The reflection radiance function must have a norm less than 1 for the recursive rendering equation to be solved, which makes sense considering the nature of light infinitely bouncing. However, my understanding is that this bounce function is the bounce function for the entire scene, meaning this condition actually states "as long as the overall scene loses energy on average with each bounce, its rendering equation is solvable." My concern is, what if two surfaces exist in the scene where no energy is lost during reflection, and any of the points on those surfaces have normal vectors pointing to each other? Would this not result in a nearly infinite amount of light, or would it simply make the function not converge (but the light itself would still be infinite), OR are there no surfaces that perfectly bounce light?

The reason why K < 1 is due to conservation of energy. The light will be absorbed into the material or turned into heat. So eventually, the light energy will be completely dissipated. Though it is interesting to imagine a scenario in which K = 1 (no energy is converted/absorbed).

Not quite sure about the convergence mathematically. Is there a proof of something like $\lim_{n\rightarrow \infty} K^n = \infty, \forall K\in \mathbb{R}^{k\times k}, ||K||\geq 1$ ?