The Lattes-Galois Identity: A Zero-Sum Unification of Transcendental Constants and the Fine-Structure Constant α

We propose the Lattes-Galois Identity (L) as a foundational phase-closure operator that establishes a zero-sum convergence between the fundamental constants of mathematics and the coupling constants of quantum electrodynamics. This paper demonstrates the derivation of L starting from a null informational state (0), expanding through the complex Eulerian manifold to incorporate e,π,i,ϕ,ℏ,c,ϵ0, and the fine-structure constant α into a single, unified algebraic expression. By isolating the zero-point of this identity, we reveal a modular link between transcendental geometry and the set of natural numbers (N∗), as previously explored in the second version of this work. This framework suggests that the numerical value of 1/α≈137 is not an arbitrary physical constant, but a geometric requirement for vacuum stability within the S5 Galois symmetry group. The resulting 'Lattes Ruler' provides a discrete metric for atomic configuration, identifying the stability of primary elements as multiples of this fundamental informational unit, according to the formula: