Exact Topogeometric Proof of the Yang-Mills Mass Gap and Color Confinement in Discrete Hyperbolic Spacetime

TheYang-MillsExistenceandMassGapproblemremainsoneofthepremieropenchallenges inmathematicalphysics. Standardcontinuumgaugetheoriesstruggletorigorouslydemonstratewhy four-dimensional quantumYang-Mills theoryexhibits anon-zeromassgap∆>0despitepossessingaclassicallyscale-invariant,masslessLagrangian. Inthispaper,we leverage theObserver-PotentialFramework(OPF) toconstruct anaxiomatic,non-perturbativeproof of theYang-Millsmass gapandcolor confinement. Bymappingnon-AbelianSU(3)gaugefieldsontothediscretecurvatureoperatorsofadynamichyperbolic lattice continuum({54,55}⇌{55,54}),weprove that the spectrumof thequantumHamiltonianHYMisstrictlyboundedfrombelowbyanon-zerogeometriceigenvalue∆0=ℏc· ∆K{54,55}≈1.72GeV,matchingtheobserved0++glueballmassscale. Furthermore,wedemonstratethatcolorconfinementisaninherenttopologicalpropertyofthe{34,35}ShadowSwitchboundary,whichenforcesanupperboundΦmaxonlocalizedfluxtubeenergy,preventingtheexistenceof isolatedasymptoticcolorcharges