A Lattice Framework for Four-Dimensional Yang–Mills Theory: Finite-Lattice Spectral Analysis, Glueball Correlators, and the Continuum Problem on R4

This manuscript presents a lattice-based framework for studying four-dimensional pure SU(N) Yang–Mills theory, with particular attention to finite-lattice spectral properties, gauge-invariant glueball correlators, and the problem of constructing the continuum theory on R⁴. The work develops the finite-lattice formulation using the Wilson measure and transfer-matrix framework. It examines the spectral representation of Euclidean two-point functions and the role of gauge-invariant observables in probing the physical spectrum. A central result is a conditional spectral theorem: if all physical gauge-invariant states satisfy a common positive exponential decay estimate for their correlation functions, then the reconstructed Hamiltonian has a strictly positive spectral gap above the vacuum. The manuscript also formulates conditional criteria connecting uniform lattice estimates and continuum spectral-gap properties. The distinction between established finite-lattice results, conditional implications, and unresolved continuum questions is made explicit. In particular, the regulator-independent uniform estimates and the construction of a non-trivial four-dimensional continuum Yang–Mills theory on R⁴ remain open problems in this manuscript. This work is not presented as a complete solution of the Yang–Mills existence and mass-gap problem. Use of AI tools: Artificial intelligence tools were used as auxiliary tools for translation and for gathering background information during the preparation of this manuscript. They were not presented as substitutes for mathematical reasoning or proof. The author is responsible for the content of the manuscript. The idea about using glueballs was choosen by me, not by AI. Yet, some of equations I solved using AI.