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% Removed from main.tex for arXiv v1 on 2026-07-26 (Hora's decision).
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% This is the "Novak--Carmichael approximation" block, formerly the second half
% of the subsection "The complete lattice of characteristics"
% (Lemma 3.3.5, Example 3.3.6, Proposition 3.3.7, Proposition 3.3.8,
%  Proposition 3.3.9 of the 2026-07-25 draft, PDF pages 14--15).
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% Why it was removed: Section 3 was restructured into three acts
% (finite arithmetic -> profinite arithmetic -> Banach fixed point), and the
% remaining part of that subsection became the preparation for the profinite
% act. This block is not used by that act: the cofinality it needs is supplied
% by the lemma SimplifyingWithFactorial (n |-> (n,n!)), which is strictly
% stronger than the unboundedness proposition here. Checked before removal:
% every \Cref to these statements was internal to the block itself.
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% What is worth keeping: prop:riegCharacteristicApprox says that the
% embeddings RiegCh -> RigCh and RiegfCh -> RigfCh have a left adjoint. That
% echoes the adjunction between the period function and the jump function in
% the Metrics subsection, so it may be worth reinstating if the lattice of
% characteristics is ever developed for its own sake.
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% This file is NOT part of the arXiv submission. Exclude it from the
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\begin{lemma}\label{lem:Shifted_Carmichaelapprox}
    For any positive integer $b$, there is the minimum Nov\'ak--Carmichael number $b'$ such that $b$ divides $b'$.
\end{lemma}
\begin{proof}
Here is a concrete algorithm:
If $b$ is Nov\'ak--Carmichael, then return $b$.
    If $b$ is not Nov\'ak--Carmichael, then take the maximum prime number $p_b$ such that $p_b$ divides $b$ but $p_b -1$ does not divide $b$.
    And (recursively) return the minimum Nov\'ak--Carmichael number $b'$ such that $\lcm (b, p_b-1)$ divides $b'$.

    This algorithm halts, since the prime number $p_b$ becomes smaller and smaller. It is easy to prove the minimality.
\end{proof}

\begin{example}[The minimum Nov\'ak--Carmichael number that is divided by $11$]
    First, start with $b=11$, we replace it with $\lcm (11, 11-1)=110$. Next, we replace $b=110$ with $\lcm (110,5-1)= 220$ and conclude $220$ is the least Nov\'ak--Carmichael number that is divided by $11$.
\end{example}

\begin{proposition}[Nov\'ak--Carmichael approximation]\label{prop:riegCharacteristicApprox}
    For any $(a,b)\in \RigCh$, there is the minimum $(a',b')\in \RiegCh$ such that 
    $(a,b)\leq (a',b')$.
    Furthermore, if $(a,b)\in \RigfCh$, then $(a',b')\in \RiegfCh$.
\end{proposition}
\begin{proof}
    If $(a,b)= \chz$, then $(a',b')=\chz$. We may assume $(a,b)\in \RigfCh$.
    Due to \Cref{lem:Shifted_Carmichaelapprox}, we can take the minimum Nov\'ak--Carmichael number $b'$ that is divisible by $b$. Then, define
    \[
        a'\coloneqq \max(a,\H(b')).
    \]
    The pair $(a',b')$ belongs to $\RiegfCh$ and satisfies $(a,b)\leq (a',b')$. Its minimality follows from the minimality of $b'$ and the necessary inequalities $a''\geq a$ and $a''\geq \H(b'')$ for every $(a'',b'')\in\RiegfCh$ above $(a,b)$.
\end{proof}
In categorical terms,
\Cref{prop:riegCharacteristicApprox} states that the two embedding (order-preserving) functions
\[\RiegCh \hookrightarrow \RigCh \]
\[\RiegfCh \hookrightarrow \RigfCh \]
have a left adjoint.
% . \memo{This data defines a \dq{closure operator} on $\RigCh$. but not preserving meets.}

As an immediate corollary, we obtain the following propositions.
\begin{proposition}
    $\RiegfCh$ is unbounded in $\N\times \N_{>}$.
\end{proposition}

\begin{proposition}[]
    For any $n\in \N$, the canonical rig homomorphism
    \[\N \twoheadrightarrow \Z/n\Z\] can be factored as
    \[\N \twoheadrightarrow \Mr{a}{b}\twoheadrightarrow \Z/n\Z,\]
    where $\N \twoheadrightarrow \Mr{a}{b}$ is a rieg homomorphism and $\Mr{a}{b}\twoheadrightarrow \Z/n\Z$ is a rig homomorphism.
\end{proposition}
\begin{proof}
    We can take $(a,b)$ by applying \Cref{prop:riegCharacteristicApprox} to $(0,n)\in \N\times \N_{>}$.
\end{proof}