Substrate condensation and the dark gauge gap: a model approach

This note presents a model treatment of the dark gauge sector’s mass gap and of the substrate condensation on which it rests, within the Spectral Quantum Gravity (SQG) framework, in their natural causal order. Part A (condensation): the symmetric configuration of the radial substrate field is shown to be a local maximum of the potential (V’’(0) = −4λv² < 0), and is not stabilized by the spectral kinetic and centrifugal terms. The symmetric phase is therefore spinodally unstable, and condensation proceeds by a finite-speed propagating front rather than by barrier nucleation. Condensation is generic and barrier-free. Part B (gap): given the resulting condensate, minimal coupling of the composite gauge field produces an effective mass that vanishes in the ultraviolet (asymptotic freedom preserved) and saturates in the infrared, reproducing the dynamical-gluon-mass structure of Cornwall and of Aguilar–Binosi–Papavassiliou. On the spectral axis the gauge two-point operator is Schrödinger-type with a strictly positive ground-state eigenvalue — an explicit gap near the dark scale Λ_dark. The note distinguishes throughout what is derived from the framework, what is computed numerically, and what is assumed or estimated. It is a model realization with an explicit dependency structure: Part B is conditional on the condensate of Part A. It is not a proof of the Yang–Mills mass gap. The assessment of the construction’s reach is left to independent analysis.