  We present an analysis of the half-filled CPT model, an analytically
tractable Kondo lattice model with Yao-Lee spin-spin interactions on a 3D
hyperoctagon lattice, proposed by Coleman, Panigrahi, and Tsvelik. Previous
studies have established that the CPT model exhibits odd-frequency triplet
superconductivity and order fractionalization. Through asymptotic analyses in
the small $J$ and large $J$ Kondo coupling limits, we identify a quantum
critical point at $J_c$, marking a transition from a superconductor to a Kondo
insulator. By estimating the vison gap energy to account for thermal gauge
fluctuations, we determine the energy scales governing the thermal breakdown of
order fractionalization. Moreover, at large $J$ the Kondo insulator undergoes
orbital decoupling, leading to the formation of a decoupled Kitaev orbital
liquid. These findings and analogies with the $\mathbb{Z}_2$-gauged $XY$ model
lead us to propose a tentative phase diagram for the CPT model at half-filling.
