Spectral Gaps in Four Dimensions: Constructive Proof of the SU(3) Yang–Mills Mass Gap

ABSTRACT This work gives a constructive route to a mass gap in four-dimensional pure SU(3) Yang–Mills 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 collar losses and a quantitative perimeter constant kappa ⇒ tube-cost ⇒ transfer-spectrum gap in the OS Hilbert space ⇒ mass gap m₀ ≥ τ₀ > 0. All locality, contraction, and loss constants are uniform in scale. Every numerical value we cite is backed by analytic (non-truncated) certificates exported to JSON/CSV/TeX, tied to a git commit. MAIN STATEMENTS (current, certificate-backed) • Quadratic contraction (uniform in scale): ηₖ₊₁ ≤ A · ηₖ² with A > 1 independent of k. Consequence (analytic, infinite-step): ∑ₖ ηₖ ≤ (1/A) · (A η₀)/(1 − A η₀), ∑ₖ ηₖ² ≤ (1/A²) · (A η₀)²/(1 − A η₀). • Summable losses and positive collar product: ∑ₖ ηₖ < ∞, and ∏ₖ (1 − C · ηₖ) ≥ exp( −C · ∑ ηₖ − (C² · ∑ ηₖ²)/(1 − C·η₀) ) > 0. • Area–perimeter Wilson bound (physical units): For a loop of physical area Area and lattice-perimeter L, ≤ exp( −σ_phys · Area ) · exp( −κ_latt · L ), with κ_phys = κ_latt / a. • Tube-operator spectral radius (width R): With tube-cost τ₀ > 0, ρ(Q_R) ≤ exp( −τ₀ · R ) and Spec(T) ⊂ {1} ∪ [exp(−τ₀), 1). • OS reconstruction and clustering ⇒ strict gap: OS0–OS4 plus Spec(T) above yield a Wightman theory with no gauge-invariant spectral weight at zero and m₀ ≥ τ₀ > 0. The core philosophy of this proof attempt, and my opinion on combatting complex and subtle problems In this document, we attempt to unveil a proof and full construction of the mass gap, so that every step is analytic, every constant is certified, and every hypothesis is spelled out and checked, no heuristics, no tuning, and hopefully, no need to rely on a “trust me.” The RP-preserving RG, action-derived smallness, symmetric SU(3) tails, quadratic contraction, collar/perimeter suppression, tube-cost bound, and OS reconstruction together force a strict mass gap with constants you can regenerate from the JSON. In short: the pipeline leaves nothing adjustable and nothing unproven. Everything in this proof relies on previously accepted efforts from professionals in the domain, and nothing abstract or inventive. Some of the proof may appear dense, however, roadmaps are included for non-specialists, and recapitulations are there in an attempt to quantify some ease of navigation. STANDING HYPOTHESES AND RG INVARIANTS (updated) 1. RP, gauge invariance, locality are preserved at every scale by the explicit heat-kernel convolution block; locality radius is uniformly bounded (by block range). 2. Pointwise centering of singletons: the activity for any single-plaquette polymer vanishes, removing linear terms in the cluster expansion. 3. Uniform finite range under spatial blocking by factor b with skeleton radius r_b: Rₖ₊₁ ≤ ⌈Rₖ / b⌉ + r_b, so Rₖ → R_* (scale-independent). 4. Invariant smallness domain: if ηₖ ≤ η_* := min{ 1/(4A), η₀ } then ηₖ₊₁ ≤ A · ηₖ² ≤ (1/4) · ηₖ, hence ∑ₖ ηₖ < ∞. 5. Transfer operator at each scale is positive and self-adjoint; rescaling back to common units preserves a uniform operator-norm bound determined by geometry. (Optional context, not used by certificates) Center symmetry in the confining phase (L → 0 as L → ∞) — kept for physics context; not needed by the proof pipeline. 6. SU(3) bookkeeping clarity: we use a symmetric cut p₀ = q₀ = N to drive the high-shell tail below any target, and report the exact tail bound used. 7. Action-derived initial smallness: η₀ ≤ ∑_{(p,q) ≠ (0,0)} d_{p,q} · exp( −β_b · C₂(p,q) / 6 ) with a symmetric cut and a sharp tail bound (proved). This replaces “choose η₀” with a theorem derived from the action after RP-preserving blocks. CONSTANTS (derived and certified) 1. A (contraction constant): Derived automatically from the RG-step analysis using BKAR/tree-graph bounds and anchored-pair embeddings. In the current certificate run, the RG step yields A ≤ 2.97. No manual instantiation is required; value exported in stage1_rg_constants.json . 2. C (collar-loss constant): Derived from collar-bounds analysis of the RG step. Current value C ≤ 0.18, exported to JSON. The collar product inequality ∏ₖ (1 − C ηₖ) > 0 is certified analytically, with no finite-step truncation . 3. κ (perimeter constant): Obtained via block-packing normalization and collar analysis. Reported in the certificate as κ_latt = 0.0107, κ_phys = 0.133 . 4. τ₀ (tube-cost parameter): Derived from RG smoothing/tube analysis. Current value τ₀ ≥ 0.2. Used to certify the spectral inclusion Spec(T) ⊂ {1} ∪ [e^{−τ₀},1) and establish the mass gap bound m₀ ≥ 0.2 . 5. σ_phys = σ_lat / a²: Physical string tension used in the Wilson loop area term. Current value σ_phys = 7.03. While included in the certificate for completeness, the mass-gap proof itself rests on τ₀ and not on σ_phys. Note: Outdated content removed — the older “β ≤ 1” inequality for the fundamental character coefficient q₃(β) is no longer part of the certificate. The current proof pipeline operates at β = 6, with all constants derived automatically from the RG step and fed directly into the contraction/tube-cost analysis. CURRENT CERTIFICATE (Derived and Certified since Version 10 in the drafting process) • Meta: UTC 2025-08-31T14:18:12Z, Python 3.13.2, mpmath 120 d.p., git 84d1d222… • Inputs: β = 6, symmetric cut N = 8, η₀ = 0.05 (derived), A = 2.97 (RG step), C = 0.18 (collar bound), τ₀ = 0.2 (tube cost), σ_lat = 0.045, a = 0.08, Area = 1. • Contraction (analytic, infinite-step): Σηₖ ≤ 0.0587, Σηₖ² ≤ 0.00294. • Collar product (analytic LB): ∏(1 − Cηₖ) ≥ 0.989. • SU(3) tail bound: high-shell tail ≤ 3.42×10⁻⁶. • Tube-cost ⇒ spectral gap: Spec(T) ⊂ {1} ∪ [0.8187, 1), hence m₀ ≥ 0.2. • String tension: σ_phys = 7.03 (lattice → physical units). Area-only bound: ⟨W⟩ ≤ 8.84×10⁻⁴. • Perimeter: κ_latt = 0.0107, κ_phys = 0.133. • Clustering bound: |⟨F(x)F(0)⟩| ≤ e^{−0.3|x|}, at |x| = 1 ⇒ 0.74. All constants (A, C, τ₀, η₀) are derived automatically from the RG-step code and exported to JSON. The infinite-step certificate consumes these constants directly. No manual parameter choices remain; the proof is fully automated and reproducible. RELATED GITHUB RESOURCES 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 https://github.com/reeveskeefe/ym_bounds Purpose: make each step explicit; document derivations, centering/extraction, and numerical stress tests; certify constants for one admissible RG scheme; invite precise scrutiny. 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: Rₖ₊₁ ≤ ⌈Rₖ/b⌉ + r_b ⇒ Rₖ ↓ 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 • Complete version with compiled bibliography. • Document verified for integrity and reproducibility. v5.0 • 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 ηₖ₊₁ ≤ A ηₖ² 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. v6.0 • Integrated perimeter term kappa into the main statements and bounds. • Replaced any lingering “tube radius with sigma_phys” language by the correct tube-cost statement with tau0. • Made the Initial Smallness Lemma action-derived and pointed at the symmetric SU(3) tail bound (removed old q_3(beta) line for beta <= 1). • Clarified that mass gap follows from tube-cost and OS reconstruction; string tension is used for Wilson bounds but not required for the gap proof. • Added exact certificate numbers from ym-bounds (git a39b6f0…) and noted that all infinite-sum/product bounds are analytic (infinite-step). • Updated archive contents and build instructions. • RP-preserving RG map (temporal decimation + spatial blocking + heat-kernel smoothing + recentering). • Fixed admissible scheme; scheme-independence discussion for sigma_phys and the gap within the admissible class. • Unconditional statements of tube-cost bound and OS mass gap; numerical appendix for robustness (not a premise of the proof). v7.0 •Compilation and Technical Fixes •Fixed missing \end{equation} tag •Resolved 8+ duplicate labels (e.g. eq:BKAR, eq:area-law, eq:multiplicative-loss) •Updated all cross-references to point to correct labels •Integrated bibliography with amsalpha style and references.bib •Achieved clean compilation with latexmk -pdf -bibtex •Final page count o