%
\label{thm:diagonal-free-protection}
Let $T \in \mathrm{HolFun}$ be a $\tau$-holomorphic function.
Then for any two compatible omega-tails $t_1, t_2$
with $T(t_1)$ and $T(t_2)$ in opposite sectors
(one purely $B$, one purely $C$),
we have:
\[
    \boxed{%
    T(t_1) \cdot T(t_2) \neq 0
    \quad\text{only if}\quad
    t_1 \text{ and } t_2 \text{ have the same sector purity
    as their outputs.}}
\]
In particular:
the zero-divisor product $e_+ \cdot e_- = 0$
cannot arise as $T(t_1) \cdot T(t_2)$
for any $T \in \mathrm{HolFun}$
and any compatible omega-tails $t_1, t_2$ ---
because $T$'s tower coherence prevents it
from mapping a compatible tower
to a pure idempotent output
unless the input was already sector-pure.

The proof combines three established results.

\emph{Step 1: Sector decomposition.}
By I.P22 (sector independence),
$T$ decomposes in sector coordinates as
\[
    T(u, v) = (g(u),\; h(v)),
\]
where $u = \chi_+(x)$ is the $B$-sector coordinate
and $v = \chi_-(x)$ is the $C$-sector coordinate.
The function $g$ acts on $B$-sector inputs
and $h$ acts on $C$-sector inputs.

\emph{Step 2: Prime-by-prime determination.}
By I.T18 (CRT coherence),
$g$ and $h$ are determined prime-by-prime:
$g$ acts only on the CRT factors
corresponding to $B$-sector primes
(those with $\mathrm{pol}(p) = +$),
and $h$ acts only on the CRT factors
corresponding to $C$-sector primes
(those with $\mathrm{pol}(p) = -$).
Each prime belongs to exactly one sector
by the Prime Polarity Theorem (I.T05).

\emph{Step 3: No simultaneous projection.}
By Proposition~\ref{prop:no-simul-projection} (I.P23),
no compatible omega-tail can project nontrivially
onto both sectors through $T$.

\emph{Conclusion.}
The zero-divisor product $e_+ \cdot e_- = 0$
would require a pure $B$-sector output
multiplied by a pure $C$-sector output.
For $T(t_1)$ to be purely $B$-sector,
we need $h(\chi_-(t_1)) = 0$:
the $C$-sector component of $T(t_1)$ must vanish.
For $T(t_2)$ to be purely $C$-sector,
we need $g(\chi_+(t_2)) = 0$:
the $B$-sector component of $T(t_2)$ must vanish.
But by I.P23, this requires $t_1$ to have trivial $C$-sector input
and $t_2$ to have trivial $B$-sector input ---
that is, $t_1$ and $t_2$ must already have
the same sector purity as their outputs.
A compatible omega-tail with nontrivial content
in both sectors cannot be mapped
to a pure idempotent output by any $T \in \mathrm{HolFun}$.
Therefore the zero-divisor product
$e_+ \cdot e_- = 0$
cannot arise from the same omega-germ ---
which I.P23 forbids.