Yang–Mills Existence and Mass Gap in Four Dimensions: Important hints and clues

Solving this problem, comes with a notorious reward of one million dollars. We presenta partial completed proof for the four-dimensional Yang–Mills existence and mass gapproblem. The manuscript is organized to separate completed results from conjecturalsteps. On the rigorous side we construct the Hamiltonian in temporal gauge on finitespatial volume and the corresponding transfer-matrix on the lattice, establish reflec-tion positivity, and prove a positive spectral gap in the strong-coupling regime on finitelattices. We derive variational lower bounds within fixed holonomy sectors and showstability of these bounds along coarse-graining maps. The single remaining leap to afull solution is isolated as a precise conjecture: a uniform lower bound on the latticemass gap along a renormalization trajectory reaching a continuum limit that satisfies theOsterwalder–Schrader axioms. Conditional on this conjecture, we prove that the recon-structed continuum Yang–Mills Hamiltonian on R3 has a nonzero spectral gap. The goalis to encourage future authors to provide a coherent, testable program. We believe theproblem is solvable now and request the author who completes the proof, to donate partof the reward to charity.