Emergent Gravitational Dynamics and SU (3) Yang-Mills from PEPS Tensor Networks: Glueball Spectrum, Running GN (μ), Einstein Identity from fPEPS, and Complete Verification Suite

This preprint presents a unified theoretical and computational framework in which non-perturbative SU (3) Yang-Mills dynamics and gravitational dynamics emerge from a single Projected Entangled Pair State (PEPS) tensor network. The QCD vacuum is modeled as a gauged PEPS on a 2D lattice combined with a Multiscale Entanglement Renormalization Ansatz (MERA) modular Hamiltonian, generating an effective potential U (Λ) U (Λ) for a collective collapse field Λ (x) Λ (x). Core Results (Version 25) Pure Gauge Sector (Nf = 0) Glueball spectrum: m (0++) =1. 71m (0++) =1. 71 GeV, m (2++) ≈2. 25m (2++) ≈2. 25 GeV, m (0−+) ≈2. 85m (0−+) ≈2. 85 GeV, m (1+−) ≈1. 95m (1+−) ≈1. 95 GeV reproduced at 2. 6–11. 3% accuracy using a single fitted parameter J=0. 54J=0. 54 One-loop beta function coefficient b0=11Nc/ (48π2) b0=11Nc/ (48π2) derived from MERA entropy flow without Feynman diagrams String tension σ=0. 194σ=0. 194 GeV² from area-law entanglement entropy, consistent with lattice QCD Speed of light derived as the Lieb-Robinson velocity of the PEPS network Emergent Gravity PEPS on the transverse plane provides an exact lattice regularization of the Nambu-Goto worldsheet CFT Independent reproduction of Lüscher term (−π/12R−π/12R), Casimir scaling, and flux-tube width w (R) w (R) confirms the mapping Fermionic PEPS: Complete Roadmap P0–P4 P0: Gap equation in PEPS background → dynamical chiral symmetry breaking, constituent mass Mcons=335Mcons=335 MeV, fπ=92. 4fπ=92. 4 MeV (chiral limit) P1: Fermionic PEPS with Grassmann variables, sign-problem handling in 2D, χeffχeff scaling P2: String breaking as topological transition in PEPS graph connectivity, Rbreak=0. 681Rbreak=0. 681 fm P3: Direct extraction of chiral condensate ⟨qˉq⟩1/3=266⟨qˉq⟩1/3=266 MeV, GMOR relation verified to 2–5% P4: Full GPU-accelerated simulation pipeline, phase diagram prediction (Tχ=150Tχ=150 MeV, Td=170Td=170 MeV for Nf=2Nf=2), baryon spectrum with RMS accuracy 6. 7% Einstein Identity Theorem MAIN RESULT v25 For any SU (3) -invariant fermionic PEPS on a 2D lattice, the renormalized energy-momentum tensor obtained via Yang-Mills gradient flow equals the Ricci curvature tensor constructed from PEPS entanglement gradients: Tμνlatt=f (η/s) ⋅RμνfPEPS+O (a2, t2, χ−1) Tμνlatt=f (η/s) ⋅RμνfPEPS+O (a2, t2, χ−1) Numerical verification: Rμν/Tμν→8πGeffRμν/Tμν→8πGeff with systematic error decreasing as 1/ (χln⁡χ) 1/ (χlnχ), reaching 2. 6% at χ=32χ=32 Finite bond-dimension corrections: F (χ) =1+c1/χ2+c2/χ4+…F (χ) =1+c1/χ2+c2/χ4+… with c1≈3. 2c1≈3. 2, c2≈8. 5c2≈8. 5 extracted numerically Sign problem advantage: fPEPS remains real-positive at μB≠0μB=0, covering the full (T, μB) (T, μB) plane including neutron star densities Verification Suite & Transparency 11+ independent tests organized in three groups: Group I (internal consistency): dimensionless combination D (χ) =O (1) D (χ) =O (1) with mean 0. 496±0. 0770. 496±0. 077, χ (MPl) ∼111χ (MPl) ∼111, RG closure via ZrenZren with ∥βeff/β1loop∥=0. 74∥βeff/β1loop∥=0. 74, running αsPEPSαsPEPS Group II (lattice validation): Casimir scaling (5–20% deviations), deconfinement temperature Tc=277Tc=277 MeV (vs lattice 270±4270±4 MeV, Δ=+2. 6%Δ=+2. 6%), topological susceptibility χtop1/4=177χtop1/4=177 MeV (vs lattice 178±9178±9 MeV, 0. 1λY>0. 1 fm 2028–2030 mgrav>10mgrav>10 MeV excluded Lieb-Robinson cc LISA, ET (GW dispersion) ∣vg/c−1∣0. 12cent>0. 12 This work establishes PEPS tensor networks as a viable bridge between non-perturbative QCD and emergent gravity, offering testable predictions for collider physics, precision gravity experiments, and cosmological surveys. Correspondence: Sergei V. Morozov, notx@bk. ru