On Chen's Extended Action for $\kappa(R,T)$ Gravity:\\ Gravity in a Polarizable Medium, Material Folds\\ and a No-Bounce Theorem

$\kappa(R,T)$ gravity is formulated at the level of its field equations. Chen's extended action, built with an adjoint multiplier field, reproduces them exactly from a local variational principle, and we show that it does physical work. In flat FLRW its Dirac reduction is consistent, and on every regular matter branch the reduced Hamiltonian is independent of $\kappa(T)$: the geometry obeys general relativity with an effective fluid, and the coupling survives only in a second-class constraint. Matter is non-conservative, exchanging energy at the rate $\Gamma=-\dot\kappa/\kappa$ through a time-symmetric balance law, yet a dressed particle number is exactly conserved. The theory is general relativity with an induced, polarizable-medium source, and the dressed number is the sum of free and bound parts. This makes precise the original analogy with Maxwell's displacement current: critical surfaces $\kappa=0$ are total screening, the repulsive phase is over-screening, and the fold of the matter map is saturation. For a linear coupling, expanding dust universes on the low-density branch begin at the fold, at a minimum volume and a maximum density $\rho_c/2$ with finite curvature, instead of a big bang. There the Dirac algorithm admits no consistent Hamiltonian evolution. No theory of the class $G_{\mu\nu}-\Lambda g_{\mu\nu}=\kappa(R,T)T_{\mu\nu}$ bounces in flat FLRW with $\kappa>0$ and $w>-1$. The linear model shares the effective Friedmann equation of Loop Quantum Cosmology, and the comparison becomes exact for dust. Effective LQC is the same energy function with the physical rather than the dressed number conserved, and the pressure $-\rho^2/\rho_c$ that produces its bounce is the compression work of the medium. The fold lies exactly at the density where LQC changes signature in the deformed-algebra approach. The reduced dust constraint quantizes to an exactly solvable transport equation of Koopman--von Neumann type. The adjoint field is the response variable of the theory and points to an open-system interpretation, with the induced medium as a candidate environment.