TCFQ Technical Notes R49–R55: Vertex Classification, Spectral–Geometric Structure, Asymptotics and Discrete Transitions in the Horn Polytope 𝑃 ( A s y m , A l s y m )

The compendium R49–R55 develops a unified and rigorous picture of the Horn polytope \(P(\mathrm{Asym},\mathrm{Alsym})\): it proves algebraically that its only vertices are the configurations \(V_k\), shows how the distinguished spectral points lie on a specific edge of the simplex, characterizes when the spectral minimum \(b_n\) matches or exceeds the geometric minimum over the vertices, derives the continuous \(n\to\infty\) limit through a unique transcendental equation for the minimizer, and identifies a discrete cascade of transitions with a critical dimension \(n^*=34\), yielding a complete geometric and asymptotic description of the system.