%
\label{prop:no-simul-projection}
Let $t = (x_k)_{k \geq 1}$ be a compatible omega-tail
(satisfying the reduction conditions of the primorial tower),
and let $T \in \mathrm{HolFun}$ be a $\tau$-holomorphic function.
Then the canonical sector projections
$\chi_+(T(t))$ and $\chi_-(T(t))$
cannot \emph{both} be nontrivial at every stage
of the primorial ladder.

More precisely:
at each primorial stage $M_k$,
the CRT decomposition
\[
    \mathbb{Z}/M_k\mathbb{Z}
    \;\cong\;
    \prod_{i=1}^{k} \mathbb{Z}/p_i\mathbb{Z}
\]
assigns each CRT factor $\mathbb{Z}/p_i\mathbb{Z}$
to exactly one sector via the Prime Polarity Theorem (I.T05):
to the $B$-sector if $\mathrm{pol}(p_i) = +$,
or to the $C$-sector if $\mathrm{pol}(p_i) = -$.
The diagonal-discipline axiom (K5, I.D03)
forbids any program from simultaneously accessing
generators from both sectors in the same application step.
Therefore, $T$'s output in the $B$-sector
depends only on $B$-sector inputs,
and $T$'s output in the $C$-sector
depends only on $C$-sector inputs.
An omega-germ cannot project nontrivially onto both sectors
simultaneously through a $\tau$-holomorphic function.

Consider a compatible omega-tail $t = (x_k)_{k \geq 1}$.
At stage $M_k$, the CRT decomposition
assigns each prime $p_i$ ($1 \leq i \leq k$)
to the $B$-sector (if $\mathrm{pol}(p_i) = +$)
or to the $C$-sector (if $\mathrm{pol}(p_i) = -$).
The $B$-sector projection at stage $k$
acts only on the $B$-sector CRT factors;
the $C$-sector projection acts only on the $C$-sector factors.

For a $\tau$-holomorphic function $T \in \mathrm{HolFun}$,
tower coherence (I.D46)
and D-holomorphy --- specifically,
sector independence (I.P22) ---
together ensure that
$T$'s $B$-sector output depends only on $B$-sector inputs
and $T$'s $C$-sector output depends only on $C$-sector inputs.

The diagonal discipline (I.D03) is the axiom-level guarantee
that programs cannot mix these sectors:
no single denotational step simultaneously applies generators
from both the $B$-sector and the $C$-sector.
This is precisely the content of axiom K5 ---
the no-diagonals constraint introduced in
Chapter~\ref{ch:diagonal-discipline}.

Combining these three ingredients:
\begin{enumerate}
    \item \emph{CRT partition}: each prime belongs to exactly one sector
          (I.T05);
    \item \emph{Sector independence}: $T$ decomposes as
          $T(u, v) = (g(u), h(v))$ in sector coordinates (I.P22);
    \item \emph{Diagonal discipline}: no program step
          can access generators from both sectors simultaneously
          (I.D03).
\end{enumerate}
It follows that an omega-germ cannot
``project nontrivially onto both sectors simultaneously''
through $T$.
If $t$ is purely $B$-sector at every stage,
then $\chi_-(T(t))$ is determined
entirely by the trivial $C$-sector input.
If $t$ has nontrivial $C$-sector content,
then the $B$-sector output is independent
of that $C$-sector content.
The two projections are structurally decoupled.