The Quantum Trapdoor: Analyzing RSA Prime Uniformity Through the Lens of Electron Delocalization and Molecular Stability

This study introduces a novel quantitative framework that bridges two disparate domains: the stability derived from Quantum Coherent Transport (QCT) in molecular systems and the security margin of RSA Public-Key Encryption. We contend that the physical principles governing molecular integrity, specifically the maximization of electron delocalization, provide a rigorous blueprint for guiding the selection of optimal large prime numbers (p and q) in cryptographic applications. Structuring our comparative analysis around the unifying concepts of strength and uniformity, we demonstrate that maximal resistance to external breakdown, whether chemical destabilization or mathematical factorization, is achieved when system uniformity is maximized (paralleling low bond deviation in molecules). By defining the Cryptographic Delocalization Measure (ℒ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</inf>), we translate these quantum principles into a practical metric for "Sustainable Security" within Consumer Technology (CT) ecosystems. This approach directly supports the theme of "Sustainable Embodied AI" by offering a computationally efficient method for pre-validating key strengths in resource-constrained environments, such as Edge AI and IoT devices, thereby reducing the energy-intensive overhead of managing compromised networks.