Emergence of Fundamental Constants from Prime Number Structure: A Mathematical Framework Connecting Number Theory, Transcendental Analysis, and Physical Parameters

This paper presents a unified mathematical framework showing that a wide range of fundamental constants from mathematics, physics, and cosmology align with a discrete structure derived from prime numbers. The framework identifies two independent but related phenomena: a “prime ladder” on which many constants cluster after simple transformations, and a narrow selection domain—termed the Shadow Corridor—within which the vast majority of prime pairs exhibit sharply enhanced precision in a specific transcendental relationship. The work demonstrates that these phenomena are not artifacts of fitting or tuning. All structural boundaries are fixed by geometric constraints, and extensive statistical analysis shows that the combined probability of the observed results arising by chance is vanishingly small. An algebraic result further shows that a specific prime value emerges uniquely from the underlying geometry, anchoring the structure. All claims in the paper are supported by exhaustive computational validation. Complete source code is provided, enabling full reproducibility of every numerical and statistical result. While the physical interpretation of the findings remains open, the mathematical structure itself is fully specified and independently testable. This paper serves as the culminating result of a sequence of studies establishing the empirical detection, geometric origin, and unified validation of the framework. For further information about the ENSO Framework, please contact Eric Needham:esnotheory1@gmail.com