Finite-Resolution Filters and Penalty Lower Bounds in Effective Field Theories

We introduce Finite Resolution Theory (FRT), a framework in which spacetime resolution is treated as an irreducible geometric constraint ϵ. Within this effective framework, filtering schematic field variables associated with a SU (N ) Yang–Mills sector through a resolution operator Πϵ yields a modified Hamiltonian containing a resolution-dependent penalty functional. The mechanism is consistent with standard Fourier uncertainty principles, according to which spatial localization is accompanied by broadening in momentum space. This is naturally interpreted as a positive localization cost for sub-resolution structure. Under the localization assumption adopted here, the framework assigns a non-zero penalty contribution to sufficiently localized non-band-limited configurations. While this work does not constitute a rigorous proof of the Yang–Mills mass gap, it establishes a formal lower bound on the penalty contribution associated with unresolved structure, and hence isolates a positivity mechanism for the effective energy of configurations with nonzero unresolved component