Emergent Gravitational Dynamics and SU(3) Yang-Mills from PEPS Tensor Networks: Glueball Spectrum, Running G_N(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 (N_f = 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 & RG Flow Running Newton constant: GN(μ)=1/[4 cent μ2 b0 ln⁡(μ/ΛQCD)]GN(μ)=1/[4centμ2b0ln(μ/ΛQCD)] with cent≈0.095cent≈0.095 reproduces the physical value GN=6.708×10−39GN=6.708×10−39 GeV⁻² at the Planck scale The historical 10381038 hierarchy between GN(ΛQCD)GN(ΛQCD) and GN(MPl)GN(MPl) is resolved as genuine RG running driven by vacuum entanglement entropy, consistent with asymptotic safety Massive emergent graviton mgrav=85mgrav=85 MeV with Yukawa range λY=2.3λY=2.3 fm, fully compatible with Eöt-Wash, GW170817, and all fifth-force constraints Strict 2D → 3+1D Mapping In the confining phase, dominant degrees of freedom are transverse flux-tube modes (D−2=2D−2=2 bosons); 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, <1%<1%) Group III (beyond-QCD phenomenology): mgrav=85mgrav=85 MeV, chiral constants, baryon spectrum RMS=6.7%, dark energy equation of state wa=+0.006wa=+0.006 Parameter audit: 1 free parameter (J=0.54J=0.54) → 11+ quantitative predictions; full error budget with parametric, systematic, and total uncertainties (68%/95% CI) Reproducibility: Open Python pipeline with fixed seeds, modular structure, GPU acceleration (cuTensorNet), and run_all_tests.py for regenerating all tables and figures Model Limitations (Transparently Stated) Anharmonicity of U(Λ)U(Λ) is ~0.3%; higher glueball states require multi-gluon operators The 2D PEPS → 3+1D map applies strictly to the confining IR regime (R≳0.7R≳0.7 fm); UV completion requires 3D PEPS or holographic lifting Chiral constants in the pure-gauge core are estimates; full Nf=2+1Nf=2+1 results require the fPEPS extension (P0–P4) The 10421042 residual discrepancy for the cosmological constant is not fully solved; the framework predicts wa=+0.006wa=+0.006, testable by Euclid/LSST Experimental Roadmap Prediction Experiment Sensitivity Timeline Falsification Criterion wa=+0.006wa=+0.006 Euclid, LSST, DESI σ(wa)∼0.03–0.04σ(wa)∼0.03–0.04 2025–2035 wa<−0.05wa<−0.05 mgrav=85mgrav=85 MeV MAGIS-100, OTIMA λY>0.1λY>0.1 fm 2028–2030 mgrav>10mgrav>10 MeV excluded Lieb-Robinson cc LISA, ET (GW dispersion) ∣vg/c−1∣<10−18∣vg/c−1∣<10−18 2030+ Frequency-dependent vgvg detected cent(Nf=2)≈0.091cent(Nf=2)≈0.091 Future fPEPS lattice χ≥24χ≥24 2027+ cent>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