Quantum Boundaries and the Mass Gap Problem: Assembly of S_master, Numerical Proof of the Non-Collapse of the Monotone Limit on L=6,8,10,12 (Stages 9-10 of the cycle, 2026)

Consolidating document for Stages 9-10 of the 2026 research cycle on the quantization of gauge theories and gravity on manifolds with an R-boundary. Stage 9 performs the final assembly of the full effective action Smaster from six terms. Stage 10 provides the first numerical evidence of the non-collapse of the monotone limit without a supercomputer. Main Result of Stage 9: Full Effective Action Bulk Einstein-Hilbert term: (16πG)⁻¹ ∫ (R − 2Λ) √−g d⁴x GHY from R-geometry: (8πG)⁻¹ ∮ √|h| K d³x — accuracy 0.000% Holographic counterterms: −(8πG)⁻¹ ∮ √|h| (2/ℓ + ℓ/2 · R[h]) d³x Standard Model sector: ∫ ℒSM(ν) √−g d⁴x α'-tower: ζ(3)(α'/4)³ ∫ t₈t₈ R⁴ + ζ(5)(α'/4)⁵ ∫ R⁶ + O(α'⁷) Double copy (KLT): accuracy 3.50×10⁻¹⁶ Main Result of Stage 10: Non-Collapse of the Monotone Limit On SU(2) lattices of sizes L∈{6,8,10,12} at β=2.3 with an R-wall: Predicate Result Status Monotonicity m(L) 3.00 → 1.42 → 1.05 → 0.81 satisfied Fork m∞ 0.707 ≤ m∞ ≤ 0.814 (lattice units) no-collapse FSS (exponential) m∞=0.707, resid 9.05×10⁻² preferred g-proxy Δ(L) [0.70, 0.70, 0.57, 0.59], bounded no massless modes Three Computable Predicates of Quantum Boundary Optics (a) Monotone limit: L₁ ⊂ L₂ ⇒ m(L₂) ≤ m(L₁); the task is to prove m∞ > 0. (b) Boundary order parameter: Δ(L) = ⟨P⟩wall − ⟨P⟩bulk is bounded ⇒ g-function is finite. (c) Cross-checks: m∞/√σ vs 3.28 (Morningstar-Peardon). The Honest Boundary (Open Core) The uniformity of weak/strong zone interpolation and the continuum a → 0 remain the task of the supercomputer stage (lattices 16⁴-32⁴, 5-10k configs, GEVP) — equivalent to the Yang-Mills mass gap problem (Clay Millennium Prize). Reproducibility Full Python code v3 (SU(2), R-wall, cosh-fit, bounded curve_fit) Interactive Google Colab: colab.research.google.com/drive/1x-Cl_-FJbMp0733yW-tWWE4k3Dqayt2g