Geometric QCD III: Exact transition amplitudes and the glueball spectrum

We complete the analysis of planar Makeenko--Migdal loop equations in the continuum limit. Using the confining twistor-string representation, we compute the quantum fluctuation determinant. In Minkowski space, this reduces to a discrete product of finite-dimensional matrix quadratures. The $\zeta$-regularized weight is independent of winding number $w$. Near the mass shell, the pole singularity is generated by $w \to \infty$, suppressing fluctuation variance as $1/w$. The path integral localizes on the classical trajectory, rendering the pole spectrum and transition residues parametrically exact in the large-$w$ WKB limit.For the open-string meson sector, we fit 40 states across five topological sectors ($h=0, \pm 1, \pm 2$). The holonomy shift $h$ dictates exact geometric degeneracies between parity families, reproducing mass splittings without phenomenological spin-orbit parameters. Evaluated one-loop residues yield relative transition cross-sections, demonstrating nontrivial topology- and flavor-dependent scaling that captures heavy-mass quenching and phase-space enhancement for high-spin light states.For the pure Yang--Mills closed string, we demonstrate dynamical stability of the pure-gauge minimal surface in the planar limit: the conformal Liouville anomaly drives the string strictly to the trigonometric minimum ($q=0$). The complex elliptic geometry analytically collapses, yielding linear Regge trajectories. The translation zero-mode measure dynamically nullifies the transition amplitude of the massless scalar ghost, providing an explicit analytic mechanism for a purely gluonic mass gap. Anchoring parameter-free glueball trajectories to the string tension natively recovers the L"uscher intercept $\alpha(0)=1/12$, yielding a macroscopic spectrum that provides a suggestive baseline for established PDG unassigned isoscalar candidates.