Deriving the Area-Term Cancelling Operator and Axiomatizing Information-Flux Dynamics

Building on quantum information flux and modular geometry, we uniquely derive a special operator that eliminates the area–divergent term purely from four axioms—self-adjointness, information conservation, vacuum stability, and area vanishing.

The operator is shown to satisfy the ``zero‐area'' extremality condition through several independent routes: the entanglement-entropy area law, the Quantum Null Energy Condition, the minimal-surface equation, and the modular Markov property.

We prove that these results hold universally in both flat and Anti–de Sitter spacetimes, irrespective of strong- or weak-coupling limits.

Furthermore, the operator coincides—up to residual terms and phase freedom—with the evolution kernel of the Unified Evolution Equation (UEE, DOI:10.5281/zenodo.15286652) and with the Information-Flux Theory (IFT, DOI:10.5281/zenodo.15399114). This establishes the functional completeness of the five-operator S5 basis and supports the vacuum-energy stabilization mechanism without external assumptions.

Consequently, the UEE/IFT framework closes autonomously on an independently constructed axiomatic system, reinforcing the mathematical foundation for broad applications such as the mass gap, the origin of gravity, and self-replicating dynamics.