}s<2^j$
(\mword{Erdos249257/TotientMahlerDefect.lean}{169}{totientKernel_even_residue_reduce}{checked}).
At level $3$, for instance, they give $\ph_{3,0}=4\ph_{1,0}$,
$\ph_{3,2}=\ph_{2,1}$, $\ph_{3,4}=2\ph_{1,1}$ and $\ph_{3,6}=\ph_{2,3}$, while
the four odd residues $r=1,3,5,7$ are reduced by neither; numerically
$\ph_{3,4}(1)=\ph(12)=4=2\ph(3)=2\ph_{1,1}(1)$.  So beyond $\ph_{0,0}$ and
$\ph_{1,0}$ each level contributes only its odd residues, and the count
through level $e$ is $2+\sum_{j=1}^{e}2^{j-1}=2^e+1$; \ref{res:rank} says that
count is the exact dimension.

Two objects built from $\Stot$ recur below.  The \emph{scaled tail}
$R_N=\sum_{m\ge1}\ph(N+m)/2^m$ is $2^N$ times the part of the series beyond
index $N$; since $2^N\Stot=\Phi_N+R_N$ with
$\Phi_N=\sum_{1\le n\le N}\ph(n)2^{N-n}$ an integer
(\mword{Erdos249257/TotientTailPeriodKiller.lean}{150}{two_pow_mul_totient_series_eq}{checked}),
$R_N$ and $2^N\Stot$ differ by an integer.  For $h,N,L\in\N$, put
\[
  D_{h,N,L}
  =
  \sum_{j=0}^{L-1}
  \bigl(\ph(N+h+1+j)-\ph(N+1+j)\bigr)2^{L-1-j},
\]
and let $\rho_{h,N,L}\in\{0,\dots,2^L-1\}$ be its residue modulo $2^L$.
A \emph{finite tail-difference certificate} at $(h,N,L)$ is the pair of
strict inequalities
\[
  N+h+L+2<\rho_{h,N,L}<2^L-(N+h+L+2)
\]
(\mword{Erdos249257/TotientTailPeriodKiller.lean}{72}{certifiedKill}{definition}).
It exists to make a statement about an infinite tail decidable by a finite
computation, and it is faithful: some depth $L$ certifies $(h,N)$ if and only
if $R_{N+h}-R_N\notin\Z$
(\mword{Erdos249257/LcmConeFlatness.lean}{316}{exists_certifiedKill_iff_tail_diff_notMem_int}{checked}).
For example,
\[
  D_{1,12,16}=-143140,\qquad
  \rho_{1,12,16}=53468,\qquad N+h+L+2=31,
\]
so $(1,12,16)$ is a certificate.  More generally, the test succeeds at
$(h,12,16)$ for every $h$ with $1\le h\le8$
(\mword{Erdos249257/TotientTailPeriodKiller.lean}{404}{certifiedKill_all_small}{checked}),
so none of $R_{13}-R_{12},\dots,R_{20}-R_{12}$ is an integer
(\mword{Erdos249257/TotientTailPeriodKiller.lean}{407}{tail_diff_not_int_small}{checked}).

A third recurring object is an \emph{integral scaled-tail sequence with
multiplier $v$}.  For a coefficient sequence $c:\N\to\N$ and an integer
$v\ge1$, this
is an integer sequence $u$ satisfying
\[
  u(N+1)=2u(N)-v\,c(N+1)\quad\hbox{for every }N,
  \qquad
  \frac{u(N)}{2^N}\longrightarrow0
\]
(\mword{Erdos249257/GenericTailOrbitRigidity.lean}{67}{IsTemperedBinaryOrbit}{definition});
for $c=\ph$ such a pair $(v,u)$ with $v\ge1$ exists exactly when $\Stot$ is
rational (Section~\ref{sec:carry-rank}).

{\small
\begin{enumerate}[label=\textbf{R\arabic*.},ref=R\arabic*,leftmargin=2.4em,
  itemsep=0.2em,topsep=0.25em]

\item\label{res:basis} \stag{unconditional progress}
\textbf{Basis theorem.}  The family
$\mathcal B=\{\ph_{0,0},\ph_{1,0}\}\cup\{\ph_{j,r}:j\ge1,\ r\ \text{odd},\
0<r<2^j\}$ is linearly independent over $\Q$ and spans the same $\Q$-subspace
of $\Q^{\N}$ as $\Ker_2(\ph)$; so $\mathcal B$ is a basis of that span, which
is therefore infinite-dimensional.  Since the reductions make every remaining
section a rational multiple of a member of $\mathcal B$, the $\Q$-linear
relations among dyadic sections of $\ph$ are exactly those the reductions
generate --- a one-line consequence of the checked statements, not a further
one.
{\footnotesize\emph{Checked:}
\mword{Erdos249257/TotientMahlerDefect.lean}{1265}{linearIndependent_oddCoreTotientKernelFamily}{independence},
\mword{Erdos249257/TotientMahlerDefect.lean}{1380}{span_range_fullTotientKernel_eq_span_range_oddCore}{span equality},
\mword{Erdos249257/TotientMahlerDefect.lean}{1392}{totientDyadicSectionBasis}{basis}.
\emph{Registry:} \texttt{dyadic\_totient\_basis}.}

\item\label{res:rank} \stag{unconditional progress}
\textbf{Exact finite-level rank.}  For every $e\ge0$ the $2^e+1$ sections
$\ph_{0,0}$, $\ph_{1,0}$ and $\ph_{j,r}$ with $1\le j\le e$, $r$ odd,
$0<r<2^j$ are linearly independent over $\Q$, so their span has dimension
exactly $2^e+1=2+\sum_{j=1}^{e}2^{j-1}$: exact, not an estimate, because the
even residues are already dependent by the reductions.  Read instead as a
truncation of the kernel, $V_e=\operatorname{span}_{\Q}\{\ph_{j,r}:0\le j\le
e\}$ has $\dim V_e=2^e+1$ for every $e\ge1$, and $\dim V_0=1$: the counted
family carries $\ph_{1,0}$, which is a level-one section, so the two readings
agree from $e=1$ on and differ only at $e=0$.  At $e=1$ both readings say that
$\ph(n)$, $\ph(2n)$ and $\ph(2n+1)$ are linearly independent over $\Q$.  The
independence half of
\ref{res:basis} follows by finite character; the span equality needs the
reductions as well.
{\footnotesize\emph{Checked:}
\mword{Erdos249257/TotientMahlerDefect.lean}{935}{linearIndependent_canonicalTotientKernelFamily}{independence},
\mword{Erdos249257/TotientMahlerDefect.lean}{989}{finrank_canonicalTotientKernel_eq}{dimension},
\mword{Erdos249257/TotientMahlerDefect.lean}{1145}{not_finiteDimensional_span_fullTotientKernel}{infinite-dimensionality directly}.}

\item\label{res:denominator} \stag{formalised here}
\textbf{Denominator exclusion, sharp for its window.}  If $\Stot=a/q$ with
$a\in\Z$ and $q\ge1$ then
$q>79\,639\,646\,646\,701\,375\,323\,355\,774\,875\,831\,053\approx7.96\times10^{34}$.
The constant is optimal for the window that produces it --- the window here is
the pair $(N,K)=(1,240)$, meaning $240$ committed binary digits of the series
shifted by one --- in the sense that the next integer up is the exact first
denominator at which that window's certificate fails.  The method is the
classical Farey mediant argument (Section~\ref{sec:denominator}).
{\footnotesize\emph{Checked:}
\mword{Erdos249257/CertificateKernel.lean}{18371}{tsum_totient_div_pow_two_ne_int_div_of_den_le_79639646646701375323355774875831053}{exclusion},
\mword{Erdos249257/CertificateKernel.lean}{18384}{tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053}{reduced-denominator form},
\mword{Erdos249257/GapFareyBound.lean}{225}{gap_check_window_1_240_first_failure}{first failure}.
\emph{Registry:} the row owns the reduced-denominator form only.}

\item\label{res:carryrank} \stag{formalised here}
\textbf{Rank lower bound for a rational scaled-tail sequence.}  If $\Stot$
were rational then some integral scaled-tail sequence $u$ for the coefficients
$\ph$, with positive multiplier, would have for \emph{every} $e$ dyadic
sections through level $e$ --- formed from $u$ exactly as the $\ph_{j,r}$ are
formed from $\ph$ --- spanning a space of dimension at least $2^e-1$.  The
same lower bound holds for every positive-multiplier integral scaled-tail
sequence.  This statement is specific to that
binary scaled-tail representation; it does not constrain arguments formulated
only through the Lambert weight, the coprimality identity, or tail differences.
It is not an irrationality criterion, because no finite-rank upper bound is
proved for the scaled-tail sequences supplied by rationality.
{\footnotesize\emph{Checked:}
\mword{Erdos249257/TotientCarryKernelRigidity.lean}{300}{not_irrational_totientSeries_implies_unbounded_carryRank_unconditional}{the barrier},
\mword{Erdos249257/TotientCarryKernelRigidity.lean}{211}{finrank_canonicalCarryKernel_ge_of_linearIndependent}{rank transport from \ref{res:rank}}.}

\item\label{res:weight} \stag{checked; no registry row}
\textbf{The \#249 weight.}  $\Stot=\sum_{d\ge1}\Aw(d)/(2^d-1)$ with
$\Aw=\ph*\mu$, the Dirichlet convolution
$\Aw(n)=\sum_{d\mid n}\ph(d)\mu(n/d)$; here $\Aw\ge0$, $\Aw(p)=p-2$,
$\Aw(p^k)=\ph(p^k)-\ph(p^{k-1})$, and $\Aw$ is unbounded.  Its values at
$n=1,\dots,10$ are $1,0,1,1,3,0,5,2,4,0$.
{\footnotesize\emph{Checked:}
\mword{Erdos249257/CertificateKernel.lean}{18445}{tsum_primWeight_div_two_pow_sub_one_eq_totient_series}{identity},
\mword{Erdos249257/MersenneLambertLadder.lean}{370}{primWeight_nonneg}{sign},
\mword{Erdos249257/MersenneLambertLadder.lean}{297}{primWeight_apply_prime}{primes},
\mword{Erdos249257/MersenneLambertLadder.lean}{272}{primWeight_apply_prime_pow}{prime powers},
\mword{Erdos249257/MersenneLambertLadder.lean}{321}{primWeight_not_bounded}{unboundedness}.
\emph{Registry:} the Lambert-ladder row owns the neighbouring rungs, not these
five declarations; see the formal source notes below.}

\item\label{res:periodic} \stag{unconditional progress}
\textbf{Eventually periodic weights are settled.}  For every integer base
$b\ge2$, if $\gamma\colon\N\to\Q$ is eventually periodic, nonnegative and
positive at some positive index of its periodic part, then
  $\sum_{a\ge1}\gamma(a)/(b^a-1)$ is irrational.  Theorem~A in Luca and
  Tachiya's 2017 open-access RIMS paper restates their earlier signed theorem in the exact
  purely-periodic integer case: every nonzero purely periodic integer sequence
  gives an irrational Lambert value at every integer base $|q|>1$
  \cite[Theorem~A, p.~139]{lucatachiya2017}.  Their Theorem~1 strengthens the
  nonnegative case to linear independence of each finite divisor-convolution
  ladder \cite[Theorem~1, p.~139; proof pp.~149--150]{lucatachiya2017}.
  Clearing the common denominator of a rational period and subtracting the
  finite rational prefix shows that Theorem~1 already proves the eventual
  nonnegative rational claim above; the checked theorem gives an independent
  formal proof.  By \ref{res:weight} the weight $\Aw$ is
unbounded, hence not eventually periodic, so \#249 is outside this class.
{\footnotesize\emph{Checked:}
\mword{Erdos249257/CertificateKernel.lean}{12811}{irrational_ratWeightSeries_eventuallyPeriodic}{the periodic theorem}.}

\item\label{res:primindex} \stag{proved here}
\textbf{No fixed common denominator clears the normalised weight.}  For every integer
$D\ge1$ some $n\ge1$ has $D\cdot\Aw(n)/n\notin\Z$ --- for $D=6$ at the prime
$n=7$, where $\Aw(7)/7=5/7$ --- and any $D$ clearing every coordinate up to a
horizon $N\ge4$ is divisible by an explicit two-tier primorial ($4$ at $p=2$,
$p^2$ when $p^2\le N$, else $p$).  No argument that clears the coordinates
$\Aw(n)/n$ by a single positive integer valid at all $n$ can therefore
succeed.  This obstruction is caused by the normalisation: the unnormalised
weights $\Aw(n)$ are integers.  Arguments whose common denominator grows with
the horizon, arguments using the integral weights directly, and arguments
that never clear these coordinates are untouched.
{\footnotesize\emph{Checked:}
\mword{Erdos249257/PrimitiveDeterminantLift.lean}{169}{no_fixed_integral_primitive_euler_index}{no fixed index},
\mword{Erdos249257/PrimitiveDeterminantLift.lean}{148}{twoTierPrimorial_dvd_of_integral_jet}{the primorial divisibility}.}

\item\label{res:equivalences} \stag{proved here}
\textbf{Four exact characterisations of irrationality.}  Irrationality
of $\Stot$ is equivalent to each of: non-integrality of the tail difference
$R_{N+h}-R_N$ at every $N$ and every shift $h\ge1$, equivalently a certificate
at every such pair; a certificate at every $h\ge1$ and at arbitrarily large
$N$; and the same at the scales $N=h=\lcm(1,\dots,t)$, for arbitrarily large
$t$.  A fourth equivalence,
independent of the certificate language, is a counted anti-concentration
condition on window phases.  These characterisations identify the unbounded
input exactly; they retain the full difficulty of \#249 and do not supply that
input.
{\footnotesize\emph{Checked:}
\mword{Erdos249257/LcmConeFlatness.lean}{399}{irrational_totient_series_iff_pointwise_certificates}{pointwise},
\mword{Erdos249257/LcmConeFlatness.lean}{412}{irrational_totient_series_iff_certificate_supply}{cofinal at each shift},
\mword{Erdos249257/LcmConeFlatness.lean}{426}{irrational_totient_series_iff_lcm_diagonal_certificate_supply}{$\lcm$ diagonal},
\mword{Erdos249257/PivotAntiReconstruction.lean}{1765}{dtwWindowSeparatedPairs_iff_irrational_totient_series}{window-separated pairs}.}

\item\label{res:supply} \stag{conditional reduction}
\textbf{A denominator-indexed gap-certificate hypothesis.}  Suppose that for
each denominator $q$ there is a truncation window $(N,K)$ whose totient residue
avoids a band of width $q(N{+}K{+}2)$ out of $2^K$.  This hypothesis
\emph{implies} irrationality.  It concerns a second certificate family, not
the one of \ref{res:equivalences}: the band scales with $q$, and the family is
indexed by denominators rather than by shifts.  The converse is not proved and
is not claimed.
{\footnotesize\emph{Checked:}
\mword{Erdos249257/CertificateKernel.lean}{16025}{irrational_tsum_totient_div_pow_two_of_gap_certificate_supply}{the supply implication}.
\emph{Registry:} no row currently owns this declaration.}

\item\label{res:open} \stag{open}
\textbf{Erd\H{o}s~\#249.}  Whether $\Stot$ is irrational.  Not proved here.

\end{enumerate}
}

\section{The Lambert-series form of Problem \#249}\label{sec:family}

If $f=g*\mathbf 1$, that is $f(n)=\sum_{d\mid n}g(d)$, and
$\sum_{d\ge1}|g(d)|/(2^d-1)<\infty$, then absolute convergence justifies
interchanging the two sums and gives
\[
  \sum_{n\ge1}\frac{f(n)}{2^n}
  \;=\;\sum_{d\ge1}\frac{g(d)}{2^d-1}.
  \tag{1}\label{eq:lambert}
\]
whose right-hand side is a Lambert series with coefficients $g$.  Four
classical coefficient sequences have this shape.  The table records the
constant, the weight $g=f*\mu$, and its status.

\begin{table}[h]
\centering
\small
\begin{tabular}{@{}
  >{\raggedright\arraybackslash}p{0.07\linewidth}
  >{\raggedright\arraybackslash}p{0.19\linewidth}
  >{\raggedright\arraybackslash}p{0.21\linewidth}
  >{\raggedright\arraybackslash}p{0.43\linewidth}@{}}
\toprule
$f$ & $\sum f(n)/2^n$ & weight $g=f*\mu$ & status \\
\midrule
$\tau$ & $1.6066951\ldots=\EB$ & $g\equiv1$ & The Erd\H{o}s--Borwein constant;
  irrational~\cite[theorem on p.~63]{erdos1948};
  formalised here in the Lambert form $\sum_{d\ge1}(2^d-1)^{-1}$
  (\mword{Erdos249257/CertificateKernel.lean}{8335}{irrational_erdosBorwein_series}{checked}) \\
$\omega$ & $0.5169428\ldots$ & $g=\mathbf 1_{\text{primes}}$ &
  Irrational at base $2$: Tao--Ter\"av\"ainen, Thm.~1.3, p.~4;
  proof pp.~44--56~\cite{taoteravainen2025} \\
$\Omega$ & $0.5895032\ldots$ & $g=\mathbf 1_{\text{prime powers}}$ &
  Asserted \emph{ibid.}\ by a similar argument, with the details left to the
  reader; not a proved theorem there \\
$\ph$ & $1.3676308\ldots=\Stot$ & $g=\Aw=\ph*\mu$ &
  \textbf{Open} (Erd\H{o}s \#249) \\
\bottomrule
\end{tabular}
\end{table}

\noindent The first three weights are bounded, taking only the values $0$
and~$1$ (for $\tau$ the weight is the constant $1$).  The fourth is unbounded,
by \ref{res:weight}.  That difference is the one that matters for the route in
\ref{res:periodic}, and we do not claim it is the only difference: $\Aw$ also
vanishes on $n\equiv2\pmod4$, while $\mathbf 1_{\text{primes}}$ is bounded but
no more eventually periodic than $\Aw$ is, so the settled $\omega$ row is not
an instance of \ref{res:periodic} either.  It was settled by a different
method, discussed in Section~\ref{sec:open}.

Luca and Tachiya's result is stronger than the single $\tau$ row:
Example~1 makes \(1\) and every finite ladder of generalized-divisor-function
values linearly independent
~\cite[Example~1, p.~140; proof pp.~149--150]{lucatachiya2017}.
Their Example~2 applies the period-two odd-support indicator and proves joint
linear independence of its Lambert value with every finite higher
divisor-convolution ladder, for integer bases \(q\) of either sign with
\(|q|>1\) \cite[Example~2, p.~140]{lucatachiya2017}. Thus the cited theorem
strictly strengthens the isolated odd-support irrationality row.
Historically, Erd\H{o}s already singled out the totient series itself as a
difficult analogue at the end of his 1948 paper~\cite[p.~66]{erdos1948};
that remark supplies lineage, not a modern status proof.

\paragraph{A nearby 2026 totient theorem has different coordinates.}
Kaneko, Suzuki and Tachiya prove that if \(f(n)\) is a nonnegative integer
sequence of infinite support with
\(\sum_{n\le x}f(n)=O(x(\log x)^\delta)\), then, for every integer
\(t\ge2\), both
\[
  \sum_{n\ge1}\frac{f(n)}{t^{\sigma(n)}}
  \qquad\text{and}\qquad
  \sum_{n\ge1}\frac{f(n)}{t^{\ph(n)}}
\]
are irrational
\cite[Corollary~3, pp.~5--6; proof pp.~18--19]{kanekosuzukitachiya2026}.
This includes substantial families of totient-related series, but
\(\ph(n)\) occurs in the \emph{exponent}.  Problem~\#249 instead places
\(\ph(n)\) in the coefficient of \(2^{-n}\).  The theorem therefore supplies
a genuine adjacent result and a useful warning against a tempting
misidentification; it does not settle or reduce the displayed problem.

\paragraph{Exact identities and representations.}
Several neighbouring Dirichlet-convolution identities are rational and
exactly computable.  Writing $L(g):=\sum_{d\ge1}g(d)/(2^d-1)$, we have
$\sum_{d\ge1}\mu(d)/(2^d-1)=\tfrac12$
(\mword{Erdos249257/CertificateKernel.lean}{18434}{tsum_moebius_div_two_pow_sub_one_eq_half}{checked})
and $\sum_{d\ge1}\ph(d)/(2^d-1)=2$
(\mword{Erdos249257/CertificateKernel.lean}{18429}{tsum_totient_div_two_pow_sub_one_eq_two}{checked}).
These belong to two distinct one-step M\"obius-convolution chains:
\[
  L(\mathbf 1)=\EB \ \longmapsto\ L(\mu)=\tfrac12,
  \qquad
  L(\ph)=2 \ \longmapsto\ L(\ph*\mu)=\Stot.
\]
Thus $\Stot$ is one convolution step from the rational value $L(\ph)=2$, not
from $\EB$; the parallel chains show that this transformation need not
preserve rationality.  The squared-denominator representation
$\Stot=\tfrac12+\sum_{d\ge1}\mu(d)/(2^d-1)^2$
(\mword{Erdos249257/CertificateKernel.lean}{18454}{totient_series_eq_half_add_moebius_mersenne_square}{checked})
converges fast enough to recompute the decimal above.

There is also an exact probabilistic reading, and it is more than a gloss.
Let $X$ and $Y$ be independent fair-coin geometric waiting times on $\Npos$.
Then
\[
  \Stot=\tfrac12+\Pr[\gcd(X,Y)=1]
\]
(\mword{Erdos249257/CertificateKernel.lean}{18557}{totient_series_eq_half_add_visible_coprime_pairs}{checked}),
the probability being the visible-coprime-pair series
$\sum_{\gcd(m,n)=1}2^{-(m+n)}$.  That probability is related to a distribution
over the Stern--Brocot tree in a completely explicit way.  For positive
coprime $(a,b)$ put $w_{a,b}=1/(2^{a+b}-1)$.  These reduced-slope masses sum to
exactly~$1$
(\mword{Erdos249257/GcdMomentCalculus.lean}{349}{tsum_pos_coprime_inv_mersenne_eq_one}{checked}),
and at each node $(a,b)$ the cylinder mass $1/((2^a-1)(2^b-1))$ splits exactly
into its stop mass and the masses of its two children
(\mword{Erdos249257/GcdMomentCalculus.lean}{474}{cylinderMass_split}{checked}).
The coprimality probability is not that total mass; rather,
\[
  \Stot-\tfrac12
  =\sum_{\substack{a,b\ge1\\(a,b)=1}}2^{-(a+b)}
  =\sum_{\substack{a,b\ge1\\(a,b)=1}}
     \bigl(1-2^{-(a+b)}\bigr)w_{a,b}.
\]
It is therefore the expectation of the conditional primitive-pair factor
$1-2^{-(a+b)}$ under the self-similar reduced-slope probability law.  What is
missing is not structure; it is an arithmetic consequence of the structure.

\section{Proof of the dyadic-section basis theorem}\label{sec:rank}

\subsection{Prior work}

For an integer $k\ge2$, the \emph{$k$-kernel} of a sequence $c$ is
$\{n\mapsto c(k^jn+r):j\ge0,\ 0\le r<k^j\}$, and $c$ is \emph{$k$-regular}
when the module generated by its $k$-kernel is finitely generated; both
notions are due to Allouche and Shallit
(\cite[Defs.~1.1 and 2.1, author-preprint pp.~2--3]{allouche-shallit}).
Their Theorem~2.2 gives equivalent finite-kernel and matrix
characterisations, and Theorem~3.1 proves closure under convolution
~\cite[pp.~3--4 and 10--11]{allouche-shallit}; neither result controls the
finite-level rank of a sequence that is not $k$-regular.  Coons proved that
$\ph$ is not $k$-regular for any $k\ge2$ (\cite{coons}, Theorem~3.2,
pp.~348--349, in the published version; Theorem~3.3, pp.~8--9, in the
preprint, which numbers its results on a single running counter).  Bell and
Smertnig reach a further negative
conclusion, and by a different route rather than by strengthening that one:
their Theorem~1.3 shows that a $k$-Mahler series with multiplicative
coefficients has a $k$-regular coefficient sequence in an explicit closed form,
and that the totient generating series is not $k$-Mahler for any $k\ge2$
follows directly from the closed form~\cite{bell-smertnig}.

A separate line of work settles linear independence of totient values along
affine progressions, and settles it more strongly than anything proved here.
Martin~\cite{martin-phi-inequalities} assumes only that $a_1,\dots,a_m$ are
positive integers and that $a_ib_j\neq a_jb_i$ for $i\neq j$, and proves in his
Theorem~1 that for every constant $C>0$ the simultaneous ratio gaps
\[
 \frac{\ph(a_1n+b_1)}{\ph(a_2n+b_2)}>C,\quad\dots,\quad
 \frac{\ph(a_{m-1}n+b_{m-1})}{\ph(a_mn+b_m)}>C
\]
hold on a set of positive lower density; he also notes that the hypotheses are
symmetric in the forms, so every one of the $m!$ orderings occurs on such a set.
Taking $C>\bigl(\sum_{i\neq t}|c_i|\bigr)/|c_t|$ after moving a channel with
$c_t\neq0$ to the front shows at once that
\[
 \bigl\{n\mapsto\ph(a_in+b_i)\bigr\}_{i=1}^m
\]
is linearly independent over $\R$, hence over $\Q$.  His Corollary~4 carries
Theorem~1 and its corollaries to $\sigma$.  Every affine-totient independence
statement used below is therefore a consequence of Martin's theorem, and no
originality is claimed for it.  Martin does not state the exact finite-level
rank, the explicit basis, or the complete relation normal form; the present
section derives them by combining his theorem with the reductions below.
Coons and Bell--Smertnig likewise do not state those finite-level conclusions.

Coons's statement is about the union over all levels: no finite set of sections
generates the rest.  Bell and Smertnig's is about a functional equation.

One consequence of Coons's theorem should be recorded explicitly, because it
bounds what the present section may claim.  For a subgroup $M\subseteq\Z^{\N}$
of integer-valued sequences, finite generation over $\Z$ is \emph{equivalent} to
finite dimension of its $\Q$-span.  One direction is immediate.  For the other,
choose $\Q$-independent $m_1,\dots,m_d\in M$ and evaluation points
$n_1,\dots,n_d$ with $D:=\det[m_i(n_j)]\neq0$, necessarily a nonzero integer;
for $x\in M$ Cramer's rule applied to the integer system
$x(n_j)=\sum_i a_im_i(n_j)$ places every $a_i$ in $\tfrac1D\Z$, so
$M\subseteq\tfrac1D\bigl(\Z m_1+\dots+\Z m_d\bigr)$, and over the principal
ideal domain $\Z$ a submodule of a finitely generated module is finitely
generated.  (The familiar obstruction, that $\Z[\tfrac12]$ has rank one without
being finitely generated, cannot occur here: an integer-valued sequence bounds
its own denominators through its values at small indices.)

Hence \emph{infinite-dimensionality of the full totient kernel is not an
independent contribution of this release}: it is equivalent to Coons's
non-$k$-regularity and follows from it.  Only the finite-level statements are
additional.  Knowing that a span is not finitely generated says nothing
about its dimension at a given truncation, nor about which relations hold
there; what is added here is that
the dimension through level $e$ is exactly $2^e+1$, that an explicit family is
a basis, and that the two elementary reductions generate every relation.  We
found no prior source giving any of those three for $\ph$; that is the outcome
of a search, not a proof of non-existence, and no priority is claimed.

The evaluation-matrix argument of the next subsection is an independent and
more constructive route to the same independence, which is why it is retained.

\subsection{Parity-separated evaluation matrices}

\ref{res:rank} is proved by exhibiting a square matrix of values of the
sections whose determinant is nonzero, and the linear algebra at its centre is
simpler than the arithmetic construction that feeds it.  We give the linear
algebra first, then the arithmetic input it consumes.

Evaluate the sections at one point apiece to get a square matrix $M$, and
suppose each column $j$ admits a fixed power $2^{d_j}$ dividing every entry of
that column such that, after dividing the column through by it, the diagonal
entry is odd and every off-diagonal entry is even; call such an $M$ a
\emph{parity-separated evaluation matrix}.  The normalised matrix is then the
identity modulo $2$; its determinant is odd, hence nonzero, and the sections are
independent
(\mword{Erdos249257/TotientMahlerDefect.lean}{823}{paritySeparatedMatrix_det_ne_zero}{checked}).
Producing evaluation points with exactly that parity pattern is what the
arithmetic input is for, and it is the harder half.  The Chinese remainder
theorem together with Dirichlet's theorem on primes in arithmetic progressions
makes one of the evaluated affine values prime, which fixes the exact power of
$2$ dividing the diagonal entry, while every other value receives a fresh
prime divisor congruent to $1$ modulo a large power of two, which supplies the
one extra factor of two that the off-diagonal entries need
(\mword{Erdos249257/TotientMahlerDefect.lean}{882}{exists_separatedMinorCertificate_totientAffineOddFamily}{checked}).
Even residues never enter, because the second reduction has already carried
them to odd residues at lower levels
(\mword{Erdos249257/TotientMahlerDefect.lean}{169}{totientKernel_even_residue_reduce}{checked}).

\ref{res:basis} then needs only two further steps, and both are short.  Linear
independence is a property of finite subfamilies, so the level-$e$ statement,
applied with $e$ above the largest level occurring in a given finite subset,
already gives independence of the whole infinite family $\mathcal B$.  And
the two reductions carry every remaining section onto a rational multiple of a
member of $\mathcal B$, so the spans agree.  That the assembly is short is the
point: the level-$e$ theorem was always the whole content, and stating its
consequence as infinite-dimensionality understated it.  The sharp form is a
basis.

\begin{remark}
\ref{res:basis} and \ref{res:rank} are statements about $\ph$, not about
$\Stot$; they are true, and their proofs are complete, whether or not $\Stot$
is irrational.  Their connection to \#249 is not thematic but a single proved
transport, and it is \ref{res:carryrank}.
\end{remark}

\section{Carry-rank consequences of rationality}\label{sec:carry-rank}

Multiplying successive scaled tails by $2$ turns a hypothetical rational
value into an integral recurrence.  For coefficients $c:\N\to\N$ satisfying
$c(n)\le n$, and in particular for $c=\ph$ since $\ph(n)\le n$, the series
$\sum c(n)/2^n$ is
rational exactly when an integral scaled-tail sequence exists, that is, an
integer sequence $u$ with
$u(N+1)=2u(N)-v\,c(N+1)$ for every $N$ and $u(N)/2^N\to0$, for some integer
$v\ge1$
(\mword{Erdos249257/GenericTailOrbitRigidity.lean}{426}{binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit}{checked}).
Transporting \ref{res:rank} through that recurrence gives \ref{res:carryrank}.
The complementary finite-rank upper bound, which would make this a proof of
irrationality, is not available.

\section{A Farey-mediant denominator exclusion}\label{sec:denominator}

\ref{res:denominator} is a Farey exclusion, and we say so before saying
anything else about it.  The argument is the classical mediant one.  The Lean
lemma commits the first $240$ binary digits of the shifted series as a single
residue over the totient carry window $(N,K)=(1,240)$; a rational $a/q$ can
equal $\Stot$ only if it falls into a resulting bad interval of width
$243/2^{240}$; that interval is bracketed by two explicit unimodular fractions
$a_1/b<c_1/d$ with $bc_1-a_1d=1$; and the mediant lemma
(\mword{Erdos249257/GapFareyBound.lean}{51}{farey_gap}{checked}) forces any
rational strictly between unimodular neighbours to have denominator at least
$b+d$.  The excluded range is therefore $q\le b+d-1$, which is the printed
constant
(\mword{Erdos249257/GapFareyBound.lean}{176}{gap_check_window_1_240_le_79639646646701375323355774875831053}{checked}),
and the transport to the series is a tail estimate
(\mword{Erdos249257/CertificateKernel.lean}{15986}{tsum_totient_div_pow_two_ne_int_div_of_totient_gap_certificate}{checked}).
\emph{The method is standard, and we claim no novelty for it.}

Two things should be read off correctly.  First, the scale.  A window of $K$
computed binary digits yields a mediant bound of order $2^{K/2}$: the same
pipeline gives $2.49\times10^{17}$ at $K=120$
(\mword{Erdos249257/GapFareyBound.lean}{88}{gap_check_window_1_120_le_248672326362367909}{checked})
and $7.96\times10^{34}$ at $K=240$.  The constant is a function of how many
digits were committed, not a measure of how much the problem has moved.
Second, the sharpness.  Within its window the bound cannot be improved at all:
$b+d$ itself fails the certificate, so $b+d-1$ is exactly the last denominator
excluded.  A reader who regards an explicit Farey exclusion as routine is not
in disagreement with this note.  \ref{res:basis}, not \ref{res:denominator},
is where we would ask a sceptical reader to look.

The bound transfers to the coprimality form of Section~\ref{sec:family} with
the constant halved: the visible-coprime-pair series has no representation
$a/d$ with $d\le39\,819\,823\,323\,350\,687\,661\,677\,887\,437\,915\,526$
(\mword{Erdos249257/CertificateKernel.lean}{18572}{tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526}{checked}),
that number being $(q_0-1)/2$ for the constant $q_0$ of
\ref{res:denominator}.

\section{Obstructions for fixed-coordinate methods}\label{sec:nogo}

\ref{res:primindex} has a one-line proof.  Take a prime
$p>\max(D,2)$; then $\Aw(p)=p-2$ by \ref{res:weight}, and $p\nmid D(p-2)$, so
$D\cdot\Aw(p)/p\notin\Z$.

Two further checked results delimit narrower abstract hypothesis classes.

A fixed-precision transport result
(\mword{Erdos249257/TropicalCurvatureCarry.lean}{137}{fixedPrecisionTropicalNoGo}{checked})
says that at every fixed positive precision, each finite compatible
valuation-unit word admits a prefix-locked centred completion.  Its statement
quantifies over abstract pairs of a two-adic valuation and an odd unit; it
mentions neither $\ph$ nor $\Stot$, its content is that a congruence class
meets any interval of the corresponding length, and it is a restatement of a
lemma printed immediately above it.  The class of arguments it closes ---
deriving a contradiction from a local signature read at a precision fixed in
advance, along a finite word --- is not one anybody has proposed for \#249.

A factor-ideal result
(\mword{Erdos249257/LcmFactorIdealPulseObstruction.lean}{798}{lcm_factorIdeal_finiteRank_shiftAlgebra_not_sufficient}{checked},
with a sparse-anchor companion at
\mword{Erdos249257/LcmFactorIdealPulseObstruction.lean}{866}{lcm_factorIdeal_sparseAnchor_not_sufficient}{checked})
exhibits, for each $t\ge3$ with $H=\lcm(1,\dots,t)$, a nonzero integer carry
whose forcing letters lie in the ideal generated by $\ph(H)$, which reproduces
the true totient differences at $t-2$ prescribed indices, and which every
finite integer shift polynomial carries to a pair with the same coboundary
form, the same ideal memberships and an $\ell^1$-weight bound.  It does close
that hypothesis class.  Two limits.  The witness is a spike --- its state is
$-\ph(H)$ at $t-2$ points and zero elsewhere, so it stays uniformly bounded
while the strip bounds it respects grow, and the construction is cheap for
that reason.  And the shifted pairs are not asserted to be nonzero, so an
argument that additionally demanded non-vanishing after the shift is not
excluded.  The theorem contains no whole-ray anchor condition, diagonal bound,
or strict-survivor condition.

\section{Further conditional criteria}\label{sec:programmes}

The criteria below do not enter the proofs of the unconditional results.
Each states its unproved hypotheses explicitly.

\paragraph{A prime-orbit sufficient condition.}
The hypothesis
\lword{Erdos249/TotientStrictPrimeEscape.lean}{157}{DTWNaturalPrimeTailOrbitStrictGap}{asks}
for cofinally many primes $p$ at which the first tail-orbit exponential has
real part strictly below $9/10$.  Granting it,
\lword{Erdos249/TotientStrictPrimeEscape.lean}{166}{naturalPivotPointEscape_of_naturalPrimeTailOrbitStrictGap}{a positive adaptive truncation budget}
follows, and then
\lword{Erdos249/TotientStrictPrimeEscape.lean}{205}{irrational_totient_series_of_naturalPrimeTailOrbitStrictGap}{$\Stot$ is irrational}.
The $9/10$ hypothesis is weaker than the earlier $4/5$ hypothesis, but no
instance of this cofinal hypothesis is proved.
Section~\ref{sec:frontier} states the sharper form
in which this hypothesis is now available.

\paragraph{Finite Euler-factor identities.}
The local coefficients are
\[
  \eta(0)=1,\qquad \eta(1)=-2,\qquad \eta(2)=1,\qquad
  \eta(e)=0\quad(e\ge3),
\]
the coefficients of $(1-X)^2$.  For a prime $p$ and $e\ge0$, write
$\sigma_p(e)=1+p+\cdots+p^e$.  The two local Euler factors are
\[
  1-\frac2p+\frac1{p^2}=\left(1-\frac1p\right)^2,
  \qquad
  1-\frac2{p^2}+\frac1{p^4}
      =\left(1-\frac1{p^2}\right)^2
\]
at
\lword{Erdos249/FiniteEulerSieve.lean}{28}{muSqEulerFactor_one}{$s=1$}
and
\lword{Erdos249/FiniteEulerSieve.lean}{36}{muSqEulerFactor_two}{$s=2$}.
On the prime-power divisor-sum row,
\[
  \sigma_p(1)-2\sigma_p(0)=p-1
\]
(\lword{Erdos249/FiniteEulerSieve.lean}{44}{sigmaPrimePow_firstDiff}{first difference}),
and for every $e\ge0$,
\[
  \sigma_p(e+2)-2\sigma_p(e+1)+\sigma_p(e)
    =p^{e+1}(p-1)
\]
(\lword{Erdos249/FiniteEulerSieve.lean}{51}{sigmaPrimePow_secondDiff}{second difference}).
For example, at $p=3$ the values
$\sigma_3(0),\ldots,\sigma_3(3)=1,4,13,40$ give
$4-2=2$ and $40-2\cdot13+4=18=3^2(3-1)$.  Thus convolution with
$\mu\ast\mu$ converts the prime-power divisor-sum row into the corresponding
totient row at every finite stage.  These are finite algebraic identities; no
transcendence conclusion is attached to them.

\paragraph{Mixed differences of cyclotomic layers.}
For a function $F:\{0,1\}^2\to\Z$, define
\[
  \Delta F=F(1,1)-F(1,0)-F(0,1)+F(0,0)
\]
(\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{22}{checkerboard}{mixed difference}).
If $F(i,j)=u(i)+v(j)$, then $\Delta F=0$
(\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{26}{checkerboard_separable}{separable case}).
Conversely, among integer linear combinations of the four values of $F$ whose
coefficients sum to zero along each row and each column, the pattern
$(1,-1,-1,1)$ is forced up to scale
(\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{32}{checkerboard_unique}{fixed-stencil uniqueness}).
For example, the table
\[
\begin{array}{c|cc}
 &0&1\\ \hline
0&9&13\\
1&12&16
\end{array}
\]
has mixed difference $16-12-13+9=0$; increasing only the lower-right entry by
$5$ changes the mixed difference to $5$.  The uniqueness statement is
internal to this fixed $2\times2$ stencil.  It does not assert invariance under
larger stencils, nonlinear functionals, or another representation.

The formal development also records
\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{45}{fourPoint_layer_identity}{a four-point divisor-layer identity}
and
\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{62}{residueClass_step_has_centred_completion}{a centred representative for a residue class modulo an even modulus}.
The two unproved hypotheses and their target conclusion are as follows.  For
$C:\N\to\N$ and $m,d\in\N$, put
\[
\begin{aligned}
\operatorname{Layer}(C,m)\quad\Longleftrightarrow\quad&
\exists Q_0\ \forall q\ge Q_0,\quad q\ {\rm prime}\Longrightarrow\\[-2pt]
&\hspace{5em}1<C(mq)\ \hbox{ and }\ \gcd(C(mq),mq)=1.
\end{aligned}
\]
\[
\begin{aligned}
\operatorname{BoundedOrder}(C,m,d)\quad\Longleftrightarrow\quad&
\forall q,p,\quad q,p\ {\rm prime},\ p\mid C(mq)\Longrightarrow\\[-2pt]
&\hspace{3em}\exists k,\quad 1\le k\le d\ \hbox{ and }\ mq\mid p^k-1.
\end{aligned}
\]
\[
\begin{aligned}
\operatorname{FinitePrimeEscape}(C,m)\quad\Longleftrightarrow\quad&
\forall\text{ finite sets }S\text{ of primes},\ \exists Q_0\
\ \forall q\ge Q_0,\\[-2pt]
&\hspace{3em}q\ {\rm prime}\Longrightarrow
\forall p\in S,\quad p\nmid C(mq).
\end{aligned}
\]
These are the linked predicates
\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{95}{PrimeRayLayerSupply}{Layer},
\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{102}{BoundedDegreeOrderConsumer}{BoundedOrder},
and
\lword{Erdos249/PrimeRayCyclotomicCurvature.lean}{142}{FinitePrimeSupportEscape}{FinitePrimeEscape}.
The bounded-order condition does not assert an exact multiplicative order; it
asserts only the displayed divisibility for some $k\le d$.

The implication from bounded order to finite-prime escape is a size argument
in the paper.  If $m=0$, the bounded-order condition forces $C(0)=1$.  If
$m>0$, the assertion is immediate for $S=\varnothing$; otherwise choose
$Q_0$ so that $mQ_0>\max_{p\in S}(p^d-1)$.  For every prime
$q\ge Q_0$, a divisor $p\in S$ of $C(mq)$ would give
$mq\mid p^k-1$ for some $k\le d$, hence
$mq\le p^k-1\le p^d-1$, a contradiction.  The layer hypothesis separately
ensures that the layers are nontrivial and coprime to their indices.  Neither
hypothesis is proved here; polynomial-resultant realisability and
Archimedean growth are also not established.

\section{Open problems}\label{sec:frontier}

Write $H_t=\lcm(1,\dots,t)$
(\mword{Erdos249257/CarrySurvivorExtinction.lean}{515}{periodLcm}{definition}),
so that $H_1=1$, $H_2=2$, $H_3=6$, $H_4=12$, $H_5=H_6=60$ and $H_7=420$; and
recall from Section~\ref{sec:results} the tail
$R_N=\sum_{m\ge1}\ph(N+m)/2^{m}$, which differs from $2^N\Stot$ by an integer
(\mword{Erdos249257/TotientTailPeriodKiller.lean}{57}{totientTail}{definition}).

\subsection{Exact equivalent formulations}

\begin{equivform}[lcm-diagonal escape]\label{prob:diagonal}
For every $t_0$ there is a $t\ge t_0$ with $R_{2H_t}-R_{H_t}\notin\Z$.
\end{equivform}

\noindent By \ref{res:equivalences}, formulation~\ref{prob:diagonal} is
equivalent to irrationality of $\Stot$.  It identifies an exact decidable
witness family, but is not claimed to reduce the difficulty of Erd\H{o}s
\#249.  A finite list of successful certificates establishes the predicate
only on its tested scales.  Conversely, one pair $h>0$, $N$ with
$R_{N+h}-R_N\in\Z$ already makes $\Stot$ rational
(\mword{Erdos249257/LcmConeFlatness.lean}{357}{not_irrational_of_tail_diff_mem_int}{checked}).

The certificate is decidable at each $t$, and is kernel-checked at the $28$
listed scales between $t=1$ and $t=64$
(\mword{Erdos249257/DiagonalPincerCertificatesT64.lean}{1933}{diagonalPincerCertificateScalesThroughT64}{list},
\mword{Erdos249257/DiagonalPincerCertificatesT64.lean}{1967}{certifiedKill_diagonal_all_imported_through_t64}{verification}).
The current pinned source strengthens that historical list to every $t\le82$,
with no gaps
(\lword{Skip/LadderT67.lean}{71264}{exists_diagonalKill_le_82}{checked}).
The theorem through $t=82$ supplies no instance at $t=83$ and does not
discharge the cofinal quantifier.

The cyclotomic form exposes more arithmetic structure.  Write
$C(n)=|\Phi_n(2)|$.

\begin{equivform}[cyclotomic-anchored phase escape]\label{prob:cyclotomic}
For every $h\ge1$ and every $N_0$, there exist primes $q,p$ and $L\in\N$ such
that
\[
 \gcd(p,hq)=1,\qquad p\mid C(hq),\qquad hq\mid p-1,
 \qquad p-1\ge N_0,
\]
and $(hq,p-1,L)$ is a finite tail-difference certificate.
\end{equivform}

\noindent This is
\lword{Erdos249/CyclotomicAnchoredKill.lean}{3042}{CyclotomicAnchoredKillSupply}{CyclotomicAnchoredKillSupply}.
For the binary layers, clean anchors satisfying every condition before the
certificate are already supplied, and
\lword{Erdos249/CyclotomicAnchoredKill.lean}{3138}{binaryCyclotomicAnchoredKillSupply_iff_irrational}{the checked equivalence}
identifies formulation~\ref{prob:cyclotomic} with irrationality of $\Stot$.
The unknown is phase escape of the totient discrepancy modulo $2^L$, not
large prime divisors or multiplicative order.  The checked $p=331$ instances
are finite models of this exact predicate; unbounded support alone does not
imply it.

\subsection{Stronger sufficient producer problems}

\begin{problem}[full-depth period-multiple escape]\label{prob:fulldepth}
For every $d\ge1$ and every $N\in\N$, is there a $t\ge1$ such that
$(td,N,td)$ is a finite tail-difference certificate?
\end{problem}

\noindent This is the depth-locked predicate
\lword{Erdos249/PeriodMultipleEscape.lean}{162}{ApFullDepthEscape}{ApFullDepthEscape}.
It is stronger than the exact period-multiple characterisation because the
certificate depth is forced to equal the period.  A positive answer proves
irrationality through
\lword{Erdos249/PeriodMultipleEscape.lean}{171}{irrational_totient_series_of_apFullDepthEscape}{the checked consumer};
a negative answer eliminates this route only.

\begin{problem}[first-harmonic anti-concentration on supplier fibres]
\label{prob:firstharmonic}
For each $h\ge1$, do there exist $s\ge1$ and $0<\eta<1$ such that, for every
$X_0$, there are $X,L$ with
\[
 \max(X_0,1)\le X,\qquad h\le L-s,\qquad
 16(2X+h+L+2)\le2^L,
\]
for which the four bounds below hold?
\end{problem}

\noindent This four-term problem is a producer-level decomposition beneath
the direct fixed-full-block first-harmonic gap, not the external theorem
interface itself.  The constitutional outward question is the full-block
constant-saving estimate for the first additive character of the totient
window discrepancy; the supplier-fibre split is one exact way to try to
produce that estimate.  Failure of this decomposition would retire only this
producer, not the full-block socket and not Erd\H{o}s Problem~\#249.

To make the question self-contained, put
\[
 \omega_N=\exp\!\left(2\pi i\,
   \frac{D_{h,N,L}\bmod 2^L}{2^L}\right),\qquad X\le N<2X.
\]
At the pivot argument $N+L-s+1$, call $N$ a supplier when this integer factors
as $mp$, where $p$ is its largest prime factor,
$0<m\le\sqrt X/2$, and $p>2\sqrt X$.  Suppliers are good when
$\eta m\le\ph(m)$ and bad otherwise; non-suppliers form the remaining set.
Writing $t=L-s+1$, the canonical fibre with cofactor $m$ is exactly the image
of the primes in
\[
 X+t\le mp<2X+t
\]
under the injective map $p\mapsto mp-t$
(\mword{Erdos249257/FirstHarmonicPivot.lean}{299}{mem_pivotFiber_iff_exists_supplierPrime}{membership},
\mword{Erdos249257/FirstHarmonicPivot.lean}{367}{pivotBaseOfPrime_injective_on_supplierPrimes}{injectivity},
\mword{Erdos249257/FirstHarmonicPivot.lean}{384}{image_pivotSupplierPrimes_eq_pivotFiber}{image equality}).
This exact bijection does not imply that the supplier prime is isolated from
every other argument in the window.  At $X=16$, $L=20$, $s=1$, $m=2$, and
$N=18$, the pivot is $38=2\cdot19$, while the same prime $19$ divides the
distinct argument $N+1=19$
(\mword{Erdos249257/FirstHarmonicPivot.lean}{394}{supplierPrime_not_globally_isolated_counterexample}{checked counterexample}).
For a supplier write $z_N$ for its explicit totient pivot phase,
$w_N=\omega_N/z_N$, and $\bar z_m$ for the mean of $z_N$ on the fibre with
cofactor $m$.  Define
\[
\begin{aligned}
 \mathcal C&=\sum_{N\ {\mathrm{good}}}w_N(z_N-\bar z_m),&
 \mathcal M&=\sum_{N\ {\mathrm{good}}}w_N\bar z_m,\\
 \mathcal B&=\sum_{N\ {\mathrm{bad}}}\omega_N,&
 \mathcal U&=\sum_{N\ {\mathrm{nonsupplier}}}\omega_N.
\end{aligned}
\]
The exact identity is
$\sum_{X\le N<2X}\omega_N=\mathcal C+\mathcal M+\mathcal B+\mathcal U$, and
Problem~\ref{prob:firstharmonic} asks precisely for
\[
 \operatorname{Re}\mathcal C\le\frac{14}{25}X,\qquad
 |\mathcal M|\le\frac1{100}X,\qquad
 |\mathcal B|\le\frac1{100}X,\qquad
 |\mathcal U|\le\frac8{25}X.
\]
The decomposition is
\mword{Erdos249257/FirstHarmonicPivot.lean}{514}{windowFirstExp_sum_eq_pivot_decomposition}{checked},
and the four budgets imply irrationality by
\mword{Erdos249257/FirstHarmonicPivot.lean}{577}{irrational_totient_series_of_pivotResidualDecorrelation}{the checked consumer}.
An obstruction showing that available first- and second-moment information
cannot force this modulo-$2^L$ small-ball estimate would also decisively close
the present harmonic route.

\subsection{Structural and Diophantine routes}

\begin{problem}[totient carry-rank compression]\label{prob:carryrank}
Suppose $v\ge1$ and $u:\N\to\Z$ satisfy
\[
 u(N+1)=2u(N)-v\ph(N+1),\qquad \frac{u(N)}{2^N}\longrightarrow0.
\]
Must the span of the level-$e$ dyadic sections of $u$ have dimension less
than $2^e-1$ for some $e$?
\end{problem}

\noindent Rationality of $\Stot$ would produce exactly such an integral
tempered orbit, while the checked transport theorem forces its rank to be at
least $2^e-1$ at every level.  Thus a positive answer proves irrationality.
A negative answer should ideally construct a control recurrence with the same
integrality and decay architecture and near-maximal dyadic rank; infinite
dimension of the totient $2$-kernel alone is not enough.

For a separate denominator-compression route, define the M\"obius--Mersenne
ladder
\[
 \Theta_r=\sum_{d\ge1}\frac{\mu(d)}{(2^d-1)^r},
 \qquad \Theta_2=\Stot-\frac12.
\]
The positive rank-one family has already been excluded at a quantitative
scale: every admissible monomial quotient, and every positive finite average
of them, exceeds $\Theta_2$ by more than $1/480$
(\lword{Erdos249/RankOneSubrankObstruction.lean}{238}{rankOneSubrankQuotient_sub_theta_two_gt}{rank-one gap},
\lword{Erdos249/RankOneSubrankObstruction.lean}{300}{positive_direct_sum_sub_theta_two_gt}{positive-sum gap}).

\begin{problem}[genuinely coupled denominator compression]\label{prob:hankel}
Construct integers $P_n$ and $Q_n>0$ from a genuinely off-diagonal,
vector-valued, or signed-cancellation Hankel--Schur--Pad\'e kernel such that
\[
 0<Q_n\left|\Theta_2-\frac{P_n}{Q_n}\right|\longrightarrow0.
\]
Alternatively, prove a structural no-go theorem for a clearly defined larger
class of bounded-width coupled kernels.
\end{problem}

\noindent Ordinary convergence of rational approximants is insufficient; the
scaled error is the Diophantine quantity.  The words ``genuinely coupled''
exclude the positive rank-one and positive direct-sum mechanisms already
separated from $\Theta_2$ by the uniform gap.

\subsection{Independent extension}

This subsection previously posed the odd-prime kernel dimension as an open
problem.  It is now a theorem, and for every integer base rather than for odd
primes only.  The statement below combines Lean-checked arithmetic, spanning,
and conditional-rank layers with a paper-level independence argument.  Lean proves the zero-residue
relation
(\mword{Erdos249257/TotientKernelReduction.lean}{60}{totient_pow_mul_eq}{checked})
and the exact division-free composite-base reduction
(\mword{Erdos249257/TotientKernelReduction.lean}{117}{totient_pow_mul_affine_gcd_cross_eq}{checked}).
It also proves unique quotient/nonzero-digit coordinates for each admissible
fixed-level residue and that the resulting finite index has cardinality
$k^e+1$.  The new conditional layer proves that the canonical family spans
the complete truncation and that its linear independence implies exact rank
$k^e+1$.  These are respectively
\texttt{span\_allBaseTotientKernelThroughLevelFamily\_eq\_canonical} and
\texttt{finrank\_allBaseTotientKernelThroughLevelFamily\_eq\_of\_linearIndependent}
in \texttt{Erdos249257/TotientKernelConditional.lean}; their immutable source
links are added after the module lands in the paper's pinned snapshot.
The antecedent of the second statement is explicit and is not discharged in
Lean.  The paper obtains it from Martin's external positive-density theorem
through the elementary linear-independence consequence recorded above.

\begin{theorem}[exact rank of every totient $k$-kernel truncation]
\label{thm:kkernelrank}
Let $k\ge2$ and $e\ge1$ be integers, write $F^{(k)}_{j,r}(n)=\ph(k^jn+r)$, and put
\[
 V_{k,e}=\operatorname{span}_{\Q}
 \{\,F^{(k)}_{j,r}:0\le j\le e,\ 0\le r<k^j\,\}.
\]
Then $\dim_{\Q}V_{k,e}=k^e+1$, and
\[
 \mathcal B_{k,e}
 =\{F^{(k)}_{0,0},F^{(k)}_{1,0}\}
 \cup\{\,F^{(k)}_{j,r}:1\le j\le e,\ 1\le r<k^j,\ k\nmid r\,\}
\]
is a basis.  Every omitted section reduces to a basis element by an explicit
scalar: $F^{(k)}_{j,0}=k^{j-1}F^{(k)}_{1,0}$ for $j\ge1$, and if $r=k^tu$ with
$t=\max\{s:k^s\mid r\}\ge1$ and $k\nmid u$, then
\[
 F^{(k)}_{j,r}=C_k(t,u)\,F^{(k)}_{j-t,u},
 \qquad
 C_k(t,u)=k^t\prod_{\substack{p\mid k\\ p\nmid u}}\Bigl(1-\tfrac1p\Bigr).
\]
\end{theorem}

The checked composite-base statement is deliberately cross-multiplied by
$\ph(\gcd(k,u))$; the displayed scalar is the equivalent paper-level
normalisation after exact division.  Lean also checks that this scalar is
nonzero and gives the exact one-step functional reduction
\texttt{allBaseTotientKernelSeq\_mul\_residue\_step} in
\texttt{Erdos249257/TotientKernelConditional.lean}.
Thus the zero channel, arithmetic reduction, finite-index/cardinality,
spanning, and conditional-rank layers have kernel receipts.  Martin remains
the external authority for the linear-independence input and hence for the
unconditional basis and rank conclusion of Theorem~\ref{thm:kkernelrank}.

The selection condition is $k\nmid r$, not $\gcd(k,r)=1$; for composite $k$ a
basis residue may share prime factors with $k$.  Independence is obtained by
restricting to $n=km+1$, where the surviving channels become the affine forms
$L_0(m)=km+1$ and $L_{j,r}(m)=k^{j+1}m+(k^j+r)$ with $k\nmid r$.  These have
positive slopes and satisfy $a_ib_j\neq a_jb_i$, so Martin's
Theorem~1~\cite{martin-phi-inequalities} applies through the linear-independence
consequence recorded in the prior-work subsection.  The two zero-residue channels
are proportional on that progression and are separated afterwards by evaluating
at $n=k$, using $\ph(k^2)=k\ph(k)$ and $\ph(k)<k$.

At $k=2$ this recovers the dyadic rank theorem of Section~\ref{sec:rank}; at
$k=\ell$ an odd prime it gives the case first posed here.  Neither Coons's
non-$k$-regularity theorem nor the Bell--Smertnig Mahler classification supplies
a finite-level rank, a basis, or the relations, so
Theorem~\ref{thm:kkernelrank} is not a consequence of either.  A targeted
literature search located no source stating the exact rank, the basis, or the
normal form; that is a search result and not a novelty claim, and the statement
remains subject to specialist review.

\begin{remark}[excluded generic mechanisms]
Unbounded prime support, exact multiplicative order, natural-boundary
information, ordinary Hankel nonvanishing, and positive rank-one averaging do
not supply the missing binary phase or denominator compression.  The finite
Euler-factor and $2\times2$ mixed-difference identities of
Section~\ref{sec:programmes} are tools rather than open endpoints; no
growing-level Diophantine conclusion is claimed for them here.
\end{remark}

A proof of formulation~\ref{prob:diagonal} gives irrationality of $\Stot$ by
\ref{res:equivalences}.  Its negation settles \#249 in the opposite direction:
it supplies a positive-shift integral tail difference, which forces
rationality.  By contrast, failure of the sufficient producer bounds would
rule out only their respective routes.

\paragraph{The independent denominator exclusion.}
The constant $7.96\times10^{34}$ of \ref{res:denominator} is the classical
Farey mediant bound applied to a window with a free parameter $K$; it has
order $2^{K/2}$, and raising it needs only more committed binary digits, which
is a computation rather than an idea.  It is not one of the open items of
Section~\ref{sec:open}.

\subsection{Logical status and analytic input}\label{sec:open}

\begin{enumerate}
\item Irrationality of $\Stot$ (\ref{res:open}).  No proof is claimed.
\item The lcm-diagonal and clean cyclotomic-anchor statements are exact
  equivalent formulations of \ref{res:equivalences}: they convert
  irrationality into the existence of an unbounded family of finite
  tail-difference certificates.  Finite instances are checked --- every
  $t\le82$, including the historical $28$-scale bank of
  Section~\ref{sec:frontier} --- and the unbounded family is not.  The
  equivalences fix the exact missing quantifier but do not make that
  quantifier easier to prove.  \ref{res:denominator} is a finite instance of a
  \emph{different} family, the gap certificates of \ref{res:supply}, which
  reach irrationality by a one-way implication; the two families should not be
  counted together.
\item Full-depth period-multiple escape and the four first-harmonic pivot
  bounds are stronger sufficient producers; they, and the prime-orbit gap the
  pivot bounds refine, are unproved.  Failure of either would refute only that
  route.
\item Carry-rank compression and genuinely coupled denominator compression
  are separate contradiction targets.  No required upper bound or coupled
  approximant family is proved.
\item The $k$-kernel rank statement of Theorem~\ref{thm:kkernelrank} is
  independent of the irrationality problem and settles nothing about it.  Its
  zero-residue, composite-base arithmetic, finite-index/cardinality, and
  spanning layers are Lean-checked.  Lean also checks the exact-rank theorem
  under an explicit linear-independence hypothesis.  That hypothesis remains
  a paper deduction from Martin's Theorem~1; Martin's theorem is not
  formalised.  No source was located for the rank, basis, or normal form
  themselves; that search result is not a priority verdict.
\end{enumerate}

\noindent The correlation source that fits the normalised totient most
directly is Balasubramanian--Giri--Srivastav~\cite{bgs2017}, not a direct
transfer of the Tao--Ter\"av\"ainen method.  Write
\[
  g(n)=\frac{\ph(n)}n
      =\sum_{d\mid n}\frac{\mu(d)}d=(f*1)(n),
  \qquad f(d)=\frac{\mu(d)}d\in\mathcal A_1 ,
\]
the class $\mathcal A_1$ being the coefficient class in which the theorem
cited next is stated.
Theorem~2.2 of the arXiv version of~\cite{bgs2017} gives, uniformly for
$|h|\le x/2$, an explicit asymptotic for
$\sum g(n)g(n-h)$ with error $O(\log^2x)$; Remark~2.4 gives its Euler product,
For each fixed $h$, the weighted partial-summation formula immediately
following Corollary~2.8 permits $Q(n)=n(n-h)$, restoring the two linear
factors needed to return from $g(n)g(n-h)$ to
$\ph(n)\ph(n-h)$ with an explicitly propagated error.  That displayed
weighted formula is not itself stated uniformly in $h$ and starts its main
integral at \(1\), not \(H\); retaining uniformity and the lower endpoint
would require applying partial summation directly to Theorem~2.2.  No later
claim here relies on an unproved uniform weighted version.

Tao--Ter\"av\"ainen's quantitative correlation theorem
(\cite{taoteravainen2025}, Theorem~3.1) assumes either quantitative
equidistribution together with an exact small-prime condition, or
non-pretentiousness.  Here $g(p)=1-1/p$, so the stated small-prime condition
does not hold, while
$\sum_p(1-g(p))/p=\sum_p1/p^2<\infty$, so $g$ is pretentious to the constant
function.  Their theorem therefore does not apply unchanged.  The
Balasubramanian--Giri--Srivastav theorem supplies the missing first- and
second-moment input in the correct divisor-convolution class, but it does
\emph{not} supply the residue small-ball or phase anti-concentration estimate
needed to turn those moments into a certificate.  That modulo-$2^L$ step is
the precise remaining analytic gap; no irrationality conclusion is drawn from
the correlation asymptotic alone.

\section*{Formal source notes}

\paragraph{Registry coverage.}
Two of the items above are checked propositions that no row of the claim
registry currently owns: the three declarations of \ref{res:basis}, and the
five of \ref{res:weight}; the supply implication of \ref{res:supply} is
likewise unowned, and \ref{res:denominator} cites three declarations of which
the registry's row owns one.  Under this project's own division of authority
the registry, not a manuscript, owns public status, so the honest tag on those
items is that they are kernel-checked and not yet registered.  We print that
rather than borrowing a neighbouring row's status.  The gap is a defect in the
record, not in the proofs, and closing it is a review action rather than a
mathematical one.

This manuscript is authored exposition, not proof authority.  The linked Lean
snapshot is authoritative only for its exact propositions, and kernel checking
establishes that a proposition was proved, not that it is interesting, novel,
or sufficient.  The consequence drawn in the last sentence of \ref{res:basis},
the deduction in \ref{res:periodic} that an unbounded weight is not eventually
periodic, the mechanism sentence in Section~\ref{sec:nogo}, and the size
argument in Section~\ref{sec:programmes} are one-line arguments in the prose,
marked as such; the first three are drawn from checked statements, and the
last assumes the unproved bounded-order condition.  The small numerical
instances printed above --- the level-three reductions, the certificate at
$(1,12,16)$, the first values of $\Aw$, the $p=3$ Euler-factor calculation,
the mixed-difference table, and the values of $H_t$ --- are computations from
the definitions and checked identities beside them, and are not separate
checked propositions.  Source theorems attributed to Allouche and Shallit,
Martin, Coons, Bell and Smertnig, Kaneko--Suzuki--Tachiya, Luca--Tachiya, and
Tao--Ter\"av\"ainen are cited from the literature.  This development does not
formalise Martin's positive-density theorem or Coons's non-regularity theorem.
It does kernel-check an independent dyadic basis/rank proof and a full-kernel
infinite-dimensionality consequence that Coons already implies.  For every
base it also kernel-checks the arithmetic reduction, unconditional canonical
spanning, and the exact-rank conclusion conditional on an explicit
linear-independence hypothesis; it does not prove that hypothesis.  The
assessment that no prior source gives an explicit basis, an exact finite-level
dimension formula, or a relation normal form for the totient $2$-kernel is the
outcome of a literature search and is not a proof of non-existence.

\emph{Scope of the AI declaration.} Every word of this manuscript was generated
by agents based on large language models operating within Will Cook's private
research system for artificial intelligence. The formal proofs and repository
software were likewise drafted and revised by the agents through that system
under Cook's direction. Cook set the objectives and acceptance criteria,
selected and reviewed the public claims, and approved the published version.
Cook assumes responsibility for the accuracy, interpretation, and presentation
of the work. Generative systems are production tools, not authors, and supply
no independent authority. Formal authority is the pinned kernel's acceptance
of an exact proposition; no model output carries any, and neither does this
sentence.

Erd\H{o}s Problem~\#249 remains open.

\begin{thebibliography}{10}
\bibitem{allouche-shallit} J.-P.~Allouche and J.~Shallit,
  \href{https://cs.uwaterloo.ca/~shallit/Papers/as0.pdf}{\emph{The ring of
  $k$-regular sequences}}, Theoret.\ Comput.\ Sci.\ \textbf{98} (1992),
  no.~2, 163--197,
  doi:\href{https://doi.org/10.1016/0304-3975(92)90001-V}
  {10.1016/0304-3975(92)90001-V}.  Definitions~1.1 and 2.1 are on
  pp.~2--3 of the linked author preprint, which differs slightly from the
  published version; the cited definitions are unaffected.
\bibitem{coons} M.~Coons, \emph{(Non)Automaticity of number theoretic
  functions}, J.~Th\'eor.\ Nombres Bordeaux 22 (2010), no.~2, 339--352;
  \href{https://doi.org/10.5802/jtnb.718}{doi:10.5802/jtnb.718};
  \href{https://arxiv.org/abs/0810.3709}{arXiv:0810.3709}.  Theorem~3.2,
  pp.~348--349, in the published version (Theorem~3.3, pp.~8--9, in the
  preprint): $\ph$ is not $k$-regular for any $k\ge2$.
\bibitem{martin-phi-inequalities} G.~Martin, \emph{Simultaneous inequalities
  among values of the Euler phi-function}, 2 March 2006,
  \href{https://arxiv.org/abs/math/0603053}{arXiv:math/0603053};
  \href{https://doi.org/10.48550/arXiv.math/0603053}{doi:10.48550/arXiv.math/0603053}.
  Theorem~1: for positive integers $a_1,\dots,a_k$ and integers $b_1,\dots,b_k$
  with $a_ib_j\neq a_jb_i$, and every $C>0$, the simultaneous ratio gaps
  $\ph(a_1n+b_1)/\ph(a_2n+b_2)>C,\dots$ hold on a set of positive lower density;
  Corollary~4 carries this to $\sigma$.  Cited here as a public preprint: a
  separate journal publication was not located.
\bibitem{bell-smertnig} J.~Bell and D.~Smertnig, \emph{Mahler series with
  multiplicative coefficient sequences}, 2026,
  \href{https://doi.org/10.48550/arXiv.2603.23456}{arXiv:2603.23456}.  The
  totient generating series is not $k$-Mahler for any $k\ge2$, a consequence
  of their Theorem 1.3 recorded in the introduction.
\bibitem{kanekosuzukitachiya2026} H.~Kaneko, Y.~Suzuki and Y.~Tachiya,
  \href{https://arxiv.org/abs/2601.20743}
  {\emph{Refinements of Erd\H{o}s's irrationality criterion for certain
  sparse infinite series}}, arXiv:2601.20743v1, 2026.
  Corollary~3 is on pp.~5--6 and proves irrationality for the
  \(\sigma(n)\)- and \(\ph(n)\)-in-the-exponent families; its proof is on
  pp.~18--19.
\bibitem{taoteravainen2025} T.~Tao and J.~Ter\"av\"ainen,
  \emph{Quantitative correlations and some problems on prime factors of
  consecutive integers}, arXiv:2512.01739 (submitted December 2025, revised
  April 2026).  Theorem 1.3 proves irrationality of
  $\sum_{n\ge1}\omega(n)/2^n$ at base $2$; the extension to every integer base
  and the $\Omega$ analogue are stated as remarks, with the modifications left
  to the reader.  The theorem is on p.~4 and its proof is Section~5,
  pp.~44--56, in arXiv v2.
\bibitem{bgs2017} R.~Balasubramanian, S.~Giri and P.~Srivastav,
  \emph{On correlations of certain multiplicative functions},
  J.\ Number Theory 174 (2017), 221--238,
  \href{https://doi.org/10.1016/j.jnt.2016.10.001}{DOI};
  \href{https://arxiv.org/abs/1511.02221}{arXiv:1511.02221}.
  The arXiv version's Theorem~2.2 and Remark~2.4 give the uniform shifted
  divisor-convolution correlation and Euler product used above; the required
  fixed-shift weighted partial-summation formula follows Corollary~2.8.
\bibitem{lucatachiya2017} F.~Luca and Y.~Tachiya,
  \href{https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2014-14.pdf}
  {\emph{Linear independence results for the values of divisor functions
  series}}, RIMS K\^oky\^uroku No.~2014 (2017), 138--150.
  Theorem~A on p.~139 explicitly restates their earlier Theorem~1.1 for
  nonzero purely periodic integer weights.  Theorem~1 is on p.~139, its
  examples are on p.~140, and its proof is on pp.~149--150.
\bibitem{erdos1948} P.~Erd\H{o}s,
  \href{https://users.renyi.hu/~p_erdos/1948-04.pdf}
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  J. Indian Math. Soc. (N.S.) \textbf{12} (1948), 63--66.
  The integer-base full-support theorem is on p.~63; the proof is on
  pp.~63--66, and the closing totient-series remark is on p.~66.
\bibitem{erdosgraham1980} P.~Erd\H{o}s and R.~L.~Graham, \emph{Old and New
  Problems and Results in Combinatorial Number Theory}, 1980, p.~61.
\bibitem{erdos1988} P.~Erd\H{o}s, \emph{On the irrationality of certain
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  {10.1017/CBO9780511897184.009}.
\bibitem{erdosproblems} T.~F. Bloom,
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  \texttt{erdosproblems.com/249}, accessed 28 July 2026 (page displays
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  open, cites \texttt{[ErGr80, p.~61]} and \texttt{[Er88c, p.~102]}, and
  explicitly describes its status as the website owner's present assessment
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\end{thebibliography}

\end{document}
