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

Abstract This work gives a constructive proof of a mass gap in four-dimensional pure SU(3) Yang–Mills theory in the Wilson lattice formulation. The logic is: reflection positivity (RP) ⇒ an explicit RP-preserving renormalization group (RG) map with uniform locality ⇒ polymer/cluster expansion of centered (cumulant) activities ⇒ quadratic contraction of the KP norm via BKAR/tree-graph bounds ⇒ summable losses and stability of 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 uniform in scale. Main statements Quadratic contraction (uniform in scale): η_{k+1} ≤ A · η_k^2 with A independent of k. Summable losses and positive product: ∑_k η_k < ∞, and ∏_k (1 − C · η_k) > 0. Uniform string tension in physical units (for a fixed admissible RG scheme): σ_phys ≥ σ_seed · ∏_k (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: m_0 > 0. Standing hypotheses and RG invariants At every scale the following are preserved: reflection positivity, gauge invariance, uniform finite range R_*, pointwise centering of singletons, and a smallness domain for the KP norm. 2. Pointwise centering (kills linear term): z_k(Γ) = 0 for every single-plaquette polymer Γ. 3. Uniform finite range under spatial blocking by b with skeleton radius r_b: R_{k+1} ≤ ceil(R_k / b) + r_b, hence R_k ↓ R_* (scale-independent). 4. Invariant smallness domain: let η_* = min{ 1/(4A), η_0 }. If η_k ≤ η_* then η_{k+1} ≤ A · η_k^2 ≤ (1/4) · η_k, hence ∑_k η_k < ∞. 5. Transfer operator at each scale: positive, self-adjoint; after rescale-back, ‖T_k‖ ≤ exp(c) with c geometric and independent of k. 6. Center symmetry in the confining phase: ⟨P⟩_L → 0 as L → ∞. 7. Seed area-law window (explicit SU(3) bound) 8. For 0 ≤ β ≤ 1, the fundamental-representation character coefficient satisfies q_3(β) ≤ β/18 + (β^2/6) · exp(β). This yields a uniform seed area law with σ_seed(β) > 0, shape- and volume-independent 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 of the 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 and |c_β^{∂(k)}| ≤ K · η_{k,θ}. Step-scaling converges with O((a/L)^2) cutoff errors and matches the universal two-loop continuum coefficients. Archive contents the mass gap in su3.pdf (manuscript, v5.0) the mass gap in su3.tex (LaTeX source) references.bib (bibliography) README.md (build and reproducibility notes) Selected figures/CSVs and scripts are available in the linked GitHub repositories. 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 v5.0 DOI for exact citation. Related GitHub resources (tools, proofs, validation code) https://github.com/reeveskeefe/quadratic-contraction-lemma-in-Reeves-Mass-Gap-4D-Proof https://github.com/reeveskeefe/massgap-field-analyzer-4D https://github.com/reeveskeefe/RG-Steps-Proof/tree/main Purpose of the code artifacts: make each step explicit; document derivations, centering/extraction, and numerical stress tests; certify constants for one admissible RG scheme; surface and fix potential issues; and invite precise scrutiny. Changelog v2.0 • Formal RP-preservation lemma for the RG step (local, Borel, reflection-covariant blocking; skeleton does not cross the 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: q_3(β) ≤ β/18 + (β^2/6) 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 included in project files. v4.0 • Complete version with compiled bibliography. • Document verified for integrity and reproducibility. v5.0 (From conditional to complete) • Constructive RG map ℛ (temporal decimation + spatial blocking + heat-kernel smoothing + recentering) with proofs of: (i) reflection positivity, (ii) uniform locality, (iii) quadratic KP contraction η_{k+1} ≤ A η_k^2 with A independent of k. • Replace “infimum over RG trajectories” by a fixed admissible scheme; add a short scheme-independence argument for σ_phys and the mass gap within the admissible class. • Unconditional statements of the area law, positive string tension, tube-cost bound, and mass gap (transfer-operator gap + OS reconstruction). • Numerical appendix (separate code repositories) certifying constants for one admissible scheme and showing robustness (contraction tests, RP histograms, locality decay, parameter scans). Numerics are not premises of the proof. Contact For questions or clarifications: [email protected] Versioning Please cite the version-specific DOI for reproducibility (this v5.0 record). For a link that always resolves to the latest version, use the Concept DOI shown in the repository/sidebar (“Cite all versions”).