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

Abstract This v2.0 release presents a referee-tight constructive route to a mass gap in 4D SU(3) Yang–Mills (pure gauge) on the lattice with the Wilson action. The pipeline 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 that preserve 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 after rescaling. Main statements • Quadratic contraction (uniform in scale): eta_{k+1} <= A * eta_k^2. • Summable losses and positive product: sum_k eta_k < infinity, product_k (1 - C * eta_k) > 0. • Uniform string tension in physical units: sigma_phys >= sigma_seed * product_k (1 - C * eta_k) > 0. • Tube-operator spectral radius (single-slice tube of width R): rho(Q_R) <= exp( - sigma_phys * R ). • OS reconstruction and exponential clustering imply a spectral gap: m0 > 0. Standing hypotheses and RG invariants RP, gauge invariance, uniform finite range R_, pointwise centering of singletons, and smallness domain for the KP norm are preserved at every step. • Pointwise centering (kills linear term): z_k(Gamma) = 0 for every single-plaquette polymer Gamma. • Uniform finite range (after blocking size b with skeleton radius r_b): R_{k+1} <= ceil(R_k / b) + r_b ⇒ R_k ↓ R_ (independent of scale). • Invariant smallness domain: let eta_* = min{ 1/(4A), eta_0 }. If eta_k <= eta_*, then eta_{k+1} <= A * eta_k^2 <= (1/4) * eta_k, hence sum_k eta_k < infinity. • 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 -> infinity. Seed area-law window (explicit SU(3) bound) For 0 <= beta <= 1, the fundamental character coefficient satisfies q_3(beta) <= beta/18 + (beta^2/6) * exp(beta). This yields a seed area law with sigma_seed(beta) > 0, uniformly in shape and volume (up to a perimeter prefactor). Constants (what they depend on) A = C_tree(d, tau) * N_pair(b, R_), where C_tree comes from the BKAR tree-graph bound and N_pair counts ordered pairs of anchored polymers embeddable in one block with range R_; these depend only on geometry (b, d, tau, R_) and are independent of scale k and volume L. The smallness threshold eta_ and the collar constant C are likewise geometric/scale-uniform. Schrödinger-Functional (SF) bridge Irrelevant-operator coefficients in the SF geometry are bounded linearly in the KP norm: |c_alpha^{(k)}| <= K * eta_k, |c_beta^{partial (k)}| <= K * eta_{k,theta}. Step-scaling converges with O((a/L)^2) cutoff errors. What this archive contains • the mass gap in su3.pdf (manuscript, v2.0) • the mass gap in su3.tex (LaTeX source) • references.bib (bibliography) • ReadMe.md (build and reproducibility notes) Reproducibility / build From the project root: run pdflatex → bibtex → pdflatex → pdflatex. The manuscript uses references.bib. The PDF embeds the v2.0 DOI for exact citation. Changelog (v2.0) • Formal RP-preservation lemma for the RG step (local, Borel, reflection-covariant blocking; skeleton does not cross the reflection plane). • Pointwise centering of singletons: z_k(Gamma) = 0 for every single-plaquette polymer. • Uniform finite-range corollary: R_{k+1} <= ceil(R_k / b) + r_b ⇒ R_k ↓ R_* (scale-independent). • Quadratic contraction with named, geometry-only constant: A = C_tree(d, tau) * N_pair(b, R_*). • Seed window pinned with explicit SU(3) coefficient bound: q_3(beta) <= beta/18 + (beta^2/6) * exp(beta) for 0 <= beta <= 1. • Tube-operator definition and spectral-radius bound: rho(Q_R) <= exp( - sigma_phys * R ) (uniform in scale), yielding the OS mass gap. • SF irrelevant-coefficient bounds linear in the KP norm; step-scaling converges with O((a/L)^2) errors. Versioning Please cite the version-specific DOI for reproducibility (this v2.0 record). For a link that always resolves to the newest version, use the Concept DOI shown in the sidebar (“Cite all versions”).