Thermal modification of $K_1(1270)\to \pi^+\pi^-K^+$ in a hot hadronic medium

We study the thermal modification of the exclusive decay $K_1^+(1270)\to \pi^+\pi^-K^+$ in a hot hadronic medium. The decay amplitude is constructed from effective hadronic interactions dominated by the $\rho$- and $K^*$-pole contributions, which enables a Dalitz-level analysis of the three-body decay in medium. Thermal effects associated with partial chiral-symmetry restoration are incorporated through a phenomenological interpolation toward vector--axial-vector degeneracy near the chiral crossover. As the temperature increases, the reduction of the parent $K_1$ mass strongly compresses the available three-body phase space, leading to substantial deformation of the Dalitz distribution and invariant-mass spectra, as well as to a pronounced suppression of $\Gamma_{K_1\to\pi^+\pi^-K^+}(T)$. As a further step toward future experimental comparison, we introduce normalized shape observables that quantify the thermal evolution of the $K^*$-dominated $\pi K$ region, the upper-edge weight of the $\pi\pi$ spectrum, and the compactification of the Dalitz population. The dominant effect identified in the present framework is therefore kinematic: thermal phase-space reduction in the strange axial-vector channel. These results suggest that the exclusive $K_1(1270)$ channel may provide a useful qualitative probe of in-medium strange axial-vector dynamics near the pseudocritical region.