Odderon exchange in high-energy $K^0_S$ regeneration at the LHC

We revisit the possibility of detecting the Odderon exchange through high-energy neutral-kaon regeneration, focusing on the in-matter $K^0_L\to K^0_S$ conversion of $(\sim\!0.2-2)$~TeV $K^0_L$ mesons originating from $pp$ collisions at $\sqrt{s}=13.6$~TeV, and on the practical constraints of realizing such a measurement in the very-forward region of an LHC interaction point. The analysis has two complementary parts. First, we reproduce the original coherent-forward-regeneration estimates for a liquid-hydrogen regenerator and extend them to realistic C, Cu and Pb regenerators of an LHC-compatible geometry, including neutral-kaon attenuation. We show that an Odderon-induced regeneration phase produces a measurable distortion of the $K^0\to\pi^0\pi^0$ decay-vertex distribution at TeV kaon energies, but that the survival of primary $K^0_S$ mesons from the interaction point imposes severe baseline requirements, while a competing electromagnetic $C=-1$ (photon-exchange) amplitude limits the interpretation of the coherent mode as a clean Odderon measurement. Second, we examine non-forward (diffractive) regeneration at lower kaon energies of $0.2-0.8$~TeV, where the primary-$K^0_S$ contamination is naturally suppressed, and estimate the competing Odderon, Pomeron--Odderon-cut, $\omega$-Reggeon and photon-exchange contributions to the regeneration amplitude. We identify neutron-induced neutral-only strangeness production and inelastic Regge backgrounds as the dominant limitations, quantify the background suppression required for an observable Odderon signal, and formulate the ingredients of an active double-regenerator subtraction strategy.