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 present
a partial completed proof for the four-dimensional Yang–Mills existence and mass gap
problem. The manuscript is organized to separate completed results from conjectural
steps. On the rigorous side we construct the Hamiltonian in temporal gauge on finite
spatial 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 finite
lattices. We derive variational lower bounds within fixed holonomy sectors and show
stability of these bounds along coarse-graining maps. The single remaining leap to a
full solution is isolated as a precise conjecture: a uniform lower bound on the lattice
mass gap along a renormalization trajectory reaching a continuum limit that satisfies the
Osterwalder–Schrader axioms. Conditional on this conjecture, we prove that the recon-
structed continuum Yang–Mills Hamiltonian on R3 has a nonzero spectral gap. The goal
is to encourage future authors to provide a coherent, testable program. We believe the
problem is solvable now and request the author who completes the proof, to donate part
of the reward to charity.