M2DE-THEAD V6.1: Einstein Field Equation with PBH Memory

Matter to Dark Energy via Black Hole Mechanism: M2DE/THEAD Theory V5.7 EFE - Complete First-Principles Chain with Einstein Equation, No-Ghost Proof and Four Falsifiable Knives V5.7 patches V5.6 (Zenodo DOI 10.5281/zenodo.23126422) with explicit Einstein Field Equation embedding. MASTER EQUATION (No modification to GR):G_μν ≡ R_μν - 1/2 R g_μν = (8πG/c^4) ( T_μν^matter + T_μν^PBH + T_μν^memory ) (1) T_μν^matter: baryons + radiationT_μν^PBH: cold asteroid-mass PBH, ρ_PBH = f_PBH ρ_DM, f_PBH=1e-8, M∈[1e15,1e17]g, p≈0T_μν^memory: emergent dark energy from BMS memory entanglement T_μν^memory = ρ_eff u_μ u_ν + p_eff (g_μν+u_μ u_ν) ρ_eff(a) = (H0/2π) n^2 ξ(a), ξ(a)=ξ0 a^2, ξ0=10 Mpc =1/(H0√f_PBH) p_eff = w(a)ρ_eff, w(a)=w0+wa(1-a), w0=-0.82, wa=-0.74 (DESI DR2 consistent) ∇^μ(T_matter+T_PBH+T_memory)=0 preserved. Buchert Averaging: ^2 = (8πG/3)(<ρ_matter>+<ρ_PBH>+ρ_eff), Q_M2DE = -8πG ρ_effAveraging scale L=ξ0=10 Mpc set by percolation nξ^3=1 at a_t=0.76 (z_t=0.32)ρ_eff=6.9e-27 kg/m3 = observed DE density, explains 5% H0 tension ΔH/H~(ξ0 H0)^2. Appendix A: HPS soft hair → Skyrmion Q via Callan-Harvey anomaly inflow L_eff=(1/Mp) C J_Skyrme f(z), operator mapping, not particle storage. Appendix B: DS/dS replica + Gibbons-Hawking dE=T_GH dS_ent replacing AdS RT, T_GH=H0/2π first principles. Appendix C: ξ(a)=ξ0 a^2 derived from D(a)∝a × λ_soft∝a, ξ0=1/(H0√f_PBH)=10 Mpc, percolation at z_t=0.32 automatically without tuning for f_PBH=1e-8 asteroid survival window (M=10^15-10^17 g). Appendix D (NEW in V5.7): Explicit EFE components G00 and Gij, no-ghost proof, S8 suppression dS8/dlnk=-0.05. Falsifiable Predictions:1. DESI DR2: w0=-0.82, wa=-0.74 within 1σ, DR3 must not find w0>-0.72. Euclid: S8 running 0.76→0.90 at k~1 Mpc^-13. GRB 221009A 18 TeV: N_coh~1e61 coherent enhancement → transparency4. LHAASO: birefringence Δφ~0.1° at 10 TeV Keywords: dark energy, PBH, entanglement entropy, DESI, S8, Einstein equation, Buchert averagingConcept DOI: 10.5281/zenodo.23132571 (cites all versions)