First Constraint on P-odd/T-odd Cross Section in Polarized Neutron Transmission through Transversely Polarized $^{139}$La

We report the first constraint on time-reversal invariance violating (TRIV) effects in polarized neutron transmission through a transversely polarized $^{139}$La target. We formulate the transmission asymmetry within the density matrix formalism, explicitly incorporating the forward scattering amplitude of $^{139}$La including tensor polarization terms up to third-rank. The formalism is applied to existing transmission data originally obtained to measure the spin-dependent cross section near the $0.75$~eV $p$-wave resonance. Since these data were not optimized for P-odd/T-odd observables, the attainable sensitivity is intrinsically limited; nevertheless, they provide a useful test of the formalism on real experimental data. No statistically significant TRIV signal is observed. By analyzing the global $\chi^2$ structure in the parameter space, we obtain an upper limit of $|W_T|<15~\mathrm{eV}$ at the 90\% confidence level. This corresponds to an upper limit on the resonance-averaged TRIV cross section of $|\Delta\sigma_{\not{T}\not{P}}|<8.3\times10^2~\mathrm{b}$. These results validate the present theoretical framework and provide guidance for future dedicated TRIV searches in polarized neutron transmission experiments.