Existence of a Mass Gap in Yang–Mills Theory via Structural Impossibility on Annular Transmission Channels

We prove that for any compact simple gauge group G, there exists a nontrivial quantumYang–Mills theory on R4 with a positive mass gap ∆ > 0.The proof does not construct the theory. It kills every alternative.We define a nonempty ambient class of continuum-limit candidates from Wilson lattice ap-proximants and eliminate both failure modes. Triviality dies at the door: a shell string-tensiontether excludes it from the ambient class. Gaplessness dies by structural impossibility. Eachannulus in R4 is represented by a canonical transmission channel—a positive operator betweenboundary Hilbert spaces whose centered part carries the non-vacuum signal. We prove thesecentered channels compose exactly, with no correction terms, as a consequence of vacuum preser-vation in the extraction formalism. Confinement then forces exponential decay of the centeredchannel via a radial area-law toll on cross-shell connected correlators, while a supermultiplicativedefect functional prevents the decay from being hidden by fragmentation. The strong-couplingclosure is unconditional; promotion to the continuum-directed regime is achieved by exact block-ing to a UV mesoscopic scale where asymptotic freedom supplies the required smallness.What survives is the desired theory. This is the third application of the Gallagher Principleof Structural Impossibility, after P̸ = NP and the Riemann Hypothesis.