Spectral Density to Physical Response: Green Functions, IR Scale, and Emergent Mass Gap in History-Dependent Field Theories (Paper LXXI)

We show that a positive spectral density ρ(μ)ρ(μ) uniquely determines the physical response of a history-dependent field theory. Its Stieltjes transform defines the quadratic effective action, whose analytic continuation yields the retarded Green function. The resulting spectral function is expressed directly in terms of ρ(μ)ρ(μ), making the theory predictive rather than phenomenological. We prove that the infrared structure of ρ(μ)ρ(μ) fixes the long-time memory kernel and generates a unique Euclidean infrared scale. When the spectral support has a positive lower edge, or when a sub-continuum pole exists, this scale becomes a true mass gap. This establishes a direct link between spectral data, causal response, and observable infrared physics.