Four-Dimensional Pure Yang–Mills Theory: Wightman Construction and Mass Gap for Compact Connected Gauge Groups with Simple Lie Algebra

Abstract

For every compact connected non-abelian gauge group with simple Lie algebra, including every finite central quotient as a distinct global form, this paper constructsa nontrivial pure Yang–Mills Wightman quantum field theory on R 4 with a unique invariant vacuum and a strictly positive Hamiltonian mass gap. A faithful deployed Wilson representation fixes one intrinsic finite-law family for each global form. On that same family the response-inverted flow constructs the asymptotically-free running coordinate and positive dimensional-transmutation scale; the cofinal finite-law catalog gives reflection positivity, Osterwalder–Schrader reconstruction, nontriviality, and a gap on the full vacuum complement; and the canonical Wilson-jet compiler supplies the curvature algebra, Yang–Mills identities, conserved stress tensor and four translation charges, one compatible contact prescription, exact actual-row finite Callan–Symanzik transport, associated-graded perturbative one-step matrices, and the stated OPE bounds. The theorem proved here is unconditional: its only quantified datum is the required global form, and every scale, catalog, spectral, reconstruction, and Yang–Millsidentification input in the displayed dependency chain is a proved theorem on the same authoritative Wilson family. Constants may depend on the fixed global form. No uniqueness across regulators, subsequences, contact prescriptions, or mixing schemes, and no numerical gap uniform across inequivalent groups, is asserted. This first volume states the theorem and contains Chapters 1–7 of the set-wide proofbearing dependency closure. Volumes II and III continue the same route without duplicating or replacing any proof. The PDF files are the authoritative rendered manuscript. The accompanying TeX files are self-contained source snapshots provided for reproducibility, inspection, and version comparison. This publication was authored and put under adversarial review with the help of the LLM model GPT 5.6 Sol. Preliminary release — quality control ongoing. This manuscript states a claimed proof of the four-dimensional pure Yang–Mills existence and mass-gap problem, according to the full Jaffe-Witten problem statement. The complete proof is being released publicly to establish an archival record and facilitate scrutiny. Systematic quality-control, proof auditing, and correction of presentation or mathematical errors are ongoing. This version should therefore be regarded as a preliminary research release rather than the final audited version.