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 ≤ θ ≤ 1, we prove that k ↦ λ_max(A_k + θD_k) is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At θ = 1 and θ = 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 ≥ 0, the normalized heat contents 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]. This versioned deposit contains the final article PDF, the reproducibility supplement ZIP, and SHA256SUMS covering those two files. The arXiv source ZIP is not part of this deposit. Licensing is file-scoped: article, prose, and documentation are CC BY 4.0; code is MIT; generated computational artifacts and data are CC0 1.0. The authoritative mapping is provided in LICENSES.md inside the supplement.