A Number-Theoretic Origin of the Fine-Structure Constant: A Derivation from the Nested Primitive Root Scheduling Model

The fine-structure constant α_em ≈ 1/137.035999 is one of the most mysterious dimensionless constants in physics. Since its introduction by Sommerfeld in 1916, no accepted theory has been able to derive its value from first principles. In this paper, we propose that the fine-structure constant may not be a free parameter requiring explanation, but rather a necessary projection of a nested primitive root scheduling structure onto the electromagnetic interaction strength. Within this model, an ongoing information–potential conversion process between the physical and information sectors is driven by an optimization agenda orchestrated by a hierarchy of nested cyclic groups (modulo 7, 13, 31, 61). We demonstrate that: (1) The generative law of the physical sector is precisely characterized by the digital root sequence 124875 (modulo 9 cyclic group) derived from cell division; (2) The optimal traversal of the information sector is captured by the cyclic number 142857 (modulo 7 cyclic group), and these two sectors are strictly isomorphic in the group-theoretic sense; (3) The inverse of the fine-structure constant α_em⁻¹ ≈ 137 exhibits a high-precision numerical association with properties of the modulo 13 cyclic group—under the Kelvin scale, this association matches the structural constant of modulo 13 with an error of about 0.117%, and the Celsius scale independently points to the same modulus; (4) This residual discrepancy can be interpreted as a mathematical expression of the unattainability of absolute zero within the cyclic group structure—a “modular remainder” that can never be closed in a finite number of steps. We also present testable scientific predictions, including a possible extremely weak cosmic evolution of the fine-structure constant, and provide rigorous falsification conditions.

fine-structure constant, nested primitive root scheduling, modulo 13 cyclic group, information-potential conversion, dual-group isomorphism, fundamental constants, number theory, cosmology