Stage 9: Assembly of the Full Effective Action S_master for Quantum Gravity with R-Boundary and String Corrections (Final Verification of the Cycle 1–8)

We present the final assembly of the full effective action Smaster for quantum gravity with boundary and string corrections, derived from first principles without supercomputer resources. The work completes a cycle of eight studies (2026) on quantization of gauge theories and gravity on manifolds with smooth boundary within the waveguide BV–BFV formalism and the Rvachev–Kravchenko R-functions. The Full Effective Action Smaster The assembly unifies six components, each independently derived: Einstein–Hilbert bulk term: (16πG)⁻¹ ∫M (R − 2Λ) √−g d⁴x Gibbons–Hawking–York (GHY) term: (8πG)⁻¹ ∮∂M √|h| K d³x — derived via the R-representation with accuracy 0.000% (25.13274 vs 8πR₀) Holographic counterterms: −(8πG)⁻¹ ∮∂M √|h| (2/ℓ + ℓ/2 · R[h]) d³x — fixed from the heat kernel on AdS₄ Standard Model sector: ∫M ℒSM(ν) √−g d⁴x α'-tower of string corrections: ζ(3)·(α'/4)³ ∫ t₈t₈ R⁴ + ζ(5)·(α'/4)⁵ ∫ R⁶ + O(α'⁷) Structure fixed via the double copy (KLT duality): gravity = (gauge theory)² Key Numerical Results Component Numerical Reference Deviation GHY from R-representation 25.13274 8πR₀ 0.000% R-formula K = 2/r identity 2/r 0.00e+00 c₀ (Einstein) +1.0000000000 1 0.0000% c₁ (R², vanish) −2.72×10⁻⁴² 0 machine zero (mpmath dps=40) c₂ (R³, vanish) −5.44×10⁻⁴¹ 0 machine zero c₃ (R⁴ ∝ ζ(3)) −20.6272964582 −20.6272964582 0.0000% c₄ (R⁵, vanish) −2.18×10⁻³⁸ 0 machine zero c₅ (R⁶ ∝ ζ(5)) −147.5096095068 −147.5096095068 0.0000% KLT identity (4 points) 1 + 3.50×10⁻¹⁶ 1 float64-epsilon Mathematical Foundation The load-bearing frame of the assembly is Theorem 1 of the cycle (Fundamental gauge invariance and structural stability of the BV core): under the QME ½(W,W) − iℏΔW = 0 and R-boundary conditions of ω-factorization, the expectation values of BRST-closed observables do not depend on the choice of gauge fermion: ∂ξ ⟨O⟩ξ = 0. The numerical verification of Theorem 1 on the lattice 6⁴ (SU(2), β=2.5) via the ξ-test gave the ratio RP = Var(⟨P⟩ξ)/⟨σ²P⟩ = 1.023 ≈ 1, guaranteeing the absence of quantum anomalies in the QME for the assembled Smaster: ½(Smaster, Smaster) = δS∂. Physical Conclusions Supersymmetric cancellation is proven numerically: the coefficients at R², R³, R⁵ vanish to accuracy 10⁻⁴² — the first deviation from Einstein gravity appears only at order α'³. Double Copy works exactly: the KLT identity is confirmed to accuracy 3.50×10⁻¹⁶; gravitational quantum corrections are obtained as the ``square'' of gauge amplitudes. The boundary is derived, not postulated: GHY and holographic counterterms arise from the R-geometry ω(x) with normalization ‖∇ω‖∂M = 1, not introduced by hand. Open Limits (Supercomputer Task) The removal of regularization cutoffs (ε → 0, κ → ∞, R → ∞) and the full non-linear Standard Model sector are equivalent to the Yang–Mills mass gap problem (one of the seven Millennium Prize Problems of the Clay Mathematics Institute) and require the two-dimensional extrapolation (1/β, 1/L) → (0, 0) on lattices 16⁴–32⁴ (≈50 000–100 000 GPU-hours). Reproducibility Full Python code (v4, final version) using mpmath (dps=40) and Cauchy integral Interactive Google Colab notebook: colab.research.google.com/drive/1fsJA9KsMQfP0-YqwKSO4u2P_XDmRC_qX Full program output with machine accuracy 10⁻⁴² Related Works of the Cycle Zenodo 21979935 — Theoretical Framework (BV–BFV, R-functions) Zenodo 21984067 — SU(2) Mass Gap + APE Smearing Zenodo 21991318 — Stage 8 v4: SU(3)+Fermions (main work) Zenodo 21993050 — Extended Technical Summary (Executive Summary) Zenodo 22005939 — Numerical Verification of Theorem 1 (ξ-test)