Yang-Mills Mass Gap: δ∞^s > 0 Spectrally Isolated

Overview

IV.T75 and III.T27 together establish the Yang-Mills mass gap in the τ-framework. IV.T75 proves that the τ-Yang-Mills Hamiltonian has a spectral gap δ∞^s > 0 — the ground state is spectrally isolated from all excited states by a positive gap. III.T27 (the Master Schema instance at E₁) provides the bridge to the orthodox Clay problem. The orthodox mass gap (existence of a positive-mass gap in 4D Yang-Mills quantum field theory) is addressed via the B ↔ I ↔ S triangle.

Detail

The Yang-Mills mass gap problem (Clay Millennium Problem) asks to prove that in 4-dimensional Yang-Mills quantum field theory, the ground state has a mass gap — a positive lower bound on the energy of all excited states. Orthodox approaches face the problem of constructing the quantum Yang-Mills theory rigorously in 4D (the theory is not known to exist as a rigorous mathematical object). Book IV approaches this through the τ-framework. IV.T75 proves the τ-Yang-Mills spectral gap: the Hamiltonian of the C-sector (strong, η-generator) Yang-Mills structure has a ground state spectrally isolated from all excitations by a gap δ∞^s > 0. The gap value is expressed as a function of ι_τ and the sector coupling. III.T27 is the Master Schema instance for Yang-Mills: it identifies the mass gap with the spectral isolation of the boundary character spectrum at the B ↔ S vertex of the Mutual Determination triangle. The bridge from τ-gap to orthodox 4D gap is marked conjectural: establishing the orthodox mass gap would require identifying the τ-Yang-Mills structure with the standard SU(N) Yang-Mills theory in 4D, which requires additional work.

Result Statement

IV.T75 + III.T27: The τ-Yang-Mills Hamiltonian has spectral gap δ∞^s > 0 (IV.T75). The orthodox Yang-Mills mass gap is addressed via the B ↔ I ↔ S Master Schema bridge (III.T27). τ-gap established; orthodox bridge conjectural.