Periodic KPZ fixed point with general initial conditions

We consider the periodic totally asymmetric simple exclusion process with a general initial condition that properly approximates a periodic upper-semicontinuous function. We find the large time limit of the rescaled space-time multipoint distribution of the height function in the relaxation time scale. The limiting functions form a consistent family of finite-dimensional distributions; thus, they define the periodic KPZ fixed point with a general upper-semicontinuous initial condition. The main technical novelty of the paper is a hitting expectation representation of the energy function and the characteristic function in the finite-time multipoint distribution formula obtained in arXiv:1912.10143. The representation of the characteristic function is partly inspired by the work of arXiv:1701.00018, arXiv:2509.03246, while the representation of the energy function is based on an extensive guess-and-check exploration.