Quantum Information Ledgers for Entanglement Flows on Planckian Causal Screens

We propose a local, finite quantum-information framework for spacetime dynamics built on Bekenstein--Hawking entropy pixels of causal screens. Each pixel carries a finite Hilbert space with geometric, field, and coarse-grained classical ledgers expressed in common entropy units. A fundamental modular-flow interval (normalized to $2\pi$) organizes a closed, reversible bookkeeping cycle, while a simple balance law shifts one unit of information among the three ledgers, tying geometric area change to modular-energy flow and to classical records. Summed over the boundary of a small geodesic ball, the reversible sector yields the local gravitational field equations with the correct coupling and an integration constant identified as the cosmological term. We then realize the balance law in two settings: a regular black-hole geometry, where modular analyticity organizes a four-quadrant modular lap through the Kruskal extension that closes with zero net area change while accumulating equal modular and classical increments; and a geodesically complete FLRW bounce, where an analogous lap pairs apparent-horizon area variations across expanding and contracting branches. This unified framework thus bridges local quantum-informational principles with gravitational phenomena, offering a finite and operational foundation for spacetime dynamics.