Emergence of Gauge Structure and Finite Quantum Field Theory from a Single Real Living Field

Living Field Theory (LFT) is a research program in which a single real field is treated as the primitive physical carrier and physical structure is described through restrictions on admissible histories. This paper isolates the gauge and quantum-field-theory emergence problem. The objective is not to assume a fundamental Hilbert space, gauge bundle, or Yang–Mills action and reinterpret them in LFT language, but to determine how much of their structure can arise from physical equivalence among real-field histories. If a local transformation changes only the representative description of an admissible history while preserving physical observables, identity-sector restrictions, and physical weighting, the physical history space is a quotient. Once equivalent representatives may be chosen independently at neighboring spacetime points, consistent comparison requires a connection. Its relational non-closure gives curvature, closed transport gives holonomy, and traced holonomy gives Wilson observables. Under explicit locality, gauge invariance, spacetime covariance, and low-curvature assumptions, the leading nontrivial semisimple curvature action has Yang–Mills form. The central exact result is finite and does not require a primitive quantum state space. For a periodic real scalar field with local Z₂ link redundancy, gauge-invariant holonomy classifies all link configurations into exactly two quotient sectors. Normalized averaging over the finite local redundancy gives Zphys=(Z++Z−)/2Zphys=(Z_+ + Z_-)/2. Only after this quotient-defined physical measure is established is a conventional transfer representation introduced, yielding Zphys=Tr(PevenKN)Zphys=Tr(PₑᵥₑₙK^N). The transfer/Hilbert description is therefore an exact representation of the prior LFT quotient in this finite model, not a primitive LFT ontology. Independent finite SU(2), SU(3), generator-curvature, variational-alignment, and local-action audits support the broader gauge bridge while preserving strict claim boundaries. Continuum interacting QFT, regulator removal, renormalization, BRST/LSZ closure, Standard Model gauge-group selection, and the Yang–Mills mass gap remain open.