Where

• G_μν[P] — operator-valued “projector Einstein tensor”. On the lattice:
  G_μν[P] = ½ π^α[∇_μ, ∇_ν] π^α + ⋯
  (Baumgarte–Shapiro-type discrete Ricci built from projected covariant differences). The expectation value with respect to the full quantum state of projectors and matter gives the emergent metric curvature.
• T_μν^m — standard stress-energy tensor of Standard Model fields, defined on the same lattice and sourced by the metric extracted from P.
• T_μν^proj — stress-energy of projector fluctuations:
  T_μν^proj = (1/σ²) ∑_α ( ∇_(μπ^α ∇_ν)π^α − g_μν ∇_ρπ^α ∇^ρπ^α ) + ⋯
• ℏ Ξ_μν — stochastic Langevin source generated by integrating out high-frequency projector modes; it encapsulates loop corrections and metric noise.
  Correlator: ⟨ Ξ_μν(x) Ξ_ρσ(y) ⟩ = (1/ℓ_Pl²) G_μνρσ(x, y; σ, a) + O(ℏ).

Limits

• Classical continuum (ℏ → 0, σ/a → ∞): Ξ_μν → 0 and T_μν^proj → 0, hence ⟨ G_μν ⟩ = 8πG ⟨ T_μν^m ⟩ (Einstein equation).
• Weak-field, lattice Newtonian: keep the leading h₀₀ term; the metric emerges from projector overlaps (effective G calibrated empirically).
• Linearised quantum gravity: expand P = P_cl + δP, yielding the quadratic action used for the graviton propagator.
• QFT on a fluctuating background: freeze G_μν[P] at its mean; propagate matter fields in that geometry while Ξ_μν supplies Hawking-like noise.

Renormalisation / Regulators

Divergent composite operators (e.g. ∇π · ∇π) are point-split by the physical UV scale σ. Counter-terms appear as higher-curvature operators ℓ_Pl² R², ℓ_Pl⁴ R_μνρσR^μνρσ, consistent with EFT fits.