\label{thm:bipolar-euler-product}
For $\mathrm{Re}(s) > 1$, the split-complex zeta function admits the Euler product
\[
\zeta_{\T}(s) \;=\; \prod_{p \,\text{prime}} \frac{1}{1 - p^{-s}} \;=\; e_+ \cdot \zeta_B(s) \;+\; e_- \cdot \zeta_C(s),
\]
where
\begin{align*}
\zeta_B(s) &\;=\; \prod_{\substack{p \,\text{prime} \\ \mathrm{Label}_p = \mathrm{B}}} \frac{1}{1 - p^{-s}}, \\[6pt]
\zeta_C(s) &\;=\; \prod_{\substack{p \,\text{prime} \\ \mathrm{Label}_p = \mathrm{C}}} \frac{1}{1 - p^{-s}}.
\end{align*}
Each prime contributes to exactly one idempotent sector. Neutral primes ($\mathrm{Label}_p = \mathrm{N}$) do not exist in this factorization, as $\mathrm{Label}_p \in \{\mathrm{B}, \mathrm{C}\}$ for all primes $p > 2$ (by Theorem~III.T14).

The classical Euler product expands as
\[
\zeta(s) \;=\; \prod_p \left(1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \cdots \right) \;=\; \sum_{n=1}^\infty \frac{1}{n^s},
\]
by unique prime factorization. In the split-complex setting, each integer $n = p_1^{a_1} \cdots p_k^{a_k}$ inherits its label from the prime labels via the polarity sum:
\[
\mathrm{polarity}(n) \;=\; a_1 \cdot \mathrm{pol}(p_1) + \cdots + a_k \cdot \mathrm{pol}(p_k),
\]
where $\mathrm{pol}(p) = +1$ if $\mathrm{Label}_p = \mathrm{B}$ and $\mathrm{pol}(p) = -1$ if $\mathrm{Label}_p = \mathrm{C}$.

The CRT decomposition (Theorem~III.T10) asserts that the ring $\mathbb{Z}/N\mathbb{Z}$ factors as
\[
\mathbb{Z}/N\mathbb{Z} \;\cong\; \bigoplus_{p^a \| N} \mathbb{Z}/p^a\mathbb{Z},
\]
and this isomorphism respects the bipolar structure: each prime power $p^a$ contributes to the B-sector if $\mathrm{Label}_p = \mathrm{B}$, and to the C-sector if $\mathrm{Label}_p = \mathrm{C}$.

Thus the Euler product factorizes into two independent products:
\[
\prod_{p} \frac{1}{1 - p^{-s}} \;=\; \left( \prod_{p \in \mathrm{B}} \frac{1}{1 - p^{-s}} \right) \times \left( \prod_{p \in \mathrm{C}} \frac{1}{1 - p^{-s}} \right).
\]
The idempotent projection maps these products to the two sectors:
\[
\zeta_{\T}(s) \;=\; e_+ \cdot \zeta_B(s) \;+\; e_- \cdot \zeta_C(s). \qedhere
\]