Spacetime Dynamics and Local Entropy Balance on Causal Horizons

We propose that spacetime dynamics can be organized by a Planck-scale bookkeeping rule, written using a modular-parameter normalization of size $2\pi$, that balances the geometric entropy increment $\delta A\mathrm{/}4G$ against a reversible modular-energy flow $\delta \text{\hspace{0.17em}⁣}⟨K⟩$ and an irreversible Landauer–Bennett cost $\ln 2\delta N_c$, where $K\equiv K_{\sigma }$ is the (dimensionless) modular Hamiltonian of the chosen region defined relative to a fixed reference state $\sigma$, and $N_c$ counts logically irreversible one-bit record updates (e.g. coarse-grained overwrites or registrations) on that screen. This ``information–geometry ledger’’ is consistent with the Bekenstein–Hawking area law and, when enforced on small causal screens under the standard entanglement-equilibrium assumptions, recovers the full nonlinear Einstein equation. In FLRW cosmology, the same bookkeeping motivates a two-component vacuum sector ${\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}={\rho }_{\mathrm{\Lambda }}+3\epsilon H^2\mathrm{/}8\pi G$ when a constant inefficiency parameter $\epsilon$ is assumed.