Discrete Concavity of Token-Graph Spectral Radii via Lorentzian Semigroups

Let F k(G) be the k-token graph of a finite graph with nonnegative edge weights, and let A k and D k be its weighted adjacency and degree matrices. For every-1 at most at most1, we prove that k λ(A k+ D k) is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At =1 and =0, this gives the signless-Laplacian and adjacency spectral-radius monotonicity conjectures of Apte, Parekh, and Sud. The spectral result follows from a finite-time theorem: for every t at least0, the heat contents k-1 1*e t(A k+ D k) 1 are log-concave in k. We encode all token levels in one Lorentzian polynomial. A four-variable operator symbol proves preservation by each edge heat gate, and the Lie-Trotter formula passes this preservation to the full semigroup. Large-time growth rates recover the top eigenvalues. The same construction also yields Lorentzian polynomials from top spectral projections. Finally, we show that the local symbol certifies exactly the parameter range [-1,1].