The Total Nevanlinna-Guaranteed Propagator Bound

This paper develops a mathematically controlled formulation of a proposed Total NevanlinnaGuaranteed Propagator Bound for the Euclidean Landau-gauge gluon propagator. The central idea is to combine an optimally truncated perturbative expansion in the ultraviolet region with a nonperturbative infrared remainder constrained by complex-analytic function theory. denotes the scalar transverse gluon propagator in Euclidean momentum space.The proposed uniform estimate .The first term represents a finite, optimally truncated perturbative sector. The second term represents a nonperturbative analytic majorant. Nevanlinna theory does not automatically imply this particular infrared function; rather, it provides the analytic interpolation framework from which an admissible envelope can be constructed after specifying the function class,interpolation data, normalization conditions, and infrared assumptions.