This release comprises two documents — Paradox Engine Mathematical Core.pdf and Paradox Engine Rosetta Stone.pdf — establishing the next-generation foundation for the Paradox Engine (PE) framework, and all applications derived from it. The Paradox Engine Mathematical Core.pdf document provides the formal derivations and operator definitions underpinning the PE, including recursive multi-index sums, integral kernels, noise terms, and reflexive boosts, it represents the foundation outlining the math everything is built upon. The Paradox Engine Rosetta Stone.pdf document presents a conceptual and structural map of the principles detailed in the Paradox Engine Mathematical Core.pdf intended for use by anyone who seeks to understand PE dynamics, including the Ultra-Dense Equation as a central formalism. This document is intended to help readers of any background or education grasp the concepts outlined in the PE framework. Together, these works propose a minimal architecture linking recursive field operators, emergent attractor dynamics, and coherence-based formalizations for universal system behavior. This Github link serves as a secondary source for all PE publications.