The Mass Gap in SU(3) Yang–Mills via Constructive Methods

Abstract This constructive proof of a mass gap in 4D SU(3) Yang–Mills theory (pure gauge) is built on the lattice formulation with the Wilson action. The logical flow is: reflection positivity (RP) ⇒ RP-preserving renormalization group (RG) blocking with uniform locality ⇒ polymer/cluster expansion with centered (cumulant) activities ⇒ quadratic contraction of activities (BKAR/tree-graph bounds) ⇒ summable losses preserving a strictly positive string tension in the continuum ⇒ tube-operator spectral-radius bound in the OS Hilbert space ⇒ mass gap. All locality, contraction, and loss constants are scale-uniform under rescaling. Main Statements Quadratic contraction (uniform in scale): η_{k+1} ≤ A · η_k² Summable losses and positive product: Σ η_k < ∞, ∏ (1 - C · η_k) > 0 Uniform string tension in physical units: σ_phys ≥ σ_seed · ∏ (1 - C · η_k) > 0 Tube-operator spectral radius (single-slice tube of width R): ρ(Q_R) ≤ exp(-σ_phys · R) OS reconstruction and exponential clustering imply a spectral gap: m₀ > 0 Standing Hypotheses and RG Invariants Reflection positivity, gauge invariance, uniform finite range R_*, pointwise centering of singletons, and the smallness domain for the KP norm are preserved at every scale. Pointwise centering (kills linear term): z_k(Γ) = 0 for every single-plaquette polymer Γ. Uniform finite range (after blocking size b with skeleton radius r_b): R_{k+1} ≤ ceil(R_k / b) + r_b ⇒ R_k ↓ R_* (scale-independent). Invariant smallness domain: let η_* = min{1/(4A), η_0}. If η_k ≤ η_*, then η_{k+1} ≤ A · η_k² ≤ ¼ η_k, hence Σ η_k < ∞. Transfer operator at each scale: positive, self-adjoint, and after rescale-back, ‖T_k‖ ≤ exp(c) with c geometric and independent of k. Center symmetry in the confining phase: ⟨P⟩_L → 0 as L → ∞. Seed Area-Law Window (Explicit SU(3) Bound) For 0 ≤ β ≤ 1, the fundamental character coefficient satisfies q₃(β) ≤ β/18 + (β²/6) exp(β). This yields a seed area law with σ_seed(β) > 0, uniformly in shape and volume (up to a perimeter prefactor). Constants A = C_tree(d, τ) · N_pair(b, R_), where C_tree arises from the BKAR tree-graph bound and N_pair counts ordered pairs of anchored polymers embeddable in one block of range R_. These depend only on geometry (b, d, τ, R_) and are independent of scale k and volume L. The smallness threshold η_ and the collar constant C are likewise geometric and scale-uniform. Schrödinger Functional (SF) Bridge Irrelevant-operator coefficients in the SF geometry are bounded linearly in the KP norm: |c_α^{(k)}| ≤ K · η_k, |c_β^{∂(k)}| ≤ K · η_{k,θ}. Step-scaling converges with O((a/L)²) cutoff errors. Archive Contents the mass gap in su3.pdf (manuscript, v4.0) the mass gap in su3.tex (LaTeX source) references.bib (bibliography) ReadMe.md (build and reproducibility notes) I’m creating a series of GitHub repositories with tools, proofs, and reproducible checks: GitHub: https://github.com/reeveskeefe/ Four-dimensional Yang–Mills is technically subtle and unforgiving of informal heuristics. To make every step explicit, these repos document derivations, assumptions, and stress tests in full. The goal is to prove it is not heuristics with rigor, surface and check and fix, or disprove any potential sources of error, and clarify points that might otherwise appear subtle or ambiguous. I will continue publishing repositories of tools, proofs, and validation code on GitHub (https://github.com/reeveskeefe/). Because 4D Yang–Mills is highly sensitive to technical details, each repository provides complete derivations, centering/extraction steps, and numerical stress tests. The aim is to eliminate heuristic gaps, make subtle points transparent, and invite precise scrutiny of any place that could be misread as abstract or ambiguous. So far there are two repositories on github of tools and proofs for researchers and referees:-https://github.com/reeveskeefe/quadratic-contraction-lemma-in-Reeves-Mass-Gap-4D-Proof-https://github.com/reeveskeefe/massgap-field-analyzer-4D Reproducibility / Build Instructions From the project root: pdflatex manuscript.tex → bibtex manuscript → pdflatex manuscript.tex → pdflatex manuscript.tex The manuscript uses references.bib. The PDF embeds the v4.0 DOI for exact citation. Changelog v2.0 • Formal RP-preservation lemma for the RG step (local, Borel, reflection-covariant blocking; skeleton does not cross reflection plane). • Pointwise centering of singletons: z_k(Γ) = 0 for every single-plaquette polymer. • Uniform finite-range corollary: R_{k+1} ≤ ceil(R_k/b) + r_b ⇒ R_k ↓ R_. • Quadratic contraction with geometry-only constant A = C_tree(d, τ) · N_pair(b, R_). • Explicit SU(3) bound: q₃(β) ≤ β/18 + (β²/6)e^β. • Tube-operator spectral-radius bound: ρ(Q_R) ≤ exp(-σ_phys R), yielding the OS mass gap. • SF irrelevant-coefficient bounds linear in KP norm; step-scaling converges with O((a/L)²) errors. v3.0 • Clarification of academic integrity and explicit statement of AI assistance. • Bibliography not compiled in the PDF, but included in the project files. v4.0 (Current) • Complete version with compiled bibliography. • Document verified for integrity and reproducibility. • Author confident in readiness for peer review and scrutiny. Contact For questions or clarifications: [email protected] Versioning Please cite the version-specific DOI for reproducibility (this v4.0 record). For a link that always resolves to the latest version, use the Concept DOI shown in the sidebar (“Cite all versions”).