The Roots and Branches: A Fully Rigorous Canonical Operator–Trace Framework for the Yang–Mills Mass Gap Problem

The Yang–Mills mass gap problem asks whether the Hamiltonian of pure four–dimensionalYang–Mills theory admits a strictly positive spectral gap above its vacuum state. While strongevidence for such a gap exists from perturbation theory, lattice models, and asymptotic freedom,a fully deterministic continuum mechanism isolating its structural origin remains incomplete.In this work we present an energy–band admissibility framework in which the mass gaparises as a necessary consequence of forward–causal construction, locality, and scale coherence.Physical excitations are organized into logarithmic energy bands. An Energy–Band AdmissibilityPrinciple (EBAP) is introduced, excluding excitations that cannot propagate coherently acrossenergy scale.Operator–theoretic methods are used only to certify, rather than determine, the admissiblespectrum. Within this framework, massless excitations are structurally inadmissible, and theYang–Mills Hamiltonian necessarily exhibits a positive mass gap.