%
\label{thm:hol-iff-idempotent}
%   II.T27, I.T05, I.D21, I.D22, I.D23
A function $f \colon \tau^3 \to H_\tau$
is $\tau$-holomorphic
if and only if it is idempotent-supported
\textup{(}Definition~\textup{\ref{def:ch38-idempotent-supported})}.
Explicitly:
\[
    \boxed{%
    f \text{ is $\tau$-holomorphic}
    \quad\Longleftrightarrow\quad
    f = e_+ \cdot f_+ + e_- \cdot f_-
    \text{ with \textup{(IS1)--(IS4)}}.}
\]

\emph{Forward direction
($\tau$-holomorphic $\Rightarrow$ idempotent-supported).}

Suppose $f$ is $\tau$-holomorphic.
We verify conditions (IS1)--(IS4).

\emph{(IS1) Bipolar decomposition.}
The Idempotent Decomposition Lemma
(II.L07, Chapter~\ref{ch:idempotent-decomposition})
gives $f = e_+ \cdot f_+ + e_- \cdot f_-$
with $f_\pm = e_\pm \cdot f$.
This is canonical and functorial (II.D48).

\emph{(IS2) Stagewise naturality.}
By the Mutual Determination Theorem
(II.T27, Chapter~\ref{ch:mutual-determination}),
$f$ is equivalent to a tower-coherent sequence
$(f_k)_{k \geq n}$ in description~(R).
Since $e_\pm$ are fixed scalars
and tower coherence is preserved
under scalar multiplication,
$(f_\pm)_k = e_\pm \cdot f_k$
are each tower-coherent.

\emph{(IS3) Channel support.}
Branch Factorization (II.L08)
applied to the $\omega$-germ transformer
associated to $f$ gives $G = G_+ + G_-$.
Prime-Split Support (II.L09)
gives $\operatorname{supp}(G_+) \subseteq \Lambda_\tau^{(B)}$
and $\operatorname{supp}(G_-) \subseteq \Lambda_\tau^{(C)}$.
Since $f_\pm$ are the holomorphic maps
corresponding to the transformers $G_\pm$,
their spectral supports
inherit the same containment.

\emph{(IS4) Polarity symmetry.}
Polarity Symmetry (II.L10)
gives $\sigma_\jj(G_+) = G_-$.
Translating to the level of maps
via the Mutual Determination equivalence:
$\sigma_\jj(f_+) = f_-$
and $\sigma_\jj(f_-) = f_+$.

\medskip
\emph{Backward direction
(idempotent-supported $\Rightarrow$ $\tau$-holomorphic).}

Suppose $f = e_+ \cdot f_+ + e_- \cdot f_-$
satisfies (IS1)--(IS4).
We must show $f$ is $\tau$-holomorphic.

By the Mutual Determination Theorem (II.T27),
$\tau$-holomorphy is equivalent to any of
the five descriptions
(R), (S), (G), (C), (H).
We verify description~(R):
$f$ is a tower-coherent refinement tail
with stabilized spectral support.

\emph{Step 1: Tower coherence.}
By (IS2), each $f_\pm$ is tower-coherent.
Since $e_+, e_-$ are constant scalars,
$f = e_+ \cdot f_+ + e_- \cdot f_-$
is tower-coherent:
\begin{align*}
    f_k
    &= e_+ \cdot (f_+)_k + e_- \cdot (f_-)_k \\
    &\equiv e_+ \cdot (f_+)_{k+1} + e_- \cdot (f_-)_{k+1}
    = f_{k+1}
    \pmod{P_k},
\end{align*}
where the congruence uses
the tower coherence of $(f_+)$ and $(f_-)$.

\emph{Step 2: Stabilization.}
By (IS3), the spectral support of $f_+$
is contained in the B-channel primes
and that of $f_-$
in the C-channel primes.
The combined spectral support of $f$
is $\operatorname{supp}(f)
= \operatorname{supp}(f_+) \cup \operatorname{supp}(f_-)
\subseteq \Lambda_\tau^{(B)} \cup \Lambda_\tau^{(C)}
= \Lambda_\tau$.
Since each $f_\pm$ is tower-coherent
with support in a fixed set of primes,
the spectral support stabilizes
at some finite stage~$n$.

\emph{Step 3: Sector independence.}
The B-channel and C-channel
of the resulting refinement tail
are independent:
the B-channel data $f_+$
depends only on $\gamma$-orbit primes,
and the C-channel data $f_-$
depends only on $\eta$-orbit primes.
These prime sets are disjoint
(by Prime Polarity, I.T05).
The bipolar channel independence
(II.P07, Chapter~\ref{ch:bndlift-construction})
is therefore satisfied.

\emph{Step 4: The five descriptions agree.}
A tower-coherent refinement tail
with stabilized spectral support
and bipolar channel independence
satisfies description~(R)
of the Mutual Determination Theorem (II.T27).
By II.T27, all five descriptions are equivalent.
In particular, description~(H)
gives a Hartogs continuation,
and description~(G)
gives a stabilized $\omega$-germ.
Therefore $f$ is $\tau$-holomorphic.

\emph{Step 5: Polarity symmetry is used.}
Condition (IS4) is needed to ensure
that the two channel components $f_+, f_-$
are \emph{compatible} with the holomorphic structure.
A decomposition $f = e_+ \cdot f_+ + e_- \cdot f_-$
satisfying (IS1)--(IS3) but not (IS4)
would have the right support properties
but might fail the holomorphic condition
because the two channels would carry
unrelated data.
The polarity symmetry $\sigma_\jj(f_+) = f_-$
ensures that $f_+$ and $f_-$
encode the same underlying holomorphic datum,
viewed from the two bipolar perspectives.
This is the condition that
the Mutual Determination equivalence
$(\mathrm{R}) \Leftrightarrow (\mathrm{C})$
(Lemma~\ref{lem:germ-character}, II.L04)
requires: the boundary character
$\varphi \colon R_\tau \to H_\tau$
must satisfy
$\sigma_\jj(\varphi^+) = \varphi^-$,
which is exactly (IS4)
at the level of boundary data.