The Infinitesimal Numberline emerges from the need to approximate continuous parameter evolution in systems where discrete steps must mimic smooth transitions. Its six sequences- Main, Complementary, Branched, Extended New, Multi-Stage, and Precision Control- provide a versatile toolkit to model linear, symmetric, cyclic, and precision-controlled progressions with high numerical stability for iterative algorithms, sensor data processing, and physical modeling in fields like quantum mechanics and engineering. This thesis details the framework's structure, validates its computational integrity, and proposes extension to enhance its applicability. I utilize and teach the AI Grok validifying my research as I teach real time so this thesis has Grok's verifications next to the equations/formulas.