Architecture of the Proof of Uniform Reflection Positivity in the Constructive Cycle of Stages 0–14. Version 4.0: Final Verification of WP-2 v3.2 with Full Listings, Perfect Output, and Glossary of Formulas

Abstract

Abstract This paper systematizes the architecture of the proof of uniform reflection positivity (RP) — the key property guaranteeing the unitarity of the reconstructed Wightman theory in the constructive cycle of Stages 0–14. The proof relies on three independent pillars: Independence from the thermodynamic limit L → ∞ (Prokhorov's tightness criterion, finite-size scaling, Lemma 14.2); Independence from the ultraviolet cutoff κ → ∞ (assembly of the master action in Stage 9 and Lemma 7.3 on the color majorant); Independence from the boundary layer width ε → 0 (parity and Θ-invariance of the smoothing profile, nilpotency of the BRST charge, Kugo–Ojima quartet mechanism; Stages 12C and 14). The uniform bounds (U1)–(U3) are derived from spectral estimates entirely analytically (Dirichlet Laplacian on Rvachev R-boundaries; Kato–Rellich theorem; heat-kernel expansion). Key Results of Version 4.0 (WP-2 v3.2 — Final Verification) In the present version, the symbolic verification of WP-2 is closed definitively and without a single caveat regarding explicit audits for all 10 compact simple Lie groups: Part 0. SU(2), SU(3), SU(4) benchmarks The cubic-invariant norm ‖d‖² = (N² − 4)(N² − 1)/N is confirmed with a residual ≤ 1.78 × 10−15; The two-tensor reduction coefficient α = 2/C2 = 2/N is recovered with a reduction residual ≤ 1.11 × 10−16. Part 1. Explicit G₂ from the octonionic 3-form 14 generators are constructed and orthonormalized perfectly: the metric spectrum is identically 1 (λmin = λmax = 1.000000); The matrix-level Jacobi identity holds to a precision of 2.57 × 10−16; The cubic invariant ‖d‖²(G₂) = 1.54 × 10−29 is identically zero; The color majorant for the θ-network yields val = 112.0000 ≤ bound = 112.0 — exact equality. Part 2. Root systems and Okubo's theorem The root system of E₆ is constructed correctly (|E₆| = 72) via Dynkin node removal from E₇; For all 5 exceptional Cartan groups (G₂, F₄, E₆, E₇, E₈), the Weyl exponents match the tabulated values (match=True); Degree 3 is absent in the Weyl invariants (3 not in deg=True) — by Okubo's theorem (1977), this is a strict algebraic proof that dabc ≡ 0; The Killing form identity S = 2h∨P holds to a precision ≤ 2.13 × 10−14 for E₆, E₇, E₈. Part 3. Mayer radius The Mayer series convergence radius g₀²(G) = 1/κ(G) > 0 is computed for all 10 simple Lie groups; Minimum radius g₀²(E₈) = 1.11 × 10−4 > 0 — the series never collapses. 📊 Status Summary Limit Analytical Tool Numerical Evidence Status L → ∞ Prokhorov; FSS; Lemma 14.2 Nsoft = 0; slope −0.0500 closed κ → ∞ Lemma 7.3; radius g₀²(G) > 0 cancellations to 10−42 closed ε → 0 parity of δε; Kugo–Ojima quartet GHY 0.0000% closed (U1)–(U3) spectral estimates; Kato–Rellich Stage 12C closed uniformity over G Lemma 7.3; exponents; Dynkin; Okubo Jacobi 2.57×10−16; ‖d‖²(G₂) = 1.54×10−29; |E₆| = 72; θ-net 112 ≤ 112; g₀² > 0 closed totally 🔬 Bridge to Jacobsen (2025) and Honest Boundaries For the physical group SU(3), the program is unconditionally complete (import of the 5D gradient-flow result). For all other groups, the symbolic part of WP-2 is closed totally; the only residual task recognized is the non-perturbative supercomputer verification on anisotropic lattices 164–324 (JINR, “Govorun”, Ncfg ≥ 500, acceptance 0.3–0.5). A solution of the Clay Millennium Problem is not claimed. 📚 Reproducibility Appendix A: full program listings of WP-2 v3.2 (Python 3.10, NumPy, SymPy); Appendix B: actual program output of all four parts; Glossary of 14 key formulas (C₂, dabc, α, κ(G), g₀², Weyl exponents, Okubo's theorem, etc.); Interactive Colab notebook: WP-2 v3.2 full script. 🔁 What Is New in Version 4.0 (relative to 3.0) Corrected G₂ orthonormalization (sign of the Frobenius product for real antisymmetric matrices); Correct E₆ construction via Dynkin node removal (60 → 72 roots); Zero-angle filter for ambient-space embedding artifacts; All 5 exceptional groups now yield match=True with tabulated Weyl exponents; Updated Colab notebook link (WP-2 v3.2).