An Entanglement–Modular Programme Toward the Four-Dimensional Yang–Mills Mass Gap

In this work we develop an entanglement–modular framework for four-dimensional Yang–Mills theory that leverages Tomita–Takesaki modular theory for gauge-invariant local operator algebras together with reflection-positivity methods. The central contribution is an explicit, logically separated implication chain—from entanglement structure to infrared clustering and a Hamiltonian mass gap—whose only additional dynamical input is formulated as a uniform spectral-weight/modular-gap hypothesis over an admissible class of regions. By isolating this single step, the manuscript provides a mathematically transparent “programme” that clarifies what remains to be established to obtain an unconditional theorem. More concretely, conditional on the stated modular-gap/spectral-weight hypothesis, we derive a flux–cut reflection inequality that converts positivity of the modular gap into exponential clustering, and then into a strictly positive Hamiltonian gap via Osterwalder–Schrader reconstruction. We also discuss the regulator-consistent setup for the entanglement area law and analyze the stability of the relevant entanglement coefficient under reflection-positive ultralocal counterterms and renormalization-group blocking, with the Wilson lattice action serving as the canonical reflection-positive discretization.