Generalized discrete Markov spectra

We develop a generalized Markov theory for the Markov--Lagrange and Markov spectra. The classical discrete Markov spectrum is governed by Markov numbers, the positive integers occurring in solutions of the Markov equation. We show that this relation admits a cluster-combinatorial extension governed by generalized Markov numbers. Replacing the Christoffel-word formalism by snake graphs, we construct generalized discrete Markov spectra attached to the generalized Markov equations $$x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz.$$ Every element of these spectra is realized simultaneously as a Lagrange constant of a quadratic irrational and as a Markov constant of a real indefinite binary quadratic form. We also prove structural results for these spectra, determine their contribution in the transition interval below Freiman's constant, and identify the boundary value obtained from regular lines of irrational slope, again realizing it both as a Lagrange constant and as a Markov constant.