Positive stability-preserving semigroups and particle capacities

This preprint classifies positive strongly continuous semigroups on summable coefficient spaces that preserve stability with nonnegative coefficients, allowing finite, infinite, and mixed coordinate capacities. In the unrestricted setting, preservation forces a canonical differential operator of order at most two and a logarithmic symbol with finitely many parameters. For conservative nonexplosive Markov chains, the classification requires no prior restriction on jump sizes and identifies the permitted immigration, migration, and death mechanisms. For finite and mixed capacities, explicit rate conditions and quadratic-contact inequalities provide exact preservation criteria. Complete preservation under inert extensions can be tested using two binary ancillary coordinates or one coordinate of capacity two. The infinite-dimensional theory distinguishes classification of an existing semigroup from generation of a formal operator, with domain results, consistent cutoffs, and weighted spaces supplying the necessary analytic foundations. The manuscript includes a disclosure of the AI tools used in its development.