Mixed Hodge Ledgers on Causal Screens

Black-hole thermodynamics and entanglement-based derivations of Einstein's equation suggest that local geometry, modular energy, and coarse-grained records on causal screens are related by simple balance laws. In parallel, for a complex surface $X$ with a SNC divisor $Y$, the complement $U=X\setminus Y$ carries Deligne's canonical mixed Hodge structure on $H^2(U,\mathbb{Q})$ with weights in $\{2,3,4\}$. Motivated by this, we propose that each sufficiently small causal diamond $D$ admits an auxiliary complexified screen $(X_D,Y_D)$ whose graded dimensions are mapped to three ledgers: geometric area, modular resources, and classical record capacity. The central conjecture is that these integers satisfy an event-wise balance at admissible discrete transitions, and that after coarse-graining over many Planckian pixels the relation reduces to the continuum ledger$\Delta\left(\frac{A_D}{4G}\right) = -\Delta\langle K_D\rangle + \Delta N_c^{(D)}$. We illustrate the bookkeeping in toy models, regular black-hole geometries, and nonsingular bounces, and interpret curvature singularities as limits where the small-diamond regime and/or admissible complexified-screen realization fails. The construction uses only causal diamonds, modular Hamiltonians in QFT, and mixed Hodge structures on complements of SNC divisors, but suggests a finite cohomological ``pixel'' organizing local geometry, entanglement, and information.