Multi-time Markov renewal chains and stratified renewal theorems

We develop a discrete Markov renewal theory on a standard Borel state space, with vector-valued sojourn times and lower-rectangle observation on $\N^d$. The Markov renewal potential is a kernel-valued convolution resolvent and yields unified representations for semi-Markov transitions, first-passage laws, occupation measures and rewards. The semi-Markov field observed on the partially ordered lattice is generally not Markov. We identify its canonical Markovian augmentation through the backward recurrence vector and give a lumpability criterion for the exceptional cases in which the augmentation can be projected back to the original state space. The lower-rectangle order leads to a stratified inverse-renewal theory: the direction simplex is decomposed into rate-determining cells, with Gaussian limits on cells having a unique active coordinate and minima of correlated Gaussian fields on their interfaces. We establish functional inverse limits, critical-interface limits and logarithmic estimates for inverse deviations. Exact-time potentials are obtained from an operator-theoretic local theorem for Fourier--Laplace perturbations of Markov-additive kernels, while a regenerative theorem gives the corresponding arithmetic lattice-class form. The results connect Markov renewal equations, multiparameter Markov structure and the local asymptotic geometry induced by rectangular observation.