Lorentzian Signature and Einstein Tensor Structure from Entropic Tick Dynamics

We present an emergent Lorentzian spacetime as the readout of a counting process. On a local discrete state machine with real, nonnegative weights, a posited tension field and two flagged posits suffice: a tick cost pricing one update at $\hbar/E$, and a spatial closure into the Hopf bundle $S^3 \to S^2$, selected on entropic grounds. The tick cost yields gravitational, kinematic and Tolman time dilation as one mechanism, and the tension field's elasticity recovers the variational structure of the Einstein equations. Near a horizon the Unruh temperature, with its coefficient $2\pi$, emerges from the tick dynamics on the closed fibre; the first law then integrates to the Bekenstein--Hawking area law with the coefficient $\tfrac14$, conditional only on the standard exterior profile. Closed-form results carry machine-precision numerical verification, including a cone-deficit counterexample showing that the exact-$2\pi$ closure is load-bearing.