%
\label{thm:regularity-criterion}
Let $f : \tau^3 \to H_\tau$ be $\tau$-holomorphic,
and let $p \in \tau^3$.
Then exactly one of the following holds:
\begin{enumerate}
    \item[\textup{(R)}]
          $p$ is $\tau$-regular for $f$:
          the $\omega$-germ sequence stabilizes at some finite stage $N$.
          Equivalently,
          $f$ has a canonical idempotent-supported extension
          through $p$,
          determined by finitely much boundary data.
    \item[\textup{(S)}]
          $p$ is genuinely $\tau$-singular for $f$:
          the $\omega$-germ sequence fails to stabilize ---
          for every $N \geq 1$,
          there exists $k > N$
          such that $\rho_{k,N}(G_f)
          \neq G_f \big|_{C_N(p)}$.
          The function requires infinitely much boundary data at $p$.
\end{enumerate}
There is no third possibility.
In particular,
there are no removable singularities:
if $p$ is not $\tau$-regular,
then no finite modification of $f$ at $p$
can restore regularity.
\[
    \boxed{%
    \text{For all } p \in \tau^3:\quad
    p \text{ is $\tau$-regular for } f
    \;\;\dot{\vee}\;\;
    p \text{ is genuinely $\tau$-singular for } f.}
\]

The proof proceeds in three steps.

\medskip
\textbf{Step 1: Exhaustive dichotomy.}
The $\omega$-germ sequence at $p$
is an inverse system
indexed by the natural numbers $N = 1, 2, 3, \ldots$
For each pair $k \geq N$,
the restriction map $\rho_{k,N}$
is either an isomorphism or not.
Define the \emph{stabilization set}:
\[
    \Sigma_f(p) = \bigl\{\, N \geq 1
    : \forall\, k \geq N,\;
    \rho_{k,N}(G_f) = G_f \big|_{C_N(p)} \,\bigr\}.
\]
Either $\Sigma_f(p) \neq \varnothing$
(case (R): take $N = \min \Sigma_f(p)$)
or $\Sigma_f(p) = \varnothing$ (case (S)).
These are exhaustive and mutually exclusive.

\medskip
\textbf{Step 2: Regularity implies canonical extension.}
Suppose $p$ is $\tau$-regular,
with stabilization at stage $N$.
Then $G_f \big|_{C_N(p)}$
contains all the germ data of $f$ at $p$.
By the Mutual Determination Theorem
(II.T27, Chapter~\ref{ch:mutual-determination}),
the $\omega$-germ data at stage $N$
determines the boundary character data at stage $N$,
which by the Global Hartogs Theorem
(I.T31, the Global Hartogs Extension (Book~I, Chapter~62))
determines the interior extension through $p$.
By the Idempotent Decomposition Lemma (II.L07),
this extension is canonically
$e_+$-/$e_-$-supported.
The extension is unique because
the Identity Theorem for $\tau$-holomorphic functions
(I.T21) ensures that
two extensions agreeing at stage $N$ must coincide.

\medskip
\textbf{Step 3: No removable singularities.}
Suppose $\Sigma_f(p) = \varnothing$ ---
the germ sequence fails to stabilize.
We claim that no finite modification
of $f$ at $p$ can produce stabilization.

Assume for contradiction
that there exists $\tilde{f}$
agreeing with $f$ on $\tau^3 \setminus \{p\}$
with $\tilde{f}(p) \neq f(p)$,
such that $\tilde{f}$ is $\tau$-regular at $p$.
Then $G_{\tilde{f}}$ stabilizes at some stage $N$.

Consider the cylinder $C_N(p)$
(Definition~\ref{def:stage-k-cylinder}, II.D10).
This is a clopen set in the ultrametric topology.
By the Cylinders Are Balls proposition
(II.P04),
$C_N(p) = B(p, 2^{-N})$
is the ultrametric ball of radius $2^{-N}$
centered at $p$.
Since $C_N(p)$ is \emph{clopen},
it has no boundary points ---
every point of $C_N(p)$
is an interior point of $C_N(p)$,
and every point outside $C_N(p)$
is an interior point of the complement.

Now $f$ and $\tilde{f}$ agree on $C_N(p) \setminus \{p\}$.
By the $\tau$-Identity Theorem (I.T21),
applied to the $\omega$-germ transformers
$G_f$ and $G_{\tilde{f}}$
restricted to $C_N(p)$:
if these agree at stage $N$
on the germ data of all points
$q \in C_N(p) \setminus \{p\}$,
then they agree on the germ data
of $p$ as well ---
because tower coherence
(via CRT reconstruction at each subsequent stage)
propagates agreement upward.
But this would mean $G_f$ also stabilizes
at stage $N$ ---
contradicting $\Sigma_f(p) = \varnothing$.

Therefore no such $\tilde{f}$ exists.
The singularity at $p$ is genuine:
it is a property of the function itself,
not of any particular presentation.