Chirality--Polarization Benchmarks in LRSM $0\nu\beta\beta$: Hard-Validated $\xi$ Opening and Dim-9 Decoupling

We study chirality and polarization systematics in neutrinoless double beta decay
($0\nu\beta\beta$) within a left--right symmetric model (LRSM), using an end-to-end
LRSM$\to$SMEFT/LEFT matching-and-running pipeline equipped with hard benchmark
validation.
Two representative benchmark points, $\xi=0$ and $\xi=\xi_{b/t}\simeq0.02424$,
reproduce the expected dim-6 chirality opening:
$|V_L^{(6)}|$ is enhanced from $6.77\times10^{-22}$ to $2.24\times10^{-12}$ while
$|V_R^{(6)}|$ remains essentially unchanged at $2.73\times10^{-11}$, and a scan yields
$|V_L^{(6)}|\propto\xi^p$ with $p\simeq1$.
With short-range dim-9 matching and injection enabled, we scan
$m_{\Delta_R}\in\{3,5,10,20,30\}\,\mathrm{TeV}$ and verify the heavy-scale behavior in a
mechanism-resolved way: $\Delta_R$-exchange coefficients exhibit the expected
decoupling $|C_9|\propto m_{\Delta_R}^{-2}$, whereas the $\nu_R$-only contribution is
$m_{\Delta_R}$ independent.
For the mixed case, where a single power law is not theoretically clean, we replace
slope assertions by monotonic non-increase with $m_{\Delta_R}$ and asymptotic approach
to the $\nu_R$ limit at large $m_{\Delta_R}$, both satisfied by the scan.
Finally, we summarize the validated outputs with minimal dimensionless chirality
indicators and phase plots in $(\xi,m_{\Delta_R})$ space, providing a compact,
operator-level polarization baseline for systematic LRSM parameter surveys and
interference studies in $0\nu\beta\beta$ phenomenology.