Emergent Gravity from Threshold Field Architecture

We present a constructive model in which spacetime geometry and matter content
emerge from a single scalar field Θ(ξ, Z), the threshold function of a probability
distribution P(ξ|Z). The metric gµν arises as the expectation of the Hessian of
Θ over P, and the stress-energy tensor Tµν arises from the gradient structure of
Θ. Numerical evidence shows that the Einstein tensor Gµν computed from this
emergent metric satisfies Gµν ≈ κTµν with correlation ≈ 0.80 and κ ≈ O(1),
stable across the physical parameter range. The Bianchi identity ∇µGµν = 0 holds
exactly in spacetime coordinates. The threshold form is consistent with a constrained
maximum entropy principle under Lorentzian signature requirement. This constitutes
a coherent, testable prototype of emergent gravitational dynamics — not a derivation
of General Relativity, but a demonstration that Einstein-like structure can arise
from information geometry without GR being assumed.