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-endLRSM$\to$SMEFT/LEFT matching-and-running pipeline equipped with hard benchmarkvalidation.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 amechanism-resolved way: $\Delta_R$-exchange coefficients exhibit the expecteddecoupling $|C_9|\propto m_{\Delta_R}^{-2}$, 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 replaceslope assertions by monotonic non-increase with $m_{\Delta_R}$ and asymptotic approachto the $\nu_R$ limit at large $m_{\Delta_R}$, both satisfied by the scan.Finally, we summarize the validated outputs with minimal dimensionless chiralityindicators and phase plots in $(\xi,m_{\Delta_R})$ space, providing a compact,operator-level polarization baseline for systematic LRSM parameter surveys andinterference studies in $0\nu\beta\beta$ phenomenology.